Study of simpli ed simulation models for slamming wave ...lib.ugent.be/fulltxt/RUG01/001/418/544/RUG01-001418544_2010_0001... · Slamming, Fluid-Structure Interaction (FSI), Numerical,

Study of simplified simulation models forslamming wave impact of floating/sailingcomposite structures

Arnaud Schoenmakers

Promotoren: prof. dr. ir. Wim Van Paepegem en prof. dr. ir. Jan VierendeelsBegeleiders: ir. Kameswara Sridhar Vepa en dr. ir. Ives De Baere

Masterproef ingediend tot het behalen van de academische graad vanMaster in de ingenieurswetenschappen: Werktuigkunde-Elektrotechniek

Vakgroep Toegepaste materiaalwetenschappenVoorzitter: prof. dr. ir. Joris Degrieck

Vakgroep Mechanica van Warmte, Stroming en VerbrandingVoorzitter: prof. dr. ir. Roger Sierens

Faculteit IngenieurswetenschappenAcademiejaar 2009–2010

Study of simplified simulation models forslamming wave impact of floating/sailingcomposite structures

Arnaud Schoenmakers

Promotoren: prof. dr. ir. Wim Van Paepegem en prof. dr. ir. Jan VierendeelsBegeleiders: ir. Kameswara Sridhar Vepa en dr. ir. Ives De Baere

Masterproef ingediend tot het behalen van de academische graad vanMaster in de ingenieurswetenschappen: Werktuigkunde-Elektrotechniek

Vakgroep Toegepaste materiaalwetenschappenVoorzitter: prof. dr. ir. Joris Degrieck

Vakgroep Mechanica van Warmte, Stroming en VerbrandingVoorzitter: prof. dr. ir. Roger Sierens

Faculteit IngenieurswetenschappenAcademiejaar 2009–2010

Permission for use of content

The author gives the permission to use this thesis for consultation and to copy parts of it

for personal use. Every other use is subject to copyright law, more specifically the source

must be extensively specified when using from this thesis.

Ghent, june 2010

Arnaud Schoenmakers

Toelating tot bruikleen

De auteur geeft de toelating deze scriptie voor consultatie beschikbaar te stellen en delen

van de scriptie te kopieren voor persoonlijk gebruik. Elk ander gebruik valt onder de

beperkingen van het auteursrecht, in het bijzonder met betrekking tot de verplichting de

bron uitdrukkelijk te vermelden bij het aanhalen van resultaten uit deze scriptie.

Gent, juni 2010

Arnaud Schoenmakers

Acknowledgements

This thesis could not be created without the help of several people. Therefore, I would like

to take the opportunity to thank them here for the help they have given me.

First, I would like to thank my promotors, Prof. dr. ir. Wim Van Paepegem and Prof. dr.

ir. Jan Vierendeels for giving me the opportunity to work on an interesting and challenging

subject.

Furthermore, I’m especially grateful for the help Sridhar Vepa and Ives De Baere have

given me regarding the numerical modelling and working with Abaqus. They always made

time to sit down and discuss whenever I had a problem. Their suggestions and critical

notes have helped me to solve numerous issues.

I also want to say thanks to Diederik Van Nuffel, for giving me a lot of information about

composite materials.

Finally, I would like to thank my girlfriend Charlotte, my brother Remi and my parents

for the support they have given me. I could not have done this without them.

May 2010

Arnaud Schoenmakers

Study of simplified simulation models forslamming wave impact of floating/sailing

composite structuresby

Arnaud Schoenmakers

Scriptie ingediend tot het behalen van de academische graad van

Master in de ingenieurswetenschappen: werktuigkunde-elektrotechniek

Promotoren: prof. dr. ir. Wim Van Paepegem en prof. dr. ir. Jan Vierendeels

Scriptiebegeleiders: ir. Sridhar Kameswara Vepa en dr. ir. Ives De Baere

Vakgroep Toegepaste Materiaalwetenschappen

Voorzitter: prof. dr. ir. Joris Degrieck

Vakgroep Mechanica van Stroming, Warmte en Verbranding

Voorzitter: prof. dr. ir. Roger Sierens

Faculteit Ingenieurswetenschappen

Universiteit Gent

Academiejaar 2009-2010

Abstract

In this thesis, the hydrodynamic impact of objects on a flat water surface is studied

using the finite-element software Abaqus™. The algorithm used to do so is a Coupled

Euler-Langrange method, in which the impacting object is modelled using a langrangian

mesh which follows the deformation of the structure, and the fluid in a eulerian mesh,

which is fixed in space and time. First, calculations on rigid bodies were performed.

