analysis of the intake grill for marine jet propulsion

Analysis of the intake grill for marine jet propulsion

Marcus Söderberg Jansson

Master of Science Thesis TRITA-ITM-EX 2019:244

KTH Industrial Engineering and Management

Machine Design

SE-100 44 STOCKHOLM

1

Examensarbete TRITA-ITM-EX 2019:244

Analys av Intagsgaller för marina vattenjetsystem


Godkänt

2019-06-11

Examinator

Ulf Sellgren

Handledare

Ulf Sellgren

Uppdragsgivare

Marine Jet Power

Kontaktperson

Tommy Hellberg

Sammanfattning Marina vattenjetmotorer har utvecklats och förfinats sedan tidigt 50-tal och har bevisats mycket

användbara för applikationer i hög hastighet med båtar i varierande storlekar. Intagsgaller är en

komponent som monteras i linje med skrovet på båtar för att förhindra oönskade föremål att

färdas genom intaget på vattenjetmotorn. Intagsgallret är påverkat av viskösa krafter, direkta

krafter och harmonisk excitation samtidigt som komponenten påverkar vattenjetmotorns

effektivitet.

I denna rapport så evalueras ett urval av metoder med målet att simplifiera utvecklingsprocessen

av intagsgaller. Ett urval av tvärsnittsgeometrier är genererade och evaluerade för att dra

generella slutsatser om effektiviteten och stabiliteten av intagsgallret. Ett par olika sorters

flödessimuleringar och finita element metoder används. Slutsatsen är att intagsgallret påverkas

av ett flertal parametrar och kan utvärderas med modal finita element metoder samt två-

dimensionella flödessimuleringar.

Nyckelord: Intagsgaller, Vattenjet, Strömning

2

Abstract Marine waterjet propulsion is a technology that has been developed and refined since the early

1950’ and is proven highly useful for high speed applications with vessels in varying sizes. The

intake grill is a component that is mounted in line with the hull to prevent debris from traveling

through the waterjet. The intake grill is affected by viscous forces, contact forces and harmonic

excitation forces all while affecting the efficiency of the waterjet.

In this report a selection of methods is evaluated and verified with the goal of simplifying the

design process of the intake grill. A selection of cross-sections is generated and evaluated to

draw general conclusions about the efficiency and stability of the intake grill. A selection of

computational fluid dynamics and modal analysis methods are utilized. It is concluded that the

intake grill is affected by many parameters and can be evaluated by modal FEM analysis and 2D

CFD analysis.

Keywords: Intake grill, Waterjet, Fluid Dynamics

Master of Science Thesis TRITA-ITM-EX 2019:244

Analysis of the intake grill for marine jet propulsion


Approved

2019-06-11

Examiner

Ulf Sellgren

Supervisor

Ulf Sellgren

Commissioner

Marine Jet Power

Contact person

Tommy Hellberg

4

FOREWORD

This thesis is carried out at Marine Jet Power in Uppsala in combination as being part of the

requirement for a master’s degree in Mechanical Engineering at the machine design department

at the Royal Institute of Technology in Stockholm, Sweden.

First and foremost, I would like to my supervisors Tommy Hellberg and Robert Thunman at

Marine Jet Power for their extensive expertise and guidance throughout this master’s thesis

project.

I would also like to thank my supervisor and examiner Ulf Sellgren for his support and guidance

at the department of machine design.

I would finally like to express my gratitude towards Philipp Schlatter at the Linné Flow Center

for his ability to assist my project with his valuable insight in fluid dynamics.


Stockholm, June 2019

6

NOMENCLATURE

Notations

Symbol Description

xa First shape control parameter

A Area

sectionA Cross-section area

yyA Added mass per unit length

yb Second shape control parameter

B Bezier curve

c Chord length

C Damping matrix

fC Skin friction coefficient

DC Aerodynamic drag coefficient

LC Aerodynamic lift coefficient

PC Pressure coefficient

fD Friction drag force

pD Pressure drag force

D Cross diffusion term

E Young’s modulus

f Frequency

cf Critical frequency

nf Modal frequency

F Force

DF Drag force

LF Lift force

G Kinetic energy

h Bernstein’s polynomial

i Binomial coefficient

I Moment of inertia

j Curve control parameter

7

K Stiffness matrix

lp Loft point

L Beam depth

m Total mass

im Partial mass

M Mass matrix

n Mode number

P Bezier point

Gr Center of mass

R Radius

0.2pR 0.2% proof stress

Re Reynold’s Number

sf Scale factor

t Time

1u Mass per unit length

Q Amplitude

q Displacement

U Velocity

*U Friction velocity

V Volume

y Non dimensional wall distance

wy First layer wall distance

Y Dissipation

z Binomial coefficient

8

Greeks Description

Reduction ratio

Frequency ratio

Effective diffusivity

Density

m Material density

fluid Fluid density

Dynamic viscosity

Angle of attack

s Tensile strength

f Endurance limit

Sheer stress

w Wall sheer stress

Poisson’s ratio

Abbreviations

CAD Computer Aided Design

CFD Computational Fluid Dynamics

FEM Finite Element Method

FFT Fast Fourier Transform

FVM Finite Volume Method

RANS Reynold’s Averaged Navier-Stokes

RPM Revolutions Per Minute

SIMPLE Semi-Implicit Method for Pressure-Linked Equations

SST Sheer Stress Transport

9

TABLE OF CONTENTS

SAMMANFATTNING ...........................................................................................................................................1

ABSTRACT ............................................................................................................................................................2

FOREWORD ..........................................................................................................................................................4

NOMENCLATURE ................................................................................................................................................6

TABLE OF CONTENTS ........................................................................................................................................9

1 INTRODUCTION ............................................................................................................................................. 12

1.1 ............................................................................................................................................. BACKGROUND

...............................................................................................................................................................................12

1.2 PURPOSE AND DEFINITIONS ................................................................................................................................14

1.2.1 PROJECT OBJECTIVES ......................................................................................................................................14

1.2.2 PROJECT DELIVERABLES .................................................................................................................................14

1.2.3 RESEARCH QUESTIONS ....................................................................................................................................14

1.3 DELIMITATIONS .................................................................................................................................................14

1.4 METHOD ............................................................................................................................................................15

1.4.1 Problem analysis ................................................................................................................................... 15

1.4.2 Problem solving method ........................................................................................................................ 15

1.4.3 Verification ........................................................................................................................................... 15

1.4.4 Design proposal .................................................................................................................................... 15

2 FRAME OF REFERENCE .............................................................................................................................. 16

2.1 LITERATURE STUDY ...........................................................................................................................................16

2.2 SOURCES OF EXCITATION ...................................................................................................................................18

2.3 BODY MECHANICS .............................................................................................................................................19

2.4 ADDED MASS.....................................................................................................................................................21

2.5 FLUID DYNAMICS ...............................................................................................................................................21

2.6 COMPUTATIONAL FLUID DYNAMICS ...................................................................................................................25

2.7 FLUID STRUCTURE INTERACTION .......................................................................................................................25

3 METHODOLOGY ............................................................................................................................................ 26

3.1 PROBLEM ANALYSIS ...........................................................................................................................................26

3.2 CROSS SECTION SHAPE GENERATION ..................................................................................................................28

3.3 PROBLEM SOLVING METHOD...............................................................................................................................30

3.3.1 Steady 2D CFD Analysis ....................................................................................................................... 31

3.3.2 Transient 2D CFD Analysis .................................................................................................................. 35

3.3.3 3D CFD Large Eddy Simulation ........................................................................................................... 35

3.3.4 Modal FEM Analysis............................................................................................................................. 36

3.3.5 Beam design effects ............................................................................................................................... 40

10

3.4 RESULT ANALYSIS ..............................................................................................................................................42

4 RESULTS .......................................................................................................................................................... 43

4.1 STEADY 2D CFD ANALYSIS ...............................................................................................................................43

4.2 TRANSIENT 2D CFD ANALYSIS ..........................................................................................................................45

4.3 MODAL FEM ANALYSIS .....................................................................................................................................46

4.4 3D CFD LARGE EDDY SIMULATION ...................................................................................................................47

4.5 BEAM DESIGN EFFECTS .......................................................................................................................................50

4.6 RESULT ANALYSIS ..............................................................................................................................................52

5 DISCUSSION AND CONCLUSIONS ............................................................................................................. 55

5.1 DISCUSSION .......................................................................................................................................................55

5.2 CONCLUSIONS ....................................................................................................................................................58

6 RECOMMENDATIONS AND FUTURE WORK........................................................................................... 59

6.1 RECOMMENDATIONS ..........................................................................................................................................59

6.2 FUTURE WORK ...................................................................................................................................................60

7 REFERENCES .................................................................................................................................................. 61

APPENDIX A: GANTT CHART ......................................................................................................................... 64

APPENDIX B: RISK ANALYSIS ........................................................................................................................ 65

APPENDIX C: ISHIKAWA DIAGRAM ............................................................................................................. 66

APPENDIX D: CROSS SECTION GEOMETRIES ........................................................................................... 67

APPENDIX E: STEADY CFD SIMULATION ................................................................................................... 72

APPENDIX F: MODAL FEM RESULTS ........................................................................................................... 75

12

1 INTRODUCTION

1.1 Background

Marine waterjet propulsion is a technology that has been developed and refined since

the early 1950’s. The technology is used to provide thrust for high speed vessels in

varying sizes by accelerating water from the inlet to the outlet. Studies have shown that

marine jet propulsion technology yields higher efficiency when compared to traditional

propeller propulsion at high speeds at 35-50 knots [1]. Marine Jet Power distributes and

develops marine waterjet propulsion units to be fitted at vessels typically ranging

anywhere between 10-50 meters in size. The typical vessel operating speed ranges

between 25-50 knots depending on vessel size and waterjet propulsion unit size. The

vessels equipped with waterjets operate in varying conditions where there is a need for

high thrust and maneuverability. Pictures of vessels equipped with Marine Jet Power

waterjets are shown in figure 1.

Figure 1. Swedish Cinderella passenger ferry (left) and IC 20 M patrol craft (right) [2].

A typical waterjet system used in marine applications can be considered as a system of

4 components. An overview of the waterjet system can be seen in figure 1. An intake

coupled to a duct provides the system with a steady flow of water that is redirected to a

horizontal direction towards the impeller that accelerates the water against the stator.

The main function of the intake is to redirect the flow of water from under the boat

towards the impeller without causing cavitation. The impeller is directly connected to a

shaft that is run by an engine of choice. The vessel steers by using a nozzle that redirects

the flow. A reversing bucket is attached after the nozzle to allow vessels to reverse by

redirecting the flow. An example of a Marine Jet Power propulsion unit is the X-series

waterjet shown in figure 2.

Figure 2. Overview of the MJP X-series waterjet [2].

13

Many different forms of waterjet propulsion units exist today. A propulsion technique

often used is called axial flow which utilizes that flow pressure is built up by axial

acceleration of the fluid with a rotor [3]. A technique called centrifugal flow accelerates

the fluid strictly radially with an impeller [4]. A combination of these two techniques is

used in most of commercial waterjets and is known as mixed-flow where the fluid both

is accelerated axially by the impeller and centrifugally to utilize the shape of a narrow

passage created between the intake and stator to accelerate the fluid. A modern mixed

flow waterjet is shown schematically in figure 3.

Figure 3. Schematic view of a modern waterjet design [5].

The intake grill is a component that is mounted in line with the hull to prevent debris

from traveling through the waterjet. The intake grill is typically included on medium to

smaller sized waterjets. Today there are a plethora of companies that manufacture and

sell waterjet propulsion units that offer intake grill arrangements but few of them give

clarification on their detailed design. There are many companies competing for the same

market shares in the waterjet propulsion unit business but not all of them specify

whether the intake grill is a purchasable option. A couple of the companies that clearly

offer intake grills are Marine Jet Power [5], Alamarinjet [6] and HamiltonJet [7]. The

grill in all of these cases consists of a series of parallel aluminum beams assembled in

line with the water flow direction that are braced to stiffen the construction if necessary.