Next, deformation of the structure was taken into consideration, for both steel and a

composite materials. This is a more realistic approach of a slamming event, because the

pressure will deform the structure, which in return will influence the pressure; this is called

Fluid-Structure Interaction (FSI). The deformation resulted in lower maximum pressures

than for rigid bodies. In general, however, the calculations overestimate the occuring

maximum pressure, and large negative pressures occur, which is not expected physically.

Implementation of air results in lower pressure, but this is caused by an overestimation

of the drag force acting on the object. Moreover, the resulting maximum pressure is still

higher than values found in literature.

Keywords

Slamming, Fluid-Structure Interaction (FSI), Numerical, Coupled Euler-Lagrange (CEL)

Study of simplified simulation models for slammingwave impact of floating/sailing composite structures

Arnaud Schoenmakers

Supervisor(s): Wim Van Paepegem, Jan Vierendeels

Abstract— In this article, calculations to estimate the hy-drodynamic pressures that arise when a structure hits a wa-ter surface are performed using the finite-element softwareAbaqus™. This phenomenon is called hydrodynamic impactor slamming, and is characterized by a high pressure loadon the structure, that is very short in time. Simulations wererun both for rigid and deformable objects, and the resultsare compared with data found in international literature. Ingeneral, the resulting maximum pressure was found to beoverestimated.

Keywords—Slamming, Fluid-Structure Interaction (FSI),Numerical, Coupled Euler-Lagrange (CEL)

I. INTRODUCTION

THE origin of this thesis lies in the contribution ofGhent University to the SEEWEC-project, a Euro-

pean research project to develop a new kind of wave en-ergy convertor (WEC). SEEWEC stands for SustainableEconomically Efficient Wave Energy Converter, and themost important objective of the project is to be able to gen-erate energy from ocean waves at a price that can competewith classical methods for energy generation.

The structure to do so resembles a large floating rig,beneath which large buoys, called point absorbers, moveup and down when a wave passes. The resulting verticalmovement is then transformed into electrical energy. TheUniversity was responsable for the design of the point ab-sorbers, which are made of lightweight composite materi-als.

When a passing wave hits a point absorber, or when itenters a water surface, it is exposed to a high hydrody-namic load. This phenomenon is called slamming, andis characterized by a large pressure that acts on the struc-ture, but lasts for only a very short period in time. Re-sistance to slamming loads is one of the major criteria forthe design of the point absorbers, because it determinesthe lifespan of the floating rig as a whole. For this rea-son, it is very important to be able to correctly estimatethe impact pressures on the point absorbers. The diffi-culty to do so lies in the fact that this is a highly coupledphenomenon, because the pressure deforms the structure,which in return influences the hydrodynamic load, and is

referred to as Fluid-Structure Interaction (FSI). In this ar-ticle, FSI calculations are performed using the commercialavailable finite-element software Abaqus™.

II. IMPLEMENTATION OF FLUID-STRUCTURE

INTERACTIONS IN ABAQUS™

The algorithm used to couple both domains is calledthe Coupled Euler-Lagrange method (CEL), in which thestructural domain is modelled using a langrangian meshthat follows the deformation of the structure, and the fluidusing an eulerian mesh, which is fixed in space and time.The fluid flows through the eulerian mesh, and the free sur-face of the fluid is reconstructed using a Volume of Fluidmethod (VOF). A general contact with a rough contact in-teraction is applied to the whole model, meaning that oncethe contact is made, it remains closed. Buoyancy is ac-counted for using this contact formulation. Also, gravity isimplemented for the complete domain. Furthermore, wa-ter and air are modelled based on an equation of state, andare both considered incompressible. Because interactionbetween different eulerian domains is not possible, bothair and water must be implemented in the same domain.

III. CALCULATIONS ON RIGID BODIES

To assess the performance of Abaqus™, calculations ongeometric shapes were performed, to be able to comparethem with results found in international literature.