A Marine Jet Power designed grill is shown in figure 4.

Figure 4. Intake grill often used with the DRB and CSU series waterjets [5].

The design of the grill is of big importance as it directly affects both the efficiency of

the entire waterjet system with respect to viscous losses as well as protecting the

robustness of operation of the waterjet by preventing component failures. Incorrect

design of the grill may also interrupt the fluid flow entering the intake.

14

1.2 Purpose and definitions

This project is initiated by Marine Jet Power AB that is an Uppsala based company that

has been manufacturing waterjet units since the 1980’s. The main purpose of this

project is to study the design of intake grills often used with waterjet propulsion units to

stop debris from entering the jet chamber. The main purpose of the project is to evaluate

causes of component failure and methods to analyze the component. A secondary

purpose in the project is to define guidelines on how to design grill components with

more ease.

1.2.1 Project Objectives

The main objectives of the project are to:

Develop a suitable methodology used to design and verify intake grills in use with waterjet propulsion units.

Study risk of component failure at variable water flow velocities.

Evaluate the grill cross section shape with respect to efficiency.

1.2.2 Project Deliverables

The main deliverables for the project are listed below;

A validated design methodology that can be used to analyze intake grills.

Provide design guidelines for marine jet propulsion unit grills.

1.2.3 Research questions

The project shall aim to answer the following research questions;

What methods can be used to analyze intake grills with respect to component

failure?

What affects the component risk of failure and component efficiency?

1.3 Delimitations

The time frame of the project is limited and is run during 20 weeks from January 2019

until June 2019. A GANTT chart of the timeline of the project is shown in appendix A.

The project consists of 30 credits at Kungliga Tekniska Högskolan in Stockholm,

Sweden. The delimitations are listed below.

The cost of the component is not considered a factor of interest in this study.

The study is only performed with the assumption of structurally homogenous

materials with isotropic properties and will not account for manufacturing.

The study is due to time limitations limited to a selection of methods of interest.

No physical measurements are performed due to time limitations and lack of necessary equipment.

Simulations are only performed at component level and will not take the full waterjet system in consideration.

No numerical CFD solution for 2D shape optimization is implemented despite

the benefits of the method in previous studies.

15

No transient 3D FSI simulation will be performed due to time limitations.

1.4 Method

This study aims to find a suitable methodology for evaluating designs of intake grills for

marine jet propulsion as well as analyzing the design of intake grill geometries. The

methodology is equally important as the acquired geometry simulation results. A risk

analysis is performed to evaluate the risks in the project and can be seen in appendix B.

1.4.1 Problem analysis

To answer the research questions a thorough understanding of similar studies is needed.

A literature study of articles of interest is performed and the parameters of interest are

documented. A parameter study is performed to identify the parameters according to

their relevance for the results. The parameter variance range is estimated and analyzed

accordingly.

1.4.2 Problem solving method

Several different methods and ways of addressing the research questions are analyzed

and ranked with a weighted criterion matrix. The selected criterion are; implementation

time, estimated result accuracy and ease of verification. It is noted that one method on

its own is unlikely to solve all parts of the problem, and therefore combinations of

methods are considered. The study aims to cover a wide variety of available methods

currently applied in the area of study and may therefore disregard more detailed

methods with respect to their implementation time. Relevant methods are investigated

through a literature study and chosen to fit the available parameter set.

1.4.3 Verification

The results from the chosen problem-solving methods are compared with known

reference data for similar geometrical shapes. The methods are evaluated against known

reference data to prove the validity of the attained results. of any more complex shapes

or forms of the grill.

1.4.4 Design proposal

After completion of the above steps, a design proposal is constructed from the acquired

results and the methods and their respective results are evaluated. The proposal aims to

include the most successful tested methods for grill analysis. A grill design proposal is

prepared for further optimization compiled from the acquired results.

16

2 FRAME OF REFERENCE

The reference frame is a summary of the existing knowledge and former performed

research on the subject. This chapter presents the theoretical reference frame that is

necessary for the performed research.

2.1 Literature Study

A literature study is performed to gain knowledge about previous research.

[8] presents general knowledge about the MJP water jet propulsion products. The

handbook provides an overview of the product series as well as analytical guidelines for

calculations of the engine thrust.

[9] investigates the possibility of further development for the thrust calculation software

currently used at MJP. The thesis proposes ways to analytically calculate the thrust of a

ship with product specific input using MJP designs. Further work with implementing

the functionality of the original software is needed to complete the original goal of the

study.

[10] thoroughly analyzes the design of a Wakejet design by using CFD. The study

includes analysis of the pump design, intake design as well as the influence of intake

grills. It is concluded that adding a grill to the water jet has a minor effect on the flow

direction inside a non-optimized intake chamber. The study does not elaborate on the

analysis on grill geometry and proposes that the intake grill geometry may be optimized

to reduce vorticity.

[11] performs a detailed analysis and investigation of the flow regime and conditions

within the intake of a waterjet unit for the company Hamiltonjet. The study is performed

with CFD and verified using field tests on a ship as well as with aeronautical wind

tunnel simulations using different methods for flow visualization.

[12] presents ways to numerically analyze and optimize shapes with respect to

aerodynamics. The thesis is mainly targeted towards use for aviation wing design but

presents a wide variety of applicable theorem for general form optimization.

[13] investigates ways to compare analytical calculations with physical measurements

of a glass fibre reinforced polymer composite beam using modal analysis. The study is

performed on a beam with constant cross section that is vibrating freely with a variety

of boundary conditions. Although the study concludes that the analytical results do not

match the measured results, the study proposes ways to express the relation between

model and reality.

[14] performs shape optimization on a 2D cross-section by using genetic algorithms

and parameterization. The study also compares the meshing methods of a 2D geometry

with hydrodynamic analysis and their influence on the resulting geometry. The report

mainly focuses on drag minimization and lift maximization for airplane wing

geometries but provides general methods that can be used for a wide variety of

optimization problems. It is shown that the study performed using adaptive meshing

converges 95% faster with 38% result improvement compared to a method using

uniform mesh.

17

[15] studies the interaction of vibrations between structures and fluids. The study

verifies the legitimacy of the ANSYS CFD model with acoustic finite elements for

submerged flat plates, circular plates and cantilever beams by physical measurements in

experimental testrigs. The study compares the mode frequencies of submerged

structures and structures in air and how they can be estimated.

[16] derives analytical formulas for calculating flexural and torsional mode frequencies

of a rectangular cantilever beam immersed in fluid as a function of the frequencies in

vacuum. The measured results show good reminiscence with the analytical model for

inviscid fluid in combination with a high Reynold’s number.

[17] analyses the influence of coupling properties between a fluid mechanical and a

structural mechanical solver during FSI simulation for a submerged Euler Bernoulli

Beam. It is shown that a weak coupling FSI simulation for a Euler Bernoulli Beam well

mimics results in air. It is also shown that an FSI analysis with strong coupling yields

very accurate mode frequencies in still fluid compared to physical measurements.

Furthermore, the damping properties of the fluid is more noticeable with the weak

coupling and influences the displacement over time.

[18] studies the dynamic response and stability of a NACA0015 symmetrical hydrofoil

by FSI simulations with a fully coupled solving method. It is concluded that numerical

viscous FSI compares well with available experimental measurements. The study also

shows that the NACA0015 hydrofoil geometry first modal bending frequency is mostly

independent of inflow velocity U in the range of 5 6Re 3.05 10 4.27 10 .

[19] compares physical measurements to a numerical solution and show that cavitation

caused by moving fluid affects the eigenfrequencies of the structure due to change in the

surrounding added mass. It is concluded that the added mass decreases as the cavitation

increases around a NACA0015 airfoil geometry and that it leads to a raise in the modal

frequency as a result.

[20] evaluates the wetted modal eigenfrequencies of a clamped-free NACA 0009

hydrofoil beam with the dry eigenfrequencies by finite element analysis in the software

COMSOL Multiphysics. The study evaluates both hydrodynamic and hydrostatic loads

at 5Re 6 10 .

18

2.2 Sources of excitation

A body-oscillator is implemented from geometrically defined object as a discrete mass

that can move with respect to a coordinate system. Any vibration can be described as a

difference in amplitude around a mean value with a harmonic interval. Not all vibrations

are considered harmonic but those that are can be described as sinusoidal.

There are many sources that may cause body-oscillator vibrations because of fluid-

structure interaction and are in some cases hard to detect and distinguish. Fluid induced

vibrations may be divided in to three separate areas of interest. Extraneously Induced

Excitations (EIE) are excitations that are considered independent of flow instabilities

originated from body geometry and movements. EIE is considered as fluctuations in

flow velocity or pressure that is independent of structure geometry. This can be caused

by turbulence of the moving fluid at certain Reynold’s numbers as well as external

excitations such as hull vibrations, engine vibrations of other mechanical component

vibrations. Typical ship vibrations may occur from many sources and cover a wide

range of the frequency spectrum. In table 1 typical sources and ranges of ship vibrations

are shown.

Table 1. Typical ship vibration sources [21].

The second source of interest is Instability-Induced Excitations (IIE) which originates as

a direct result of fluid flow past the body-oscillator. This type of excitation includes

study of different kinds of vortex shredding and types of boundary layer interactions

between the fluid and body. Fluid induced vibrations (FIV) is a broad subject and can be

described as the study of induced motion as a function of harmonic irregularities in the

flow boundary regime between the object and fluid.

Movement of the body-oscillator itself may cause force fluctuations that affect the

body-oscillator vibrational behavior. Effects of this nature are categorized as

Movement-Induced Excitations (MIE) and may affect mode shapes and

eigenfrequencies of the body-oscillator. MIE can be influenced by fluid coupling, mode

coupling and multiple-body coupling of several body-oscillators in near proximity. If a

body-oscillator has apparent mode coupling, a phenomenon called frequency merging

must occur such that two flow-dependent modal frequencies forms a common value at a

certain velocity. If multiple-body coupling is apparent, the motion of neighboring body-

oscillators influences the fluid-dynamic coupling. Multiple-body coupling can play an

essential role in the excitation. A visual representation of the three excitation sources are

shown in table 2.

19

Table 2. Body Excitation sources [22].

2.3 Body Mechanics

A general dynamic model for body elastic vibrations in can be defined as (1)

Mq Cq Kq F (1)

where M is the mass matrix, C is the damping matrix K is the elastic matrix and q is

the body displacement vector as a function of time [23]. By determining the matrices

and assuming harmonic motion the system is solvable with second order differential

equations (ODE). Harmonic motion in its simplest form can be defined as a time

dependent function according to (2)

( ) sin( )q t Q t (2)

where Q is the amplitude, is the frequency and t is the time. An important nearby

EIE to monitor is the engine shaft rotational frequency 2s sf as it induces

vibrations in the ship hull of harmonic nature. Another factor to consider when

evaluating the possible relevant EIE’s is the number of impeller blades as they could

produce vibrations with multiples of the shaft frequency. Resonances are created by the

periodic passing of the impeller blades in the pump are transferred to surrounding

components [24] in the waterjet and may be calculated according to (3)

c sf z f (3)

where z are the number of impeller blades and sf is the shaft frequency. The range of

revolutions per minute for the shaft that are of consideration has been determined to 0 to

3000 rpm which corresponds to maximum 50Hz. The number of impeller blades of

interest are 4 or 5 with most typical waterjet configurations.

The dimensionless damping coefficient is defined as . Varying the damping

coefficient heavily affects amplitude amplification around resonance frequencies. An

example of this is shown in figure 5 where the amplitude at different dimensionless

damping coefficients are shown when the excitation frequency is varied against the

modal frequency n . If the dimensionless frequency ratio is larger than 2 there is

no apparent harmonic amplification at the modal frequency [25] .

20

Figure 5. Resonance frequency magnification at varying damping coefficients due to harmonic excitation.

The preferred minimum allowable first modal bending frequency to not acquire

harmonic amplification is defined to be larger than the critical frequency according to

(4)

2 2allowed c sf f zf (4)

The modal eigenfrequency of a clamped-clamped beam in vacuum, symmetrical around

the transversal direction may be calculated according to (5)

2

12

nn

K EIf

L (5)

where E is young’s modulus, I is the beam moment of inertia in the transversal

direction, 1 is the mass per unit length, n is the mode shape index and L is the beam

length. The coefficient nK is a reference number defined as

1 22.4K for the first mode

shape and 2 61.7K for the second mode for fixed-fixed beams [26].

The total mass m of a 3D body of homogeneous material may be calculated as the

material density m times the volume of the body V . For a 2D cross-sectional shape the

mass per unit length 1u is therefore be expressed as the material density m times the

cross-sectional area sectionA times the unit depth 1L according to (6)

1 m section m sectionu A L A (6)

The center off mass Gr for a generalized object is calculated according to (7)

i iG

m rr

m

(7)

where im is the partial mass of the studied section, ir is the coordinate center of the

studied section and m is the sum of all partial masses also known as the object total

mass. An objects area moment of inertia at a parallel axis with respect to the center of

mass can be calculated according to Steiner’s theorem [27] in equation (8)

21

2

y cmI I Ad (8)

where yI is the moment of inertia of a partial section and d is the distance from the

partial section to the objects center of mass.

2.4 Added Mass

A phenomenon that affects the modal frequency of submerged objects is the added

mass. Essentially the added fluid mass around a body in movement a

Extensive research has been performed on hydrodynamic vibrations in static fluid [16].

Modal analysis of structures submerged in static fluid can be simulated by using

ANSYS fluid nodes [15] with added mass. In this representation, no fluid motion is

implied, and the structure vibrations are assumed to be one-directional corresponding to

the first bending mode. In its simplest form, the eigenfrequencies nf of a vibrating

structure submerged in fluid is compared with the eigenfrequencies in vacuum ,n vacf .

The ratio of the frequencies can be described as the frequency reduction ratio 1 0

according to (9).

,

n

n vac

f

f (9)

Previous studies have shown that the frequency reduction ratio is less significant for

higher mode frequencies [28]. The frequency reduction ratio has been analytically

determined for uniform beams with symmetrical cross-sectional shapes in stationary

fluid from previous studies [22] as according to (10)

1

1

1 yyA u

(10)

where yyA is the added mass per unit length. For a cylindrical cross-section the mass per

unit length is defined as (11).

2

,yy cyl fluidA r (11)

For an elliptical cross-section the mass per unit length is defined as (12)

2

,yy ellipse fluidA a (12)

where a is half of the width of the ellipse.

2.5 Fluid dynamics

In this chapter the relevant underlying theory of fluid mechanics is introduced to

analyze the grill component. The Reynold’s number is a dimensionless number

according to (13)

ReUc

v (13)

where u is the average mean flow velocity, c is the chord length and v is the viscosity

of the fluid [29]. The Reynold’s number is considered a ratio of the inertial forces

22

divided by the viscous forces interacting on an object during flow and describes the

characteristics of fluid motion. It is widely acknowledged that the flow turns turbulent at

numbers above the lower critical Reynold’s value of 2320 [30]. The normalized vortex

shedding frequency is known as the Strouhal number and is a dimensionless number

according to (14)

l

Stu

(14)

that can be used to determine the vortex shredding frequency vf according to (15).

v

Df

USt (15)

Various studies have been performed on the relation between the Strouhal number and

the Reynold’s number [31] . The Strouhal number is shown as a function of the

Reynold’s number for a smooth circular cylinder according to a compilation of available

experimental measurements shown in figure 6.

Figure 6. Strouhal number as a function of Reynold’s number on a circular cylinder [32].

When the Reynold’s number reaches the critical range at 5Re 3 10 the Strouhal

number jumps from approximately 0.2 to about 0.45 due to critical boundary layer

separation [31]. At approximately 6Re 1.5 10 the boundary layer becomes fully

turbulent at one side of the cylinder which causes a big jump in the Strouhal number.

The different stages of flow regimes around circular cylinders in steady current is

described as a function of the Reynold’s number shown in table 3.

23

Table 3. Regimes of flow around a smooth circular cylinder in steady current [32].

The Strouhal number is highly dependent of the surface roughness as the boundary layer

transition shifts and has previously been studied for circular cylinders [33]. Similar

effects can be observed on various objects when put through fluid flow although far

from all objects have analytical guidelines.

The drag force DF is the force component acting in the direct opposite direction of the

flow direction on an object affected by a relative flow velocity. The amount of drag that

affects an object can be determined with use of the dimensionless drag coefficient DC

that is defined according to (16)

212

DD

fluid

FC

U A (16)

where fluid is the fluid density, U is the average stream-wise flow velocity and A is the

reference area. The reference area is widely accepted as the object frontal area for most

objects but is defined as the chord length c times the depth L for reference airfoil

geometries. The lift coefficient LC is defined by a dimensionless number as a function of

lift force LF acting perpendicular to the drag force. It is defined according to (17).

24

212

LL

fluid

FC

U A (17)

The drag and lift coefficients are based on the integrated pressure and sheer forces

acting over a profile surface [29]. Measured values exist for many simple geometrical

cross-sectional shapes of the coefficients DC & LC as a function of the Reynold’s

number. The pressure drag is defined as the sum of the pressure integral in the normal

direction over the surface according to (18)

cosF

A

D p A (18)

where is defined as the relative angle between the flow direction and the normal of

the surface. The friction drag is calculated similarly according to (19)

sinf

A

D A (19)

as the sum of the sheer stress that acts tangentially to the surface as seen in figure 7.

12

Figure 7. Normal pressure and tangential sheer forces on an object during flow.

The sum of the pressure drag pD and the frictional drag

fD corresponds to the total drag

according to (20).

D p FF D D (20)

is drastically affected due to Reynold’s number [30] but are determined for more

complex shapes by using CFD. The geometry skin friction constant fC is calculated

according to (21).

2.3

102log (Re) 0.65fC

(21)

The wall sheer stress w is calculated according to (22)

0

w

y

u

y

(22)

which also can be expressed as a function of w , the fluid density and the mean

stream fluid velocity according to (23)

21

2w f fluidC U (23)

The friction velocity *U is calculated as a function of w and according to (24).

25

*

w

fluid

U

(24)

The first layer wall distance wy is then calculated according to (25)

*

w

fluid

yy

U

(25)

where y is a reference numerical value.

2.6 Computational Fluid Dynamics

The practice of numerical calculation of the fluid behavior around bodies is defined as

Computational fluid dynamics (CFD) and is today an industry standard tool for a

plethora of engineering applications. CFD is considered an effective tool for analyzing

and optimizing performance of components affected by fluids and has been evolving

rapidly under the 21th century. The method provides a detailed analysis of fluid

traveling past structures by observing velocity and pressure as a function of time. Forces

as well as flow field data from performed simulations provides essential information to

evaluate designs. The methods in available CFD softwares numerically solve the

Navier-Stokes or Euler equations in 2 or three dimensions. The k sheer stress

transport (SST) model developed by Menter [34] blend the two popular turbulence

models k , that performs well in the near-wall regions with the k model often

used with laminar flow due to its ability of modeling free-stream fluid [35]. The

underlying equations utilized with the k SST model in Ansys FLUENT can be

observed in equation (26)

j

j

i k k k

i

i

i

kk ku G Y

t x t x

u G Y Dt x t x

(26)

where kG represents the generation of turbulence kinetic energy due to mean velocity

gradients, Grepresents the generation of , & represent the effective

diffusivity of &k . &Y Y represent the dissipation of &k due to turbulence. D

represents the cross-diffusion term. The model is thoroughly described in the FLUENT

6.3 Users guide.

2.7 Fluid Structure Interaction

The interaction between fluids and solids is a phenomenon that occurs in many places in

nature and engineering applications. The movement of a body and the response of the

fluid is studied with laws and equations from different physical origins and are often

combined with numerical methods to evaluate performance. Fluid Structure interaction

couples the characteristics of mechanical dynamics with fluid dynamics by transferring

a displacement of a body in movement to the fluid dynamics problem which affects the

pressure distribution on the surface of the body [15]. This change in pressure then alters

the external force acting on the body.

26

3 METHODOLOGY

In this chapter the chosen methodology is introduced and explained.

3.1 Problem analysis

The proposed methodology for this thesis is described in figure 8.

Figure 8. Methodology flowchart

First an Ishikawa diagram is created to identify potential causes of component failure.

The diagram can be seen in appendix C. The component risk of failure is assessed by

initially performing a thorough literature study of previous analyses of waterjet intake

grills and related reports of component failures in similar environments. An Ishikawa

diagram analysis is performed to break down the main causes of the grill component

failure [36] and is shown in appendix B. As shown in the diagram, there may be many

causes to the component failure with varying level of importance.

The grill component is located in a complex environment with both EIE originating

from water turbulence and overall ship vibrations that have the potential to cover a wide

range of frequencies. The most prominent harmonic excitation on a motor driven vessel

are the frequencies generated by the engine itself and the driveshaft that it is connected

to [21].

It is of big importance to determine parameters and results to evaluate the grill

performance as is targeted towards several goals. Two direct design goals are

acknowledged as; reducing the component risk of failure and the component efficiency.

As these optimization targets can result in contradictory design it is important to

evaluate the causes and effects of both targets individually. The monitored results from

the performed study are: reduction of the mode frequencies nf and reduction of the

drag & lift forces DF &

LF alongside observing the operating conditions and loads the

component are exposed to. There are several vibration sources of interest in this thesis

as the grill is seen as a vibrating body under turbulent flow. The chosen design variables

for this study are observed with a P-diagram shown in figure 9. Parameters of interest

for this experiment are studied with a thorough literature study in combination with a

parametrized component design.

The water density fluid is kept constant at 10203/kg m , while the dynamic viscosity

is dependent of the water temperature corresponding to water at 20 degrees Celsius [30].

As the design of experiments (DOE) and simulation time is highly dependent on the

number of variables, only the most necessary input parameters are considered.

The selected material for the grill is LM6M aluminum manufactured in one piece by

casting. Material properties for the grill component is shown in table 4 below.

27

Figure 9. P-Diagram for the optimization problem.

Table 4. Material Properties for Aluminum Alloy LM6M.

Parameter Description Unit Size

E Young’s Modulus Pa 971 10

Poisson’s Ratio - 0.33

m Material Density 3/kg m 2650

0.2pR 0.2% Proof Stress Pa 660 10

s Tensile Strength Pa 6160 10

f Endurance Limit 75 10 cycles

Pa 651 10

The controllable factors of interest used for design variation are shown in table 5 below

and the noise factor is shown in table 6.

Table 5. Control Factors

Control Factors Description Unit Range

U Flow velocity m/s 5-25

L Beam Length m 0-2

,x ya b Shape Control

Parameters

- -

Table 6.

Noise factor

Noise factor Description Unit Range

Inflow angle of

attack

deg 5

28

3.2 Cross Section Shape generation

The grill cross section is created with various methods. One simple option is to create

splines using Bezier curves with control points. The benefit of using Bezier spline

curves is that the curve boundary conditions easily are specified by coordinate

placement. The curve is created by using a standard cubic Bezier curve [37]. The curve

can be represented as a series of points in for an arbitrary number i of points iP

according to (27)

,

0

( ) ( )m

i m i

i

B j h j P

(27)

where 0,1j is a curve control parameter that follows the curve such that 0j

returns the first point coordinates 1P and 1j returns the last point

nP coordinates. The

with Bernstein’s polynomial , ( )i zh j is constructed according to (28)

,

0

( ) (1 )z

z i i

i z

i

zh j j j

i

(28)

where 'z i are the binomial coefficients. The cross-section curves are created by 6

control points that are specified as according to table 7 that are steered by implementing

shape control parameters &x ya b where 0.1,1.0xa & 0.12,0.30yb . The control

parameters are implemented in such a way that xa controls the overall shape of the

cross section and yb operates the profile height. All shapes operate as a function of the

chord length to enable scalability of any chord length c. A generated geometry is shown

in figure 10.