A rigid body constraint is applied to the impacting ob-jects in order to prevent deformation. The results for awedge were compared with an analytical formula for themaximum pressure (Equation 1), proposed by Zhao andFaltinsen [1].

pmax =1

2ρV 2 π2

4tan(α)2(1)

Where ρ is the density of the water, V is the impact ve-locity and α is the angle between the wedge an the wa-ter surface, called the deadrise angle. For α = 10° andV = 5.2m/s, a maximum pressure of 10.9 bar was ob-tained, which is higher than 10.7 bar obtained by usingEquation 1, but the difference is small.

The results for a cone were compared with experimen-tal and numerical results by Peseux [2]. Again using animpact velocity V = 5.2m/s, a pmax of 10.30 bar wasreached, while Peseux obtained pmax = 5.5 bar in exper-iments and pmax = 7 bar from numerical simulations, sothe values obtained with Abaqus™ are very conservative.

Calculations for a rigid cylinder and sphere were com-pared with numerical results by Battistin and Iafrati [3].Their results are presented using a pressure coefficientCp = p/(1/2ρV 2) in function of the dimensionless arclength y/R, with R the radius of the sphere or cylinder.The time after impact is made dimensionless using z/R,where z denotes the submersion of the rigid body. Figure1 shows the results of the calculations for a cylinder and asphere with a radius of 150 mm. The shape of the curvecorresponds quite well with their results, but the values ofCp are very high: for a cylinder, Cpmax = 168.74 and fora sphere Cpmax = 240.0, while Battistin and Iafrati ob-tained respectively Cpmax = 107 and Cpmax = 78. Also,the resulting pressures are higher than experimentaly ob-tained results by Lin and Shieh [4]. Moreover, in all simu-

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4−20

0

20

40

60

80

100

120

140

160

180

y/R

Cp

cylindersphere

Fig. 1. Dimensionless pressure for sphere and cylinder. z=0.01

lations there was a very high level of numerical noise, andthe pressure in a single element reached extremely highvalues in several timeframes. Furthermore, a shockwave,with negative pressures which had the same order of mag-nitude as the positive pressure peak, was propagated in thewater when the object hit the free surface.

IV. CALCULATIONS ON DEFORMABLE BODIES

When deformation was taken into account, the resultingmaximum pressures were lower than for the calculationsperformed on rigid bodies, and the part of the object thathits te water was de-accelerated faster. For the wedge, theresulting maximum pressure was 8.64 bar. The results fora deformable cone were compared with experimental annumerical results by Peseux [2], and the pressures were

also found to be higher.Because the point absorbers are made out of glass fibre

reinforced composite, slamming simulations on a cylinderof the same material were performed, and compared withresults for the same cylinder made of steel. The defor-mation of the composite cylinder was higher than for thesteel cylinder, due to its lower stiffness. Figure 2 showsthe pressure for both a steel and a composite cylinder, fora dimensionless submersion z/R = 0.01. The resultingpressures are still much higher than results found in [3]and [4].

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−2

0

2

4

6

8

10

y/R

Pres

sure

[bar

]

SteelComposite

Fig. 2. Pressure for steel and composite cylinder. z=0.01

The Tsai-Wu failure criterion was applied to see if thecomposite would fail under impact. From these results itcould be concluded that the stresses are too low to causeany damage, altough there is a severe deformation of thecylinder.

V. CONCLUSION

The performance of Abaqus™ to simulate fluid-structure interactions has been investigated. The defor-mation of the impacting predicts lower pressures than forrigid bodies, but the values of the maximum pressure arestill higher than values found in literature.

REFERENCES

[1] R. Zhao and O.M. Faltinsen Water entry of two-dimensional bod-ies, Journal of Fluid Mechanics, 1993.

[2] B. Peseux, L. Gornet and B. Donguy Hydrodynamic impact: Nu-merical and experimental investigations, Journal of Fluids andStructures, 2005.

[3] D. Battistin and A. Iafrati, Hydrodynamic loads during water entryof two-dimensional and axisymmetric bodies, Journal of Fluids andStructures, 2003.

[4] M.-C. Lin and L.-D. Shieh Flow visualization and pressure charac-teristics of a cylinder for water impact, Applied Ocean Research,1997.