Table 7. Bezier point placement

Point x-coordinate y-coordinate

1 0 0

2 0

2

yb c

3

2

xac yb c

4 xa c

20

c

5 c 40

c

6 c 0

29

Figure 10. Generated shape at 0.8, 0.26x ya b

The cross-sectional shape is generated in MATLAB and is exported as coordinates to a

text file. Every cross-section is imported to Ansys Design Modeler as design points. The

shape of the curves is then recreated through the points by using the Ansys command

3D Curve. The trailing edge of every cross-section shape is terminated with a vertical

edge. The generated cross-sectional shapes are seen in appendix D. Well documented

reference symmetrical NACA0024 airfoil geometry is used to verify some of the

simulations and are shown in figure 11.

Figure 11. Reference airfoil geometry NACA0024 (airfoiltools, 2019)

30

3.3 Problem solving method

The method addressed to solve the research question requires detailed evaluation as the

method has a big effect on the validity of the result. Due to limited working time of 20

weeks, the choice of solving method also has a big influence on the target of the study.

A selection of methods and criterion are based on the performed literature study. No

FEM stress analysis is considered as it is easily performed at a later stage in the design

process.

The methods are ranked according to three criterion in weighted criterion matrix that is

seen in table 8. The first criteria “Implementation time” approximate the time to

successfully perform a study with the selected method. The selection “Low” yields the

method a score of 3 points, while “Medium” yields 2 points and “High” yields 1 point.

The implementation time criteria weigh the ease of verifying the result with the selected

method where the choice “High” yields 3 points, “Medium” yields 2 points and “Low”

yields a score of 1 point. The result accuracy criteria estimate the validity of the

acquired results and their usability in this study. The result accuracy is considered the

most important criteria and the choice “High” is therefore given 6 points, “Medium” 4

points and “Low” is given a score of 2 points. The sum of these criterion are multiplied

horizontally for each row and is displayed as one scalar result where higher is better.

The 4 chosen methods to analyze in this study are: 2D CFD analysis, 3D CFD Analysis,

Modal FEM in fluid and Analytical Modal Analysis.

31

Table 8. Weighted criterion evaluation matrix for the solving method.

3.3.1 Steady 2D CFD Analysis

The 2D CFD Analysis provides a computational friendly way of calculating the drag

coefficient and force as a function of the cross-sectional shape of the grill profile as well

as the flow parameters. The 2D cross sectional shape is analyzed in ANSYS FLUENT

[38] with guidelines from ITTC [39]. The cross section is observed as an independent,

infinitely thin slice of a fully submerged rigid body with a constant flow of water. There

have been performed extensive research on turbulence models and their applications for

certain case studies. The flow is considered turbulent as the Reynold’s number is much

larger than 4000 for flow velocities 5 25 /U m s [40]. A general overview of the

2D CFD Analysis is shown in figure 12.

32

Figure 12. 2D CFD Analysis

The analysis domain is constructed as a planar geometry according to guidelines and is

shown in figure 13. The chord length c of the geometry is kept constant at 0.1m. The

inlet is modeled as a half circle with radius 5c. The distance from the hydrofoil

geometry center to the outlet is fixed at 20 times the chord length c.

Figure 13. CFD analysis domain.

The fluid domain is meshed with a structured C-mesh with a constant number of 454000

finite elements by using Ansys Design Modeler. The mesh is controlled with fixed

numbers of element divisions at all edges of the model to eliminate remeshing

differences as a factor between the tested cross-sectional shapes. A high bias factor is

implemented to refine the mesh to the preferred 1y value. The mesh is constructed

solely from quadrilateral elements and is shown in figure 14 for a NACA0024 reference

geometry. The y value is shown over the chord length in figure 15 for the NACA0024

reference geometry.

Figure 14. Fluid domain mesh overview (left) and mesh near studied geometry (right).

33

Figure 15. y value for meshed NACA0024

ANSYS FLUENT is solver is set to pressure-based with absolute velocity. Gravity is

not considered a factor of interest as the symmetric behavior of the geometry is studied.

The fluid is considered incompressible and Newtonian. The simulation is performed

with a 2-equation k SST-turbulence model which is proven to give accurate results

for Reynold’s number values in applicable ranges for ship hydrodynamics [39]. A 4-

equation SST transition turbulence model is compared to the regular k SST model.

The simulation is run in saltwater with a constant fluid density at 10203/kg m and a

constant dynamic viscosity of 0.001003 kg/m s . All walls in the domain are

considered stationary with a no-slip sheer condition. The inlet is modeled with a

constant flow at velocity U in the x-direction. The inlet turbulent intensity is set to 5%

and the turbulent viscosity ratio is kept to the default setting 10. The solution method

SIMPLEC is selected for the k SST model. A coupled solver is used for the 4-

equation SST transition model. The ANSYS FLUENT solution convergence is

monitored by observing the residuals. A convergence criterion of 310 is imposed for the

velocity residuals and 410 for the continuity residual according to ANSYS FLUENT

guidelines [41]. Most simulations are run approximately 1500-3000 iterations to reach

this convergence criteria. The drag force DF and lift force LF shown in figure 6 is

monitored and saved at the final solution iteration of continuity residual convergence.

All geometries are evaluated with a static analysis at fluid velocity 20 /U m s with

0 and 2 . A study is also performed with a fixed shape parameter 0.4xa

where the shape parameter yb is varied and the fluid velocity U is varied in the range

of 5 25 /m s at 0 .

The FLUENT CFD-model is verified by performing simulations on the reference

NACA0024 geometry with 0 at 5,10[ / ]U m s corresponding to 5Re 5 10 and 6Re 1 10 . The results are compared with known reference data to verify the model.

The results for the k SST model be seen in table 9 and the 4-equation SST

transition model is shown in table 10.

Geometry DC value at

LC value at DC value at

LC value at

34

Table 9. Ansys fluent k SST model verification.

Table 10. Ansys fluent k SST transition model verification.

5Re 5 10 , 2 5Re 5 10 , 2 6Re 1 10 , 2 6Re 1 10 , 2

NACA0024

reference

0.01106 0.19780 0.00883 0.20832

NACA0024

k SST

0.01748 0.15610 0.01535 0.16309

absolute error 0.00642 0.04170 0.00652 0.04523

relative error +58.0% -21.1% +73.8% -21.7%

Geometry DC value at

5Re 5 10 , 2 LC value at

5Re 5 10 , 2 DC value at

6Re 1 10 , 2 LC value at

6Re 1 10 , 2

NACA0024

reference

0.01106 0.19780 0.00883 0.20832

NACA0024

SST

transition

model

0.01152 0.18785 0.00844 0.19660

error 0.00046 0.00995 0.00039 0.01172

relative error +4.2% -5.0% -4.4% -5.6%

35

3.3.2 Transient 2D CFD Analysis

A transient unsteady 2D CFD analysis is performed at one geometry with varying

relative flow angle . The study aims to monitor the drag and lift forces as a function

over time with the 4-equation k SST Transition-turbulence model. The same

analysis domain and mesh used in the steady simulation is used in the transient analysis.

The study is performed to evaluate how the transient solver compares to the steady

solver. Ansys Fluent is configured identically as for the case with the steady solver but

with a transient solver at a fixed timestep of 0.001 seconds. The simulation is run for

approximately 2.5 seconds of simulation time and the results are compared to the results

from the steady 2D CFD simulation.

3.3.3 3D CFD Large Eddy Simulation

A 3D CFD simulation is performed to analyze the grill performance. The Large Eddy

simulation turbulence model is chosen due to its proven ability to capture shedding

frequencies over time over blunt bodies [42]. The purpose of the analysis is to study

differences between the 2D analysis case as well as investigating the ability to simulate

turbulent boundary layer shedding effects over a specified beam geometry. The analysis

is performed on an extruded beam geometry with chord length 0.1c m at a fixed

angle of attack 2 to evaluate the drag force DF and lift force

LF as a function of

time. The studied geometry is enclosed in the exact same fluid domain specified in the

2D CFD case but with a third dimension determined by L . The domain is setup

similarly with an inlet, outlet and the shape geometry as an extruded cutout according to

figure 16. All other surfaces are determined as walls with no-slip condition according to

figure 17.

Figure 16. 3D LES analysis domain with boundary conditions.

36

Figure 17. 3D LES analysis domain with boundary conditions.

The mesh density is decreased compared to the 2D case due to the implementation of a

third dimension. The mesh consist of a structured mesh with 474516 elements and is

shown in figure 18. The simulation is performed with a beam depth 0.2L m at fluid

velocity 10 /U m s . Another simulation is performed for 1.0L m & 20U m/s. The

simulations are run with a timestep of 0.001 seconds and up to 20 iterations per time

step for a duration of over 2.5 seconds.

¨ Figure 18. 3D LES analysis mesh at L=0.2m.

3.3.4 Modal FEM Analysis

The Modal FEM analysis aims to study the mode shapes and eigenfrequencies of

extruded body geometries from cross-sectional shapes. The study is performed on a

single beam with fixed-fixed end conditions with the Ansys Workbench Modal module

and is conducted on the 100 generated cross-section shapes to study the undamped

harmonic oscillation in vacuum. Only the first two modal frequencies 1 2&f f are

considered of interest as they coincide with the range of ship vibrations for typical beam

lengths of 1m. The mode shapes are shown in figure 19.

Figure 19. First (left) and second (right) bending mode shapes.

37

The model is meshed with SOLID186 elements. A mesh convergence study is

performed where the mesh is refined until the monitored modal frequencies converge

and a change in mesh element quantity yield no significant change in the modal

frequencies.

The study is also performed for the 100 cross sectional shapes on single beam with

fixed-fixed end conditions in still fluid to study the effect of added fluid mass with the

Ansys Workbench Modal Acoustics module. The model is divided in one structural

domain simulating the aluminum beam and one fluid domain simulating still water. The

fluid domain is modeled as an extruded D-profile enclosing the beam on interacting

sides as shown in figure 20. This fluid domain shape allows that quadrilateral mesh

mapping is valid for all studied cross-sectional grill shapes.

Figure 20. FEM modal analysis domain in fluid.

The structural domain is meshed with SOLID186 elements and the fluid domain is

meshed with SOLID30 elements. A fluid-structure-interaction interface is defined on

the faces between the structural and fluid domain with a fully coupled method.

FLUID30 elements are only compatible with in this configuration in ANSYS

Workbench if all elements in the fluid domain are meshed as quadrilateral 8 node 3D

elements that share node positions with the SOLID186 elements. As in the previous

study the structural beam domain is modeled with fixed-fixed end conditions. Figure 21

38

shows the analysis setup and mesh for one of the tested cross-sectional geometries.

Figure 21. FEM modal analysis domain (left) and mesh (right) in fluid.

The solution method is verified for an ellipse with the length L of 1m due to the

known analytical models for the added mass. The ellipse structural domain is modelled

in ANSYS design modeler with a width of 100mm and a height of 25mm. A fluid

domain study is performed to analyze the effect of the surrounding fluid domain size.

The fluid domain size is incrementally increased to ensure that the results converge. The

model setup and mesh are shown in figure 22 at a width of 5 times the width of the

ellipse and 10 times the height of the

ellipse.

Figure 22. FEM modal analysis on ellipse setup (left) and mesh (right)

A mesh convergence analysis is performed on the ellipse in vacuum where the first

modal frequency is monitored as the mesh is refined. As seen in figure 23 the first

modal frequency 1,vacf converges as the mesh is refined. The fluid domain is equally

scaled in the x and y direction from 2 times bigger than the ellipse geometry to 40 times

bigger while the first modal frequency is monitored. The results are shown in figure 24.