Studie naar eenvoudigere rekenmodellenvoor de slamming van golven op

drijvende/varende composiet constructies

Arnaud Schoenmakers

Nederlandstalige samenvatting

1. Situering

Deze thesis vindt zijn oorsprong in de medewerking van de Universiteit Gent aan het

SEEWEC-project, een Europees research project met als bedoeling elektriciteit te gener-

eren uit golven op zee. Belangrijk hierbij is dat de kostprijs van deze elektriciteit moet

kunnen concurreren met de klassieke methodes van energie-opwekking. Het voorgestelde

concept lijkt op een drijvend olieplatform, waar aan de onderkant meerdere cilindervormige

boeien aan bevestigd zijn, point absorbers genaamd. Deze bewegen op en neer wanneer

een golf voorbijkomt. Daarna wordt deze beweging omgezet in elektriciteit via een hy-

draulische motor die een elektrische generator aandrijft.

De Universiteit was verantwoordelijk voor het ontwerp en produceren op grote schaal

van de point absorbers, die gemaakt zijn uit composietmaterialen. Een van de belangrijk-

ste ontwerpparameters hierbij was het bestand zijn tegen slamming-krachten. Slamming

treedt op wanneer een voorwerp zich losmaakt van het wateroppervlak, waarna het terug

op het wateroppervlak valt, of wanneer het geraakt wordt door een grote golf. Hierbij

treedt er een grote hydrodynamische druk op op het lichaam, die echter slecht zeer kort is

in duur. Bovendien zorgt deze hydrodynamische belasting ervoor dat de constructie ver-

vormt, wat op zijn beurt de hydrodynamische druk beınvloedt. Dit gekoppeld probleem

wordt omschreven als Fluid-Structure Interaction (FSI). In deze thesis worden slamming

simulaties uitgevoerd met behulp van het commercieel beschikbare eindige elementen-

pakket Abaqus™.

2. Implementatie van FSI-berekeningen in Abaqus™

Het algoritme dat gebruikt wordt om beide deelproblemen - de stroming van het water

en de vervorming van de structuur- te koppelen heet de Coupled Euler-Langrange methode

(CEL). Dit houdt in dat het structureel gedeelte beschreven wordt met een langrange mesh,

vii

die de vervorming van de structuur volgt, en het fluıdum gedeelte met een euler mesh,

waarvan de knopen op dezelfde plaats blijven gedurende de simulatie. Het fluıdum stroomt

als het ware door de mesh, en het vrije vloeistofoppervlak wordt gereconstrueerd met

behulp van de Volume Of Fluid (VOF) methode. De contactformulering voor het volledige

model is ruw, wat wil zeggen dat eens er contact is tussen het fluıdum en de structuur,

het contact gesloten moet blijven. Door deze formulering te gebruiken, is er rekening

gehouden met de opwaartse Archimedeskracht. Zwaartekracht is geımplementeerd voor

het volledige model.

De fluıda worden gemodelleerd op basis van een Equation of State (EOS), en worden

beide als onsamendrukbaar beschouwd. Aangezien de optredende fluıdumsnelheden aan

de lage kant zijn bij het simuleren van slamming, is dit aanvaardbaar. Interactie tussen

verschillende euler-domeinen is niet mogelijk. Als lucht geımplementeerd wordt, moet dit

dus in hetzelfde domein gebeuren als voor het water.

3. Berekeningen met starre lichamen

Om de performantie van Abaqus™ na te gaan worden berekeningen uitgevoerd op voor-

werpen waarover informatie terug te vinden is in de internationale literatuur, zoals een

wig, een conus of een cilinder. Op deze manier kunnen de resultaten vergeleken worden

met de waarden die daar gevonden worden.

Om er voor te zorgen dat het lichaam niet vervormt tijdens de analyse, wordt een

rigid body constraint toegepast op het structurele gedeelte. Belangrijk hierbij is wel dat

randvoorwaarden, die voordien geldig waren op heel het object, of op bepaalde delen van

het object, nu moeten toegepast worden op een referentiepunt, dat een gekozen vast punt is

op het structurele gedeelte. Voor een bespreking van de randvoorwaarden wordt verwezen

naar Paragraaf 4.2.1 in de thesis. Figuur 1 geeft een overzicht van het rekendomein.

viii

Figure 1: Overzicht van rekendomein voor wig

De resultaten voor een wig werden vergeleken met een analytische uitdrukking voor de

maximum optredende druk, voorgesteld door Zhao en Faltinsen [1] (Vergelijking 1).

pmax =1

2ρV 2 π2

4tan(α)2(1)

Met ρ de dichtheid van het water, V de intredesnelheid en α de wighoek. De bekomen

resultaten voor intredesnelheid V = 5.2, m/s worden vergeleken met Vergelijking 1 in

Tabel 4.1 voor verschillende waarden van de wighoek α. De berekende drukken zijn eerder

aan de lage kant, en komen vrij goed overeen met de berekende waarden met Vergelijking

1. De afwijking wordt groter als de wighoek kleiner wordt. Wel moet opgemerkt worden

dat soms, door numerieke instabiliteiten, de waarde voor de druk in een enkel element vele

malen hoger kan liggen dan de getabelleerde waarden.

Table 1: Vergelijking tussen de berekende maximum druk en Vergelijking 1

α pmax calculated [bar] pmax Zhao [bar]

6 20.96 30.20

10 8.76 10.73

14 4.91 5.73

De resultaten voor de conus worden vergeleken met experimentele en numerieke re-

sultaten voorgesteld door Peseux et al. [2]. Opnieuw wordt een intredesnelheid van 5.2

m/s opgelegd, en de waarde van de conushoek α is 10. Een maximale druk van 10.30

bar werd berekend, terwijl Peseux [2] een maximale druk van 5.5 bar behaalde tijdens

ix

experimenten, en een waarde van 7 bar als numeriek resultaat. Er kan dus gesteld worden

dat de drukken berekend met Abaqus™zeer conservatief zijn.

Voor een cilinder en een bol werden de resultaten vergeleken met numerieke resultaten

van Battistin en Iafrati [3]. Hun resultaten zijn gegeven in de vorm van een dimensieloze

drukcoefficient Cp in functie van de dimensieloze booglengte op de cilinder y/R, voor een

bepaalde waarde van de dimensieloze onderdompeling z/R. Hierbij is R de straal van de

cilinder of de bol, en z is de onderdompeling van het structureel gedeelte. Figuur 2 toont

de drukcoefficient voor de cilinder voor zowel z = 0.01 en z = 0.05. De vorm van de

curve komt vrij goed overeen met de vorm in [3], maar de waarde voor Cp is vele malen

hoger. Voor de bol wordt hetzelfde fenomeen gezien, alhoewel de druk iets lager is door

driedimensionele effecten (Figuur 3). Ook wanneer de drukken vergeleken worden met de

experimentele resultaten van Lin en Shieh [4], blijkt dat de berekende drukken heel hoog

zijn.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−20

0

20

40

60

80

100

120

140

160

180

y/R

Cp

z/R = 0.01z/R = 0.05

Figure 2: Dimensieloze druk voor cilinder bij verschillende waarden van z/R

x

Figure 3: Dimensieloze druk voor bol bij z/R = 0.01

Voor al deze voorwerpen werden ook simulaties uitgevoerd met de implementatie

van lucht als een tweede fase. Voor het simuleren van een onvervormbaar object, met

wighoeken groter dan 5 zou dit geen verschil mogen geven voor de resultaten: lucht

kan niet opgesloten worden in concave holtes, en de veroorzaakte weerstandskracht bij de

beschouwde snelheiden kan verwaarloosd worden ten opzichte van de zwaartekracht. De

resulterende drukken zijn echter wat lager, wat als volgt kan verklaard worden: door het

gebruik van een ruw contact, remt het object reeds af wanneer het nog in vrije val is, wat

fysisch gezien niet verwacht wordt. Daardoor is de snelheid bij impact, en dus ook de

maximale druk, lager.

Ook werd in alle simulaties een grote hoeveelheid ruis in de resulterende drukken gezien,

waarbij de druk soms extreem hoge waarden kan aannemen in een enkel element. Voor een

cilinder wordt dit geıllustreerd in Figuur 4, waar ook het optreden van een schokgolf, met

negatieve drukken waarvan de amplitude even groot kan zijn als bij de positieve drukpiek,

getoond wordt. Dit fenomeen wordt niet teruggevonden in de internationale literatuur.

xi

Figure 4: Zeer hoge druk in element door numerieke instabiliteit. Niet-uitgemiddelde waarden.

De weergave van het wateroppervlak bij de impact van een cilinder wordt vergeleken met

experimentele resultaten van Lin en Shieh [4], die een hoge-snelheids camera gebruikten

om de vorming van de jet vast te leggen. In de simulatie met Abaqus™ wordt de vorm

en de richting van de jet vrij goed benaderd, enkel het aantal druppeltjes dat zich vormt

ligt lager. Dit is te wijten aan het feit dat de kleinste van deze druppeltjes niet opgelost

worden in de mesh.