The modal mass per length unit is analytically calculated for the ellipse according to

(12). The ellipse modal frequency in vacuum is then analytically calculated according to

(5) where the reduction ratio is acquired according to (10) and is multiplied to the

analytical result in vacuum. The results are shown in table 11. Although there is a

present error in the acquired results, the model follow does follow the underlying theory

for the ellipse.

39

Figure 23. Number of mesh elements effect on modal frequency in vacuum.

Figure 24. Fluid domain size effect on modal frequency in fluid.

Table 11. Ellipse FEM model verification

Study First modal

frequency in

vacuum 1,vacf

First modal

frequency

in fluid

1, fluidf

Frequency

reduction

ratio1

Analytically calculated modal frequency Hz 108.30 68.43 0.6319

FEM modelled modal frequency Hz 109.01 69.37 0.6363

Difference Hz 0.7247 0.9397 0.0044

Relative error [%] 0.67% 1.37% 0.70%

40

3.3.5 Beam design effects

A number of typical ways to reinforce the beam mounting and their effect on the modal

frequency in vacuum are evaluated. This study aims to evaluate small changes to the

ordinary extruded beam design affect the modal frequency, and as sharp edges generally

are avoided due to the risk of stress concentration spots. A reference NACA0024

geometry is extruded to a depth of L with a chord length 0.1c m while the first and

second modal frequencies are monitored as is seen in figure 25. The geometry is

meshed with quadrilaterals according to the simulation method in vacuum used in

chapter 3.2.4 and is setup with fixed-fixed boundary conditions. The ANSYS

Workbench modal module is used to solve the simulation for the first modal frequency.

Note that the chosen methods are evaluated individually. Combinations of the methods

may further increase the modal frequencies but are not performed in this study. The

chosen beam design methods are shown in table 12.

Figure 25. Beam modal frequency as a function of beam length

Table 12. Beam design methods

Name Picture Design parameter Range

Transition

section

Scale factor sf

Loft point lp

10-50%

0.1-0.5 [m]

41

The transition section design parameters are defined such that the centered cross-section

profile retains its scale and position for all changes to the design parameters &sf lp .

The parameter sf scales the NACA0024 cross-section uniformly in all directions

around its middle of the chord line ( 0.05c m ). The parameter lp changes the loft

position from the edge of the beam towards the center of the beam resulting in a longer

lofted section. The two design parameter are visualized in figure 26. The end radius

study simply varies the beam length in addition to adding a uniform radius at the end of

the beam fixed support that varies from 0-20mm. The studies are performed in matrix

form and calculates combinations of the independent variables.

Fillet

radius

Beam length L

Fillet radius R

0-1 [m]

0-20 [mm]

42

Figure 26. Design parameters lp and sf.

3.4 Result analysis

Initially the acquired modal frequencies of the suspended beams are compared with the

2D CFD analysis case. As the drag and modal frequency of the beams are related via the

cross-sectional geometry, the geometries can be evaluated as a function of the results.

As the viscous forces are target for minimization and the eigenfrequencies are target for

maximization it is possible to evaluate the cross-sectional geometries by dividing the

drag force with the modal frequency.

The best performing cross-sectional geometry is further evaluated by performing 2D

CFD simulations by varying between -5 to 5 degrees and U between 5-40m/s. The

center of mass Gr , mass Gm and moment of inertia GI at the center of mass is calculated

for the cross-section according to (7), (8). The cross section is also analyzed with the

modal FEM analysis and is mapped for varying beam length L and chord width c . A

constraint is constructed as the maximum allowable first bending modal frequency

according to (4) where the shaft frequency sf is determined to 50Hz and the number of

impeller blades z are determined to 5. This yield a minimum modal frequency

constraint allowedf of 354Hz for the first bending mode. With good data of the drag

coefficient the external viscous forces is calculated on these beam configurations that

fulfill the constraints by modifying the formula for the drag coefficient according to

(29)

2 21 1

2 2D D fluid D fluidF C U A C U Lc (29)

where the reference area A is expressed as the beam chord length c times the beam

length L . This force can be utilized as input in future FEM analyzes.

43

4 RESULTS

In this chapter the results from the performed analyzes are presented according to their

order in the methodology chapter.

4.1 Steady 2D CFD analysis

The steady 2D CFD analysis of the studied geometries for 0 at 20U m/s are

shown in figure 17. The results for geometries with 1.0xa are excluded due to

meshing incompatibility with the rounded trailing edge with the structured mesh

method. As seen in figure the geometries with shape parameter 0.8 0.9xa perform

well due to the low monitored drag. It is concluded that the yb shape parameter scales

the drag force close to linearly as shown in figure 27.

Figure 27. Drag force (left) and coefficient (right) as a function of shape control parameters &x ya b at

0 , 20U m/s .

The steady analysis of the geometries for 2 at 20U m/s is shown in figure 28.

The results can be seen in matrix form in appendix E. It is clearly shown that the

geometries with a low xa value yields both a higher DC and LC value at 2 &

20U m/s corresponding to 6Re 2 10 . It is also observed that the lift force and

coefficient reduce for higher yb values at 2 and 20U m/s.

44

Figure 28. Drag & lift as a function of shape control parameters &x ya b at 2 , 20U m/s .

The drag force and drag coefficient is monitored when the shape control parameter xa is

fixed to 0.4xa . The results are seen in figure 29 as a function of shape control

parameter yb and the inflow velocity U at 0 . As expected, the drag coefficient

decreases as the inflow velocity increases. No out of the ordinary response is acquired

as the chord height parameter yb increases. As seen in figure 28 the drag force on the

studied cross-sectional shape approximately increases as a function of the squared

inflow velocity U in the range of 5 25U m/s according to the drag equation (16)

due to the small change in drag coefficient.

45

Figure 29. Drag force (left) and drag coefficient (right) as a function of shape control parameter yb and

inflow velocity U .

4.2 Transient 2D CFD analysis

The analysis is performed on the cross-section shape with 0.4, 0.24x ya b . The velocity

contour over the geometry at 20 /U m s and 5 is shown in figure 30. The

monitored drag and lift coefficients are shown as a function of time in figure 31. No

time dependent harmonic behavior is observed for any of the solutions as the solution

stabilize.

Figure 30. Velocity contour at 20U m/s, 2 and t=2.5s.

46

Figure 31. Drag & lift as a function of shape control parameters &x ya b at 2 , 20U m/s .

4.3 Modal FEM analysis

The acquired modal frequencies as a function of the shape parameters &x ya b is shown

in figure 32. The first row shows the acquired first and second modal frequencies in

vacuum and the second row show the first and second modal frequencies in still water.

47

The results can be viewed in matrix form in appendix F.

Figure 32. Cross section modal resonance frequencies as a function of shape control parameters &x ya b .

The frequency reduction ratio n for the first and second mode frequencies are shown in

figure 33. The result clearly shows that the beam shapes with a higher yb value yields a

significantly higher frequency reduction ratio and that shapes with a higher xa value

perform better in all cases. Both the first and second modal frequency reduction ratios

follow the same exact trend, but the first modal frequency reduction ratio is more

significant as can be seen in appendix F.

Figure 33. Frequency reduction ratio n as a function of shape control parameters &x ya b .

4.4 3D CFD Large Eddy Simulation

The first simulation is run for 3.600 seconds at a timestep of 0.001 seconds. The

pressure and velocity contour at 3.600 seconds for 10 /U m s and 0.2L m is seen

in figure 34. The monitored drag and lift coefficient are seen in figure 35. The

frequency response of the drag coefficient at 2.5 seconds is seen in figure 36.

48

Figure 34. Pressure (left) and velocity (right) contour at 3.600 seconds for 10U m/s and 0.2L m at

2 .

Figure 35. Large Eddy Simulation drag coefficient (left) and lift coefficient (right) as a function of time at 6Re 10 .

Figure 36. Large Eddy Simulation drag coefficient frequency response plot at 6Re 10 .

49

It is concluded from this simulation that there is no significant occurring harmonic

release in the monitored drag and lift coefficient. There is a slight indication around 150

Hz in the frequency response plot that harmonic content is present, but it is not

considered significant due to the amount of noise in the other parts of the spectrum.

The monitored drag and lift coefficient for the second LES simulation performed at

1L m, 20U m/s at 2 is seen in figure 37 as a function of time. The FFT

analysis of the drag coefficient at 0.6 seconds is seen in figure 38. This simulation

shows clear behavior of harmonic content up until 1.2 seconds. It is not determined why

this behavior suddenly stops as shown in figure 37. The harmonic shedding frequency is

observed in the range of 50Hz.

Figure 37. Large Eddy Simulation drag coefficient (left) and lift coefficient (right) as a function of time at 6Re 2 10 .

Figure 38. Large Eddy Simulation lift coefficient frequency response plot at 6Re 2 10

50

4.5 Beam design effects

The transition section results factor analysis results are shown in figure 39 where the

first and second modal frequencies are shown as a function of the scale factor sf and

length factor lp .

Figure 39. First (left) and second (right) modal frequency as a function of design parameters &sf lp

The end radius study results are shown in figure 40 where the first modal frequency is

shown as a function of the beam length and the uniform radius at the beam boundary.

The dimensionless frequency ratio is used to visualize the relative change in modal

frequency for different beam lengths L as the fillet radius R is varied. The relative

frequency increase is shown in figure 41 as a function of the beam length and the fillet

radius size where the reference value is set to the beam length and fillet radius 0R

51

Figure 40. First modal frequency as a function of beam length L and fillet radius R .

Figure 41. First modal frequency relative increase as a function of beam length L and fillet radius R .

52

4.6 Result analysis

The geometries are evaluated by dividing the wetted modal frequency with the drag

coefficient shown in figure 42. As shown in the figure the cross sectional shapes with a

high xa and

yb value performs best out of the tested shapes. The chosen geometry for

further detailed studies is picked as the best performing geometry in the 1 / Df C surface

plot. The geometry is generated for 0.8xa , 0.3yb and is seen in figure 43. The

mass per unit length, moment of inertia and center of mass of the cross-section is shown

in table 12.

Figure 42. The first modal frequency divided by the drag coefficient (left) and lift coefficient (right) at

2 and 20 /U m s .

Figure 43. The chosen geometry for detailed analysis.

Table 12. Chosen geometry details.

Variable Amount Unit

Mass per unit length 6.132 kg/m

Moment of inertia 1.244610

4m

Center of mass 0.42 c -

53

The cross-sectional shape is analyzed by the 2D CFD k SST transition. The drag

coefficient DC is shown as a function of the angle of attack at Reynold’s numbers in the

range of 6 61 10 Re 4 10 in figure 44. The lift coefficient DC is shown as a function

of the angle of attack at Reynold’s numbers in the range of 6 61 10 Re 4 10 in figure

45. The dotted lines represent the cubic spline interpolation of the acquired datapoints

shown with circles.

Figure 44. Drag coefficient DC as a function of the angle of attack at 6 61 10 Re 4 10

Figure 45. Lift coefficient LC as a function of the angle of attack at 6 61 10 Re 4 10

54

As expected, the drag coefficient DC reduces as the Reynold’s number increases. It is

observed that the variation of the lift coefficient LC reduces as the Reynold’s number

increases. The first bending modal frequency in fluid is shown as a function of the beam

length L and the cross-section chord length c in figure 46. The constraint 1 allowedf f

is implied at 354Hz as previously calculated and show the beam variations that pass the

criteria as a function of beam length L and chord length c in figure 47.

Figure 46. First modal bending frequency in fluid as a function of L and c .

Figure 47. Suitable beam geometries at 1 354allowedf f Hz.

55

5 DISCUSSION AND CONCLUSIONS

All of the performed methods and their acquired results are discussed and evaluated

with robustness and result validity. The reference simulations for known geometries in

chapter 3 and the results in chapter 4 provide the necessary material for discussion.

5.1 Discussion

All of the methods evaluated in this study gives insight to specific parts of the behavior

of the intake grill it is clear that one single method fails to evaluate all areas of interest.

A combination of the most useful methods are therefore suggested as an engineering

alternative when designing intake grills. As no physical measurements are performed on

any of the generated beams it is unfortunately not possible to fully verify nor falsify the

simulation results acquired either by CFD or modal FEM analysis which provides a

level of uncertainty to the evaluation of the methods in this study.

The performed cross section shape generation is not performed based on any previous

study and may yield results that share little to no practical use for industry applications.

The general design does however mimic typical symmetrical airfoil designs with a

smooth leading edge and a sharp trailing edge which is an industry standard technique

for hydrofoils and airfoil applications. As the aim of the shape generation is to cover a

big selection of general to quickly spot trends and yield indications as to what shapes

are viable while evaluating the modal frequency versus drag and lift forces the study

does not imply that an optimal solution is intended to be acquired. The shapes are

generated with 50 datapoints that are stitched with the Ansys Workbench Design

Modeler 3D curve with point interpolation and no study on how many datapoints that is

considered adequate is performed which might lead to misleading results. More

datapoints could be used for further detailed analysis and its effect on the acquired

results. An alternative choice of cross section geometries could be to inspect and chose

geometries from the Airfoiltools database due to their heavily tested and optimized drag

and lift coefficients at specified Reynold’s numbers but this is not performed in this

study.

The steady 2D CFD simulation result verification with the NACA0024 shows that there

is a significant error in the k SST model at 5Re 5 10 and 6Re 1 10 . It is

concluded that this implies that the boundary layer transition does not reach fully

turbulent wall boundary separation for the geometry or that the model is unable to

operate in this Reynold’s range around the studied geometry. The 4-equation k SST

transition model results yields seemingly more accurate results in this Reynold’s

number range but still fails to acquire exact results compared with the reference data.

There are many other parameters that affects the drag and lift coefficient of the studied

geometry which makes the exact reason for this phenomenon hard to distinguish to a

single cause. This may include mesh type, mesh mapping, mesh distribution in the

analysis domain, analysis domain size, local and global mesh aspect ratio, analysis

boundary conditions, solver type, solution convergence, and solution control

parameters. Other types of mesh configurations are tested but yield inaccurate values for

the lift force due to uneven element placement around the cross-section geometry upper

and lower surface. The fully structured C-mesh succeeds to eliminate the lift coefficient

error but may influence the result of the tests with angles of attack 0 . All solutions

converge to a continuity residual of 410 but an even lower residual convergence criteria

could yield more accurate results for detailed optimization. It is noticed during

56

simulation setup that larger mesh aspect ratio results in risk of solution divergence with

the coupled solver. This is most easily corrected by implementing a more homogeneous

mesh to keep y around 1 which could greatly increase the element amount and

simulation time but is not performed in this study. Several attempts of mesh refinement

with constant element sizes are performed to find a level where the y value is low

enough without increasing the global mesh aspect ratio out of proportion without

success with the structured mesh. During the study it is noted that small changes to the

mesh or input parameters yield drastically different results for the drag and lift forces which gives reason to believe that the CFD model is not as robust as it may seem for the

studied geometries.

The transient 2D CFD analysis with a fixed timestep of 0.001 seconds shows that the

drag and lift force does vary a bit with time as the solution is initialized but then

stabilizes around an equivalent value attained with the steady solver. It is uncertain if

the drag and lift force stabilization purely is related to the k SST model or the

choice of timestep length. The transient k SST model is therefore considered to be

of little use in this study apart from verifying that the steady solver provides reliable

results.

The Large Eddy Simulation results show a clear behavior of transient turbulence around

the studied geometry at 2 with a 0.001s timestep which provides evidence that the

2D k SST model is failing to evaluate vortex shedding. The two Large Eddy

Simulation are run for approximately 48 hours and it is noticed that the simulation

requires much more computational resources to traditional RANS methods. As the analysis domain transitions incorporates a third dimension the mesh element amount

naturally increases to a much larger number. The simulations are therefore not run at

similar y values which influences the fluid behavior around boundary layer transition.

More detailed analysis can be performed by lowering the timestep and increasing the mesh element size at the expense of longer simulation time. As the simulations only are

performed for a maximum total simulation time of 3.6 seconds it is not proven if there is

occurring frequency dependent shedding after the simulation ends.

The FFT dataset analysis is performed in MATLAB with a gaussian window size

adjusted to fit the dataset at manually selected timesteps to provide the best

representation of harmonic behavior. As this is an iterative manual analysis there is

possible that the data could be analyzed differently. Filtering the dataset in the

frequency domain can provide more accurate frequency response plots but is not

performed due to the alteration of the result. External software such like Audacity or

similar that is specialized at performing FFT analysis is an option that could be

compared to verify or falsify the acquired method.

The modal FEM analysis provides the most accurate results out of all of the tested

methods when verified to known reference data for symmetrical ellipses. The model is

providing a valid option to regular FEM modal simulations. The performed mesh

convergence analysis converges nicely, and the domain size comparison verifies that the

model can be considered robust. It is noted that the computational time increases fast

when increasing the mesh element amount at the FSI boundary between the fluid and

solid when performing the modal FEM analysis in fluid.

Many assumptions are made as to what externally induced excitations that are of

interest for this component. To be absolutely sure that the EIE frequencies fall into the

studied range of problematic frequencies, a thorough study on hull vibrations should be

performed with the MJP waterjet system. This study makes no regard for what type of

57

engine and shaft frequency that is used with various MJP installations so the range of

studied problematic frequencies and the conclusions that are made may have to be

altered according to new input data.

No other material than LM6M is investigated in this study purely to limit the number of

parameters to study. The mechanical properties of LM6M are not among the highest of

cast aluminum alloys and may therefore be changed to a better performing alloy with

respect to fatigue and tensile strength. A change in material directly affects the modal

FEM studies but the general trends of the solutions are kept intact. As the CFD

simulations assume a fixed geometry uncapable of deforming it is certain that the results

are to be unchanged for a different alloy. The effect of the material roughness surface

parameter could be implemented in the CFD study but is disregarded in this study due

to time limitations. This parameter is not estimated to have a major impact on the result

but

As no simulation with the intake geometry is performed it cannot be concluded if any of

the suggested geometries would perform differently in a such an environment. The

turbulence at varying locations in the intake would first have to be determined either by

physical measurements or detailed CFD simulations at varying inflow velocities to

further evaluate the grill design.

Unfortunately, no FSI-simulation is performed in this study that could have given

valuable insight to the translational and rotational movement behavior of the beam

under fluid flow. This is a time-consuming process and is tested at a small timestep but

due to insufficient convergence per timestep and due to the considerably big mesh that

has to be used in order to achieve small y values it is concluded that the scope of this

study would have been too big for this study in combination with the already tested

methods.

The 4-equation k SST model provides results that are of the same magnitude of

reference airfoil data with a structured mesh but could be greatly improved as the

monitored drag coefficient has a large error of ±5%. The FEM in fluid analysis

performed yields good results for symmetrical ellipses but would need to be verified by

physical measurements on actual grill geometries to be proven fully valid in the use case

performed in this study.

The grill component is located in a very complicated environment and is not an easy

component to design and evaluate. Combining many models may yield indications

towards an optimal design but ideally physical tests will determine the performance of

the intake grill. This study started off as very narrow with a limited scope and quickly

emerged to a wide array of time-consuming method investigations but there are many

available options for further analysis of the grill component that are yet to be tested. If

the thesis is to be repeated with the knowledge the outcome would surely have been of

other nature as most of the time is spent verifying the CFD models against reference

data.

Cavitation patterns are not directly monitored but all simulations that are run fulfill the

requirements that no negative pressure is present. The effect of cavitation is not

evaluated and may affect the acquired drag and lift values for large angles of attack.

58

5.2 Conclusions

The cross-sectional shape, chord width c and beam length L drastically affects

the component performance and risk of failure.

A lofted profile with scaled boundaries can increase the first bending frequency up to 50% at a 1m long extruded NACA0024 beam.

A fillet radius of 20mm can increase the first modal bending frequency up to 4% at a 1m long extruded NACA0024 beam.

The 2D CFD model drag coefficient value yields consistent results on the

NACA0024 geometry with a static error of about 5% at Reynold’s numbers 6 6Re 5 10 &1 10

The modal FEM analysis in fluid performs well with an error of 0.7% for the reduction ratio of a symmetrical ellipse.

Cross-sections with elliptical profiles tend to contribute to a smaller lift force

when the angle of attack compared to the other studied geometries.

Tall cross sections are preferable with respect to modal vibrations as they

acquire a larger reduction ratio .

Dimensioning can be performed against the first modal frequency as the higher mode frequencies are significantly higher up in the frequency spectrum.

59

6 RECOMMENDATIONS AND FUTURE WORK

In this chapter, recommendations on more detailed solutions and/or future work in this

field are presented.

6.1 Recommendations

Methods to terminate the beam trailing edge and its effect on harmonic release could be

of big interest to study after an effective stable grill geometry is developed. Further

simulations with the LES model using a finer mesh and a smaller timestep is a good

option that could yield more reliable results.

A FSI simulation on the effect of the mechanically induced vibrations could be

performed to further analyze harmonic behavior. A FSI simulation could ideally be used

to calculate the added mass and reduction factor at varying inflow velocities U and

varying angles of attack . This is time-consuming study to perform as the fluid-

structural coupling further complicates the solution validity. As the FSI simulation also

is structurally time-dependent compared to the performed simulations where the

structural domain is static, there is a necessity to remesh the analysis domain at every

timestep which introduces more risk of error and a much larger simulation time.

The grill design could be further analyzed with a 3D Finite Volume Method (FVM)

simulation with the full waterjet intake geometry to simulate the component behavior in

its actual environment. Due to the many factors of ship design, this is also a complicated

study that requires many estimations and simplifications to yield generalized results.

The computational time of such a study heavily depends on mesh and y values of the

target areas of interest but would most likely be very time intensive even with RANS

models. If the flow in close proximity of the impeller is accurately modelled this

method would most likely be the best computational option to verify a complete grill

design with. This type of study would also be useful to analyze the grill effect on intake

cavitation patterns which heavily affect the waterjet performance.

Cross-sections with a smaller c yield smaller forces from drag and lift but need careful

evaluation due to increased sensitivity to external loads when dimensioned.

There might be additional theory that can be applied in addition to the performed

simulations that provide drastically faster and more reliable results than the analysis in

this study that still is to discover.

As the grill is a component that consist of a finite number of parallel beams it would be

interesting to study the parallel beam-to-beam distance effect on the drag and lift

coefficients. It would also be interesting to study the beam-to-beam distance effect on

the added mass effect of the beam cross section as this is not included in this study.

Methods that potentially could be of use when performing this study would be a 2D

CFD simulation and possibly a fully coupled 3D FSI analysis for exact results and

understanding of the beam behavior at certain fluid velocities.

The effect of temperature deviations could be evaluated to see how much of an impact it

has on the acquired results as it drastically changes the viscosity of the fluid.

Physical measurements on a boat would be a good final verification to prove that the

methods provide valid results when a design is finished.

60

6.2 Future work

Below is a list in bullet form of potential future work. Some of the proposed topics are

very time-consuming but could yield interesting and valuable results.

Continue development of shedding analysis model.

Perform trailing edge analysis to evaluate the impact of minor design changes and the shape robustness.

Transient FSI analysis with hydrodynamic mass effects.

Multiple parallel beam added mass coupling analysis.

Full waterjet system analysis.

Analyze the effect of grill placement in the intake.

Physical measurements on a vessel with waterjets equipped with an intake grill.

Measurements in cavitation chamber on the waterjet with intake grill to verify cavitation patterns caused by the grill.