(a) t = 0.0285 − 0.0342s (rising up) (b) t = 0.03s (rising up)

xii

4. Berekeningen met vervormbare lichamen

De optredende drukken zijn lager wanneer de vervorming in rekening wordt gebracht,

en het deel van het object dat op het water slamt ondervindt een grotere deceleratie. Voor

de wig is de maximum optredende druk gelijk aan 8.64 bar. Ook hier worden ietwat lagere

drukken bekomen wanneer lucht geımplementeerd wordt, maar de invloed van lucht is

groter door de vervorming. Aangezien de gebruikte wighoek groot genoeg is om cushioning

(waarbij lucht vastgeraakt in concave holtes onder het voorwerp) te voorkomen, kan hieruit

geconcludeerd worden dat lucht toch enigzins vastgeraakt, wat er op kan wijzen dat lucht

te stijf gemodelleerd is.

Figuur 5 toont de druk in het element waar de eerste impact plaats vindt, in functie van

de tijd en voor verschillende waarden van α. Om de figuur wat duidelijker te maken, werd

aan de curves een zekere offset in de x-richting meegegeven. In Tabel 2 is een vergelijking

gemaakt met experimentele en numerieke resultaten van Peseux [2] voor een vervormbare

conus. Ook hier blijkt dat de berekende drukken hoger zijn dan verwacht.

0 1 2 3 4 5 6 7

x 10−4

−8

−6

−4

−2

0

2

4

6

8

Time [s]

Pressure [bar]

! = 6°! = 10°! = 14°

Figure 5: Druk in element waar eerste impact plaats heeft, voor verschillende waarden van α

xiii

Table 2: Vergelijking van resultaten van Peseux [2] met berekende resultaten voor vervormbare

conus

α Peseux Experimental [bar] Peseux Numerical [bar] Abaqus™[bar]

6 1.4 2.2 6.67

10 1.2 1.3 4.83

14 0.6 0.6 3.66

Omdat de point absorbers uit glasvezel versterkte composiet gemaakt zijn, zijn ook simu-

laties voor een cilinder met hetzelfde materiaal uitgevoerd, en vergeleken met de resultaten

voor dezelfde cilinder gemaakt uit staal (ρ = 7800 kg/m3, E = 210GPa, ν = 0.3). Beide

cilinders hebben een dikte van 3 mm. De vervorming voor zowel een stalen als een com-

posiet cilinder is weergegeven in Figuur 6. Let wel dat hier een schalingsfactor 20 is

gebruikt om de vervorming te kunnen visualiseren. In werkelijkheid blijft het onderste

punt van de cilinders het onderste punt gedurende de analyse, zodat eigenlijk een ellip-

tische vervorming verkregen wordt. De procentuele afwijking van de originele diameter

wordt getoond in Figuur 7. Voor een stalen cilinder is de afwijking op het einde van de

simulatie 1.29%, voor een composiet cilinder 2.05%, wat te verklaren is door de grotere

stijfheid van het staal.

xiv

(a) staal (b) composiet

Figure 6: Vervorming voor stalen en composiet cilinder. t na impact is 1.25 ms

0 0.5 1 1.5 2 2.5

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

DOD [%]

(a) staal

0 0.5 1 1.5 2 2.5

x 10−3

0

0.5

1

1.5

2

2.5

Time [s]

DOD [%]

(b) composiet

Figure 7: Procentuele afwijking van originele diameter

Aangezien de stijfheid van het composiet materiaal lager is, treedt er grotere vervorming

op, wat ook tot een lagere druk leidt. Figuur 8 toont Cp voor een stalen cilinder. De waar-

den van de dimensieloze onderdompeling zijn z/R=0.01 en z/R=0.05. De drukcoefficient

Cp is lager dan voor een starre cilinder, maar is nog steeds veel hoger dan de waarden

gevonden in [3]. Ook komt de vorm van de curve minder overeen door het optreden van

grote schommelingen in de druk.

xv

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−40

−20

0

20

40

60

80

100

120

140

160

y/R

Cp

z/R = 0.01z/R = 0.05

Figure 8: Dimensionless pressure for deformable cylinder

Om te kijken of het composietmateriaal zou falen bij de impact, werd het Tsai-Wu

criterium toegepast. Dit is een schadecriterium dat gebaseerd is op de spanningstoestand

in het materiaal, bepaald door σ11, σ22, σ12. Om niet beschadigd te geraken, mag de

Tsai-Wu waarde IF niet groter worden dan 1. Voor de onderste en bovenste laag is de

waarde van IF weergeven in Figuur 9. Omdat de cilinder buiging ondergaat, zijn dat de

twee meest bepalende lagen. Zoals te zien op de figuur blijft IF overal kleiner dan 1, en de

zones waar de cilinder het water raakt zijn het meest kritiek. De spanningen veroorzaakt

door de vervorming zijn dus echter te klein om schade te veroorzaken.

xvi

(a) onderste laag (b) bovenste laag

Figure 9: Tsai-Wu failure criterion

5. Conclusie

De performantie van Abaqus™ om FSI-berekeningen uit te voeren bij hydrodynamische

impact is onderzocht. De vervorming van de objecten geeft aanleiding tot een lagere druk,

maar de waarden voor de maximale druk zijn toch groter dan de waarden die gevonden

zijn in de internationale literatuur. Op sommige tijdstippen wordt de druk in een bepaald

element van de mesh extreem hoog. Bovendien heeft te implementatie van lucht als een

tweede fase een grotere invloed heeft dan verwacht: het object wordt al afgeremd door de

lucht terwijl het nog in vrije val is. Ook heeft de lucht een grotere invloed bij vervormbare

objecten waarvoor geen cushioning effect verwacht wordt, wat er op kan wijzen dat lucht

te stijf gemodelleerd is.

xvii

Contents

Contents xviii

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The SEEWEC-project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Slamming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Goal of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Structure of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Literature Study 5

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Analytical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Early Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.1 Recent developments . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.2 Fluid-Structure Interactions (FSI) . . . . . . . . . . . . . . . . . . . 14

2.3.3 Volume Of Fluid Method (VOF) . . . . . . . . . . . . . . . . . . . . 18

3 Implementation of a CEL-model in Abaqus™/CAE 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Explicit dynamic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.2 Stable increment time . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.3 Application to CEL . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Material definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Contact interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Predefined fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Boundary conditions and loads . . . . . . . . . . . . . . . . . . . . . . . . . 28

xviii

Contents

3.7 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.7.1 Displacement hourglass scaling factor s . . . . . . . . . . . . . . . . 29

3.7.2 Bulk viscosity scaling factors b1 and b2 . . . . . . . . . . . . . . . . . 29

3.7.3 Mesh size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Calculations for rigid bodies 31

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2 Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Effect of deadrise angle α . . . . . . . . . . . . . . . . . . . . . . . . 40

Influence of the implementation of air . . . . . . . . . . . . . . . . . 43

4.3 Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Comparison with experimental results . . . . . . . . . . . . . . . . . 51

4.4 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Implementation of air . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Comparison with experimental results . . . . . . . . . . . . . . . . . 60

Comparison with numerical model . . . . . . . . . . . . . . . . . . . 62

Formation of the jet . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Calculations on deformable bodies 70

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2 Wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.3 Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

xix

Contents

5.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4.3 Composite cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Modelling of the composite lay-up . . . . . . . . . . . . . . . . . . . 83

5.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Failure of the material . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6 Conclusion 93

Bibliography 95

List of Figures 98

List of Tables 102

xx

Nomenclature

Abbreviations

CEL Coupled Euler-Lagrange

DOD Deviation of Original Diameter

FSI Fluid Structure Interaction

SEEWEC Sustainable Economically Efficient Wave Energy Converter

VOF Volume Of Fluid

WEC Wave Energy Converter

Units

α Deadrise angle

c0 wave speed in material m/s

Cp Pressure coefficient -

E Elastic modulus Pa

G Shear modulus Pa

M Mass kg

µ Dynamic viscosity Pa · s

ν Poisson ratio -

p Pressure Pa

R Radius of cylinder/sphere m

ρ Density kg/m3

s Stress tensor Pa

σ11 Stress in fibre direction Pa

σ22 Stress perpendicular to fibre direction Pa

σ12 Shear stress Pa

v Velocity m/s

V Impact velocity m/s

y Arc length on cylinder/sphere m

z Penetration depth m

Chapter 1

Introduction

1.1 Introduction

In the last few years it has become very clear that the reserves of fossil fuels are about

to run out in the coming decades, whereas demand keeps increasing due to fast growing

economies such as China, India and Brazil. Moreover, ecological concerns arise about

the emission of greenhouse gasses related to the burning of fossil fuels. Furthermore, the

western world, and the European Union in particular, is dependent on the import of fossil

fuels from politically instable regions (54% of the current demand), and this dependence is

expected to increase to 70% in the years to come if policy does not change [5]. The world

we live in today is addicted to fossil fuels. Hence, research in renewable energy resources is

conducted in order to become less dependant on this classic sources of energy. Renewable

energy is energy that can be considered inexhaustable. Classic examples are solar, wind

and geothermal energy.