61

7 REFERENCES

[1] The Royal Institution of Naval Architects, Waterjet Propulsion 5, London: The

Royal Institution of Naval Architects, 2008.

[2] Marine Jet Power, "Marine Jet Power," Marine Jet Power, 11 06 2019. [Online].

Available: www.marinejetpower.com. [Accessed 11 06 2019].

[3] Central Cargo Ship, "Axial flow pumps - Procedure for marine pumps," Central

Cargo Ship, 12 06 2019. [Online]. Available: http://generalcargoship.com/axial-

flow-pumps.html. [Accessed 12 06 2019].

[4] Ceneral Cargo Ship, "Centrifugal Pumps," Ceneral Cargo Ship, 12 06 2019.

[Online]. Available: http://generalcargoship.com/centrifugal-pumps.html.

[Accessed 12 06 2019].

[5] Marine Jet Power, "MJP DRB CSU Designers Guide," 01 03 2019. [Online].

Available: https://marinejetpower.com/uploads/2019/03/MJP-DRB-CSU-

Designers-Guide.pdf. [Accessed 04 06 2019].

[6] Alamarinjet, "Alamarinjet Designer's Manual," Alamarinjet, 01 03 2017. [Online].

Available: http://www.alamarinjet.com/wp-content/uploads/2017/03/Designers-

manual-Ver.-1.3_low.pdf. [Accessed 05 06 2019].

[7] HamiltonJet, "HamiltonJet Manufacture Installation," HamiltonJet, 05 06 2019.

[Online]. Available: https://www.hamiltonjet.com/global/manufacture-installation.

[Accessed 05 06 2019].

[8] Marine Jet Power, MJP Designer's Guide, Uppsala, 2016.

[9] M. Bergsek, "Thrust Prediction Program for Marine Jet Power," Stockholm, 2011.

[10] H. Martinsson and J. Varosy, "CFD Simulation of the Wakejet flow," Göteborg,

2017.

[11] B. Gavin John Griffith-Jones, "Investigation of Incompressible Flow Through an

Intake Duct with Applications to Waterjet Propulsion," , Christchurch, 1994.

[12] O. Amoignon, "Numerical Methods for Aerodynamic Shape Optimization,"

Department of Information Technology, Uppsala, 2005.

[13] O. Adediran, "Analytical and Experimental Vibration Analysis of Glass Fibre

Reinforced Polymer Composite Beam," Department of Mechanical Engineering,

Karlskrona, 2007.

[14] M. Kouhi, "Aerodynamic Shape Optimization using Adaptive Meshing,"

International Conference on Engineering Optimization, Lissabon, 2010.

62

[15] J. A. N. Hengstler, "Influence of the Fluid-Structure Interaction on the Vibrations

of Structures," University of Stuttgart, Stuttgart, 2013.

[16] C. A. V. Eysden and J. E. Sader, Resonant frequencies of a rectangular cantilever

beam immersed in a fluid, Melbourne: Journal of Applied Physics, 2006.

[17] I. A. A. V. R. H. S.M. Hosseini Z., "On The Elastic Beam-Fluid Interaction Based

On Structural Mode Shapes," Delft University of Technology, Delft, 2016.

[18] E. J. Chae, "Dynamic response and stability of flexible hydrofoils in

incompressible and viscous flow," Naval Architecture and Marine Engineering,

Michigan, 2015.

[19] X. Liu, L. Zhou, X. Escaler, Z. Wang, Y. Luo and O. De La Torre, "Numerical

Simulation of added mass effects on a hydrofoil in cavitating flow using acoustic

fluid-structure interaction," Journal of Fluids Engineering, vol. 139, 2017.

[20] A. Cortona, "Structural and Hydrodynamic Characterization of a NACA 0009

Hydrofoil by finite elements," in Conference: 18th Numerical Towing Tank

Symposium (NuTTS’15), Italien, 2015.

[21] V. Bertram, Practical Ship Hydrodynamics, Elsevier Science, 2012.

[22] E. Naudascher and D. Rockwell, Flow-Induced Vibrations - An Engineering

Guide, Lehigh University: Dover Publications, 1994.

[23] Y. Wu, L. Shengcai , L. Shuhong , D. Hua-Shu and Q. Zhongdong , Vibration of

Hydraulic Machinery, Peking: Springer, 2013.

[24] J. F. Gulich, "Centrifugal Pumps," in Centrifugal Pumps, Springer Science &

Business Media, 2010, p. 629.

[25] U. Sellgren, "Lecture 1," in MF2068 Machine dynamics, Stockholm, 2017.

[26] K. L. Richards, Design Engineer's Sourcebook, CRC Press, 2018.

[27] N. Apazidis, Mekanik II: Partikelsystem, stel kropp och analytisk mekanik,

Stockholm: Studentlitteratur AB , 2011.

[28] J. S. Carlton, Marine propellers and propulsion. – 2nd ed., Oxford : Elsevier, 2007.

[29] P. K. Kundu, I. M. Cohen and D. R. Dowling, Fluid Mechanics, London: Elsevier,

2016.

[30] Y. Nakayama and R. F. Boucher, Introduction to Fluid Dynamics, Butterworth-

Heinemann, 1998.

[31] G. Schewe, On the force fluctuations acting on a circular cylinder in crossflow

from subcritical up to transcritical Reynolds numbers, vol. 133, Journal of Fluid

63

Mechanics, 1983.

[32] B. M. Sumer and J. Fredsøe, Advanced Series on Ocean Engineering, Volume 26:

Hydrodynamics around Cylindrical Structures (Revised Edition), World Scientific,

2006.

[33] H. E. Achenbach E., "On vortex shedding from smooth and rough cylinders in the

range of Reynolds numbers 6 x lo3 to 5 x lo," vol. 109, 1981.

[34] F. R. Menter, "Zonal Two Equation Kappa-Omega Turbulence Models for

Aerodynamic Flows," in 24th AIAA Fluid Dynamics Conference, Orlando FL,

1993.

[35] ANSYS, "sharcnet," ANSYS, [Online]. Available:

https://www.sharcnet.ca/Software/Fluent6/html/ug/node487.htm. [Accessed 01 06

2019].

[36] B. G. Dale, Managing Quality p. 357, John Wiley & Sons, Incorporated, 2007.

[37] G. Farin, Curves and surfaces for Computer Aided Geometric Design, Arizona:

Academic Press Inc., 1993.

[38] Ansys, "Ansys," ANSYS inc, 29 01 2009. [Online]. Available:

http://www.afs.enea.it/project/neptunius/docs/fluent/html/ug/main_pre.htm.

[Accessed 19 02 2019].

[39] ITTC, "ITTC – Recommended Procedures and Guidelines - Practical Guidelines

for Ship CFD Applications," Copenhagen, 2014.

[40] D. S. Henningson and M. Berggren, Fluid Dynamics: Theory and Computation,

Stockholm, 2005.

[41] "Setting Up the CFD Simulation in ANSYS Fluent," ANSYS INC, 2017. [Online].

Available: https://www.sharcnet.ca/Software/Ansys/15.0.7/en-us/help/flu_tg/x1-

47200028.9.html. [Accessed 19 02 2019].

[42] J. Smagorinsky, "General Circulation Experiments with the Primitive Equations,"

Monthly Weather Review, p. 99–164, March 1963.

[43] J. P. Den Hartog, Mechanical Vibrations, Massachusetts: Dover Publications,

1985.

[44] B. Dodson, P. C. Hammett and R. Klerx , Probabilistic Design for Optimization

and Robustness for Engineers, John Wiley & Sons, Ltd, 2014.

[45] ANSYS, "4.6. Transition SST Model," ANSYS, [Online]. Available:

https://www.sharcnet.ca/Software/Ansys/17.0/en-

us/help/flu_th/flu_th_sec_turb_transition.html. [Accessed 01 06 2019].

64

APPENDIX A: GANTT CHART

65

APPENDIX B: RISK ANALYSIS

66

APPENDIX C: ISHIKAWA DIAGRAM

67 APPENDIX D: CROSS SECTION GEOMETRIES

72

APPENDIX E: STEADY CFD SIMULATION

Drag force DF at inflow velocity 20U m/s and 0

Drag coefficient DC at inflow velocity 20U m/s and 0

xa

yb 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.12 186.06 169.02 152.21 135.03 119.30 106.17 96.99 93.67 95.58

0.14 194.17 178.06 162.47 147.11 132.83 119.89 109.80 104.35 102.93

0.16 203.20 187.15 172.07 157.96 144.83 132.10 121.49 114.70 111.15

0.18 213.15 196.27 181.00 167.59 155.30 142.81 132.06 124.73 120.25

0.20 224.18 205.35 189.00 175.58 163.74 151.56 141.20 134.35 130.46

0.22 236.13 214.48 196.34 182.35 170.65 158.80 149.21 143.65 141.55

0.24 248.54 223.85 203.79 189.15 177.54 165.90 157.02 152.87 152.78

0.26 261.24 233.56 211.61 196.40 184.91 173.32 164.94 162.06 163.92

0.28 274.40 243.52 219.53 203.69 192.26 180.60 172.67 171.17 175.21

0.30 288.01 253.73 227.57 211.01 199.58 187.73 180.19 180.19 186.65

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.12 0.0091 0.0083 0.0075 0.0066 0.0058 0.0052 0.0048 0.0046 0.0047

0.14 0.0095 0.0087 0.0080 0.0072 0.0065 0.0059 0.0054 0.0051 0.0050

0.16 0.0100 0.0092 0.0084 0.0077 0.0071 0.0065 0.0060 0.0056 0.0054

0.18 0.0104 0.0096 0.0089 0.0082 0.0076 0.0070 0.0065 0.0061 0.0059

0.20 0.0110 0.0101 0.0093 0.0086 0.0080 0.0074 0.0069 0.0066 0.0064

0.22 0.0116 0.0105 0.0096 0.0089 0.0084 0.0078 0.0073 0.0070 0.0069

0.24 0.0122 0.0110 0.0100 0.0093 0.0087 0.0081 0.0077 0.0075 0.0075

0.26 0.0128 0.0114 0.0104 0.0096 0.0091 0.0085 0.0081 0.0079 0.0080

0.28 0.0135 0.0119 0.0108 0.0100 0.0094 0.0089 0.0085 0.0084 0.0086

0.30 0.0141 0.0124 0.0112 0.0103 0.0098 0.0092 0.0088 0.0088 0.0091

73

Drag force DF [N] at inflow velocity 20U m/s and 2 Lift force

LF [N] at inflow velocity 20U m/s and 2

Drag coefficient DC at inflow velocity 20U m/s and 2 Lift coefficient

LC at inflow velocity 20U m/s and 2

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.12 184.14 165.22 149.99 139.45 130.59 118.66 109.89 109.49 115.73

0.14 193.62 177.94 163.79 151.31 140.03 128.61 119.88 116.55 117.72

0.16 203.79 189.51 175.58 161.66 148.80 137.73 129.06 123.79 121.58

0.18 214.67 199.92 185.37 170.52 156.90 146.02 137.44 131.21 127.30

0.20 226.38 208.39 191.96 176.96 163.84 153.00 144.52 138.61 135.21

0.22 238.80 215.70 196.55 181.90 170.11 159.16 150.80 146.20 144.98

0.24 251.51 224.22 202.73 188.10 177.14 165.99 157.77 154.55 155.65

0.26 264.39 234.74 211.69 196.48 185.42 173.98 165.94 163.86 166.90

0.28 277.56 246.46 222.24 206.13 194.48 182.64 174.81 173.93 179.04

0.30 291.03 259.38 234.38 217.05 204.30 191.97 184.37 184.77 192.08

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.12 0.2222 0.2129 0.2059 0.2018 0.1986 0.1939 0.1898 0.1878 0.1874

0.14 0.2240 0.2186 0.2131 0.2074 0.2018 0.1967 0.1921 0.1879 0.1843

0.16 0.2255 0.2233 0.2191 0.2124 0.2050 0.1995 0.1940 0.1873 0.1798

0.18 0.2269 0.2270 0.2239 0.2166 0.2082 0.2024 0.1958 0.1861 0.1740

0.20 0.2281 0.2294 0.2271 0.2203 0.2118 0.2059 0.1979 0.1844 0.1666

0.22 0.2292 0.2308 0.2290 0.2232 0.2155 0.2094 0.1998 0.1821 0.1578

0.24 0.2302 0.2318 0.2303 0.2252 0.2179 0.2112 0.1997 0.1782 0.1484

0.26 0.2312 0.2327 0.2311 0.2262 0.2186 0.2106 0.1969 0.1724 0.1387

0.28 0.2321 0.2332 0.2313 0.2262 0.2181 0.2083 0.1922 0.1650 0.1283

0.30 0.2330 0.2334 0.2307 0.2253 0.2163 0.2042 0.1854 0.1560 0.1172

xa

yb 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.12 0.0090 0.0081 0.0074 0.0068 0.0064 0.0058 0.0054 0.0054 0.0057

0.14 0.0095 0.0087 0.0080 0.0074 0.0069 0.0063 0.0059 0.0057 0.0058

0.16 0.0100 0.0093 0.0086 0.0079 0.0073 0.0068 0.0063 0.0061 0.0060

0.18 0.0105 0.0098 0.0091 0.0084 0.0077 0.0072 0.0067 0.0064 0.0062

0.20 0.0111 0.0102 0.0094 0.0087 0.0080 0.0075 0.0071 0.0068 0.0066

0.22 0.0117 0.0106 0.0096 0.0089 0.0083 0.0078 0.0074 0.0072 0.0071

0.24 0.0123 0.0110 0.0099 0.0092 0.0087 0.0081 0.0077 0.0076 0.0076

0.26 0.0130 0.0115 0.0104 0.0096 0.0091 0.0085 0.0081 0.0080 0.0082

0.28 0.0136 0.0121 0.0109 0.0101 0.0095 0.0090 0.0086 0.0085 0.0088

0.30 0.0143 0.0127 0.0115 0.0106 0.0100 0.0094 0.0090 0.0091 0.0094

xa

yb 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.12 0.2222 0.2129 0.2059 0.2018 0.1986 0.1939 0.1898 0.1878 0.1874

0.14 0.2240 0.2186 0.2131 0.2074 0.2018 0.1967 0.1921 0.1879 0.1843

0.16 0.2255 0.2233 0.2191 0.2124 0.2050 0.1995 0.1940 0.1873 0.1798

0.18 0.2269 0.2270 0.2239 0.2166 0.2082 0.2024 0.1958 0.1861 0.1740

0.20 0.2281 0.2294 0.2271 0.2203 0.2118 0.2059 0.1979 0.1844 0.1666

0.22 0.2292 0.2308 0.2290 0.2232 0.2155 0.2094 0.1998 0.1821 0.1578

0.24 0.2302 0.2318 0.2303 0.2252 0.2179 0.2112 0.1997 0.1782 0.1484

0.26 0.2312 0.2327 0.2311 0.2262 0.2186 0.2106 0.1969 0.1724 0.1387

0.28 0.2321 0.2332 0.2313 0.2262 0.2181 0.2083 0.1922 0.1650 0.1283

0.30 0.2330 0.2334 0.2307 0.2253 0.2163 0.2042 0.1854 0.1560 0.1172

74

Drag coefficient DC at varying inflow velocity U

and 0 for 0.4xa

U

yb

5 10 15 20 25

0.12 0.0096 0.0083 0.0075 0.0068 0.0067

0.14 0.0100 0.0086 0.0078 0.0074 0.0070

0.16 0.0105 0.0090 0.0081 0.0079 0.0073

0.18 0.0110 0.0094 0.0085 0.0084 0.0077

0.20 0.0115 0.0098 0.0089 0.0087 0.0080

0.22 0.0120 0.0102 0.0093 0.0089 0.0084

0.24 0.0127 0.0107 0.0097 0.0092 0.0087

0.26 0.0134 0.0113 0.0102 0.0096 0.0092

0.28 0.0145 0.0119 0.0101 0.0101 0.0096

0.30 0.0161 0.0125 0.0109 0.0106 0.0102

75

APPENDIX F: MODAL FEM RESULTS

First mode frequency 1, [ ]vacf Hz in vacuum as a function of shape control parameters &x ya b .

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.12 57.85 58.74 59.57 60.35 61.07 61.75 62.39 62.99 63.56 64.09

0.14 64.90 66.01 67.04 68.00 68.89 69.71 70.49 71.21 71.89 72.52

0.16 72.05 73.38 74.61 75.74 76.78 77.75 78.65 79.49 80.27 81.01

0.18 79.26 80.81 82.23 83.53 84.73 85.83 86.86 87.81 88.69 89.52

0.20 86.53 88.29 89.90 91.37 92.71 93.95 95.09 96.15 97.14 98.05

0.22 93.84 95.81 97.60 99.23 100.72 102.09 103.35 104.51 105.59 106.59

0.24 101.18 103.36 105.33 107.12 108.76 110.26 111.63 112.90 114.07 115.15

0.26 108.55 110.93 113.09 115.03 116.81 118.42 119.91 121.27 122.53 123.70

0.28 115.92 118.51 120.84 122.94 124.85 126.59 128.19 129.65 131.00 132.25

0.30 123.31 126.10 128.61 130.86 132.90 134.76 136.46 138.02 139.46 140.79

Second mode frequency 2, [ ]vacf Hz in vacuum as a function of shape control parameters &x ya b .

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.12 158.63 161.16 163.51 165.72 167.77 169.69 171.49 173.18 174.76 176.24

0.14 177.86 181.01 183.93 186.63 189.15 191.48 193.66 195.69 197.58 199.34

0.16 197.34 201.11 204.57 207.77 210.72 213.45 215.98 218.33 220.52 222.55

0.18 216.99 221.36 225.37 229.04 232.41 235.53 238.40 241.07 243.54 245.83

0.20 236.76 241.73 246.26 250.40 254.18 257.67 260.87 263.84 266.58 269.11

0.22 256.62 262.18 267.22 271.81 276.00 279.84 283.37 286.62 289.62 292.39

0.24 276.53 282.67 288.21 293.24 297.85 302.04 305.89 309.42 312.67 315.68

0.26 296.51 303.21 309.24 314.71 319.68 324.21 328.36 332.16 335.67 338.90

0.28 316.46 323.72 330.25 336.13 341.48 346.34 350.79 354.86 358.61 362.05

0.30 336.41 344.22 351.23 357.54 363.24 368.43 373.17 377.51 381.49 385.15

76

First mode frequency 1, [ ]fluidf Hz in water as a function of shape control parameters &x ya b .

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.12 27.71 28.62 29.47 30.27 31.04 31.75 32.42 33.04 33.59 34.05

0.14 32.10 33.27 34.36 35.39 36.36 37.28 38.13 38.91 39.62 40.21

0.16 36.68 38.11 39.46 40.72 41.91 43.03 44.07 45.03 45.88 46.62

0.18 41.44 43.14 44.75 46.26 47.67 48.99 50.23 51.36 52.36 53.20

0.20 46.34 48.33 50.20 51.94 53.59 55.13 56.54 57.84 59.00 59.96

0.22 51.38 53.66 55.80 57.81 59.66 61.40 63.02 64.48 65.79 66.87

0.24 56.54 59.10 61.52 63.78 65.86 67.81 69.61 71.26 72.71 73.90

0.26 61.80 64.66 67.35 69.86 72.18 74.34 76.34 78.15 79.75 81.06

0.28 67.17 70.33 73.31 76.06 78.61 81.00 83.18 85.15 86.88 88.31

0.30 72.63 76.10 79.33 82.33 85.13 87.72 90.10 92.24 94.11 95.65

Second mode frequency 2, [ ]fluidf Hz in water as a function of shape control parameters &x ya b .

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.12 158.63 161.16 163.51 165.72 167.77 169.69 171.49 173.18 174.76 176.24

0.14 177.86 181.01 183.93 186.63 189.15 191.48 193.66 195.69 197.58 199.34

0.16 197.34 201.11 204.57 207.77 210.72 213.45 215.98 218.33 220.52 222.55

0.18 216.99 221.36 225.37 229.04 232.41 235.53 238.40 241.07 243.54 245.83

0.20 236.76 241.73 246.26 250.40 254.18 257.67 260.87 263.84 266.58 269.11

0.22 256.62 262.18 267.22 271.81 276.00 279.84 283.37 286.62 289.62 292.39

0.24 276.53 282.67 288.21 293.24 297.85 302.04 305.89 309.42 312.67 315.68

0.26 296.51 303.21 309.24 314.71 319.68 324.21 328.36 332.16 335.67 338.90

0.28 316.46 323.72 330.25 336.13 341.48 346.34 350.79 354.86 358.61 362.05

0.30 336.41 344.22 351.23 357.54 363.24 368.43 373.17 377.51 381.49 385.15

77

Frequency reduction ratio1 for the first mode frequency as a function of shape control parameters &x ya b .

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.12 0.4790 0.4872 0.4946 0.5016 0.5082 0.5141 0.5196 0.5244 0.5284 0.5313

0.14 0.4946 0.5040 0.5125 0.5205 0.5278 0.5347 0.5410 0.5465 0.5511 0.5544

0.16 0.5091 0.5194 0.5289 0.5377 0.5459 0.5535 0.5603 0.5664 0.5715 0.5755

0.18 0.5228 0.5339 0.5442 0.5538 0.5627 0.5708 0.5783 0.5849 0.5903 0.5943

0.20 0.5356 0.5473 0.5584 0.5685 0.5780 0.5868 0.5946 0.6015 0.6074 0.6116

0.22 0.5476 0.5600 0.5717 0.5826 0.5923 0.6014 0.6098 0.6170 0.6230 0.6274

0.24 0.5588 0.5718 0.5840 0.5954 0.6055 0.6150 0.6236 0.6312 0.6375 0.6417

0.26 0.5694 0.5829 0.5955 0.6073 0.6179 0.6278 0.6367 0.6445 0.6509 0.6553

0.28 0.5794 0.5934 0.6066 0.6187 0.6297 0.6398 0.6489 0.6568 0.6632 0.6678

0.30 0.5890 0.6035 0.6169 0.6292 0.6406 0.6509 0.6602 0.6683 0.6749 0.6794

Frequency reduction ratio2 for the second mode frequency as a function of shape control parameters &x ya b .

xa

yb

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.12 0.4871 0.4958 0.5037 0.5111 0.5181 0.5244 0.5302 0.5353 0.5395 0.5426

0.14 0.5027 0.5125 0.5215 0.5300 0.5378 0.5451 0.5517 0.5575 0.5624 0.5659

0.16 0.5172 0.5279 0.5379 0.5472 0.5558 0.5639 0.5711 0.5776 0.5830 0.5873

0.18 0.5307 0.5424 0.5532 0.5634 0.5727 0.5813 0.5892 0.5961 0.6019 0.6061

0.20 0.5435 0.5558 0.5674 0.5781 0.5881 0.5973 0.6056 0.6129 0.6191 0.6236

0.22 0.5555 0.5685 0.5808 0.5922 0.6025 0.6121 0.6208 0.6285 0.6348 0.6395

0.24 0.5668 0.5803 0.5932 0.6051 0.6157 0.6257 0.6348 0.6428 0.6493 0.6540

0.26 0.5774 0.5914 0.6047 0.6170 0.6282 0.6386 0.6479 0.6561 0.6628 0.6676

0.28 0.5875 0.6021 0.6159 0.6285 0.6400 0.6507 0.6602 0.6685 0.6753 0.6802

0.30 0.5971 0.6122 0.6262 0.6391 0.6510 0.6619 0.6716 0.6801 0.6871 0.6919

analysis of the intake grill for marine jet propulsion

Documents