1.2 The SEEWEC-project

From 2005 until 2009 , the European Union supported a project to develop a wave energy

convertor, called SEEWEC, which stands for Sustainable Economically Efficient Wave

Energy Converter. The goal of this project was to design a construction that could con-

vert the energy comming from ocean waves into electrical energy, at a price that could

compete with classical methods for power generation. The initial idea was to construct

a large floating platform, under which a number of floating cylindrical buoys called point

absorbers, could move up and down under the influence of passing waves. On the plat-

form, the vertical motion of the point absorbers was converted into electrical energy using

a hydraulic engine coupled to a electrical generator. Figure 1.1 shows a rendering of the

1

Chapter 1. Introduction

wave energy convertor (WEC).

Figure 1.1: WEC platform concept

Several partners were chosen to design different aspects of the project, of which Ghent

University was responsable for the composite material design and the large scale manufac-

turing of the point absorbers. One of the most important design criteria was the durability

of the point absorbers, as violent waves can cause severe loads during heavy storms.

1.3 Slamming

The point absorbers must be designed in order not to fail under slamming loads. Slamming

occurs when a structure, for example a ship, is released from the water surface after which

it hits the water surface again. The object is then exposed to a hydrodynamic impact (see

Figure 1.2). The same phenomenon occurs when a structure, for example an oil rig, is hit

by a large wave. Slamming is characterized by a high pressure load that is very short in

time, but can cause severe damage to the structure. Moreover, in several cases, slamming

is identified as the cause of failure of ships and floating structures [2]. Because of this,

slamming research is mainly focused on marine applications.

2


Figure 1.2: Slamming of composite hull during the 2008 Volvo Ocean Race

The pressure distribution on a object that impacts a water surface is characterized by

a large and small pressure peak, as can be seen on Figure 1.3. Point absorbers can be

exposed to the same kind of load when they are released from the water surface and slam

back onto it. This can be caused by a resonance of the vertical movement of the point

absorber with the passing wave, in which the wave amplifies the motion of point absorber,

or when there is a phase shift between the movement of the point absorbers and the wave,

meaning that the point absorber moves down when the water surface rises, or vice versa.

In order to remain operative for several years and thus survive sometimes severe storms,

the magnitude of the occuring pressures must be known precisely to select a material

with the appropriate characteristics. This is rather complex, because the pressure has an

influence on the deformation of the structure, which in return influences the flowfield and

the pressure; this coupled problem is referred to as Fluid-Structure Interaction (FSI).

3


Figure 1.3: Pressure distribution on an impacting wedge [6]

1.4 Goal of the thesis

A numerical approach to solve Fluid-Structure Interactions was presented in a previous

thesis by Koen Stoop and Steve Vermeulen [7]. They simulated slamming events using

a coupling algorithm between Fluent™, which is a fluid solver, and Abaqus™, which is a

structural solver, to simulate slamming events. The pressures calculated with Fluent™ are

implemented on the structure in Abaqus™, which results in a deformation of the structure.

This deformation is then implemented back in Fluent™.

However, since the release of version 6.7 of Abaqus™, it is possible to include fluids in

the simulations, using only Abaqus™. The goal of this thesis is to assess the performance

of Abaqus™ for modelling slamming situations, which is done by comparing the results

from these calculations with results presented in international literature. In this thesis,

the Extended Functionality version 6.9-EF is used.

1.5 Structure of the thesis

First, a history of the research conducted on hydrodynamic impact is presented, and the

most important models, both analytical and numerical, are discussed. Next, the model

parameters of the simulation in Abaqus™ are explained, and the choice for the values

of those parameters is argumented. The results of calculations on rigid objects, such as

wedges and cylinders, are given in Chapter 4. In Chapter 5, the results for calculations

on deformable objects is presented, where both steel and composite materials are used as

materials for the impacting objects. Finally, a conclusion about the obtained results is

given in Chapter 6.

4

Chapter 2

Literature Study

2.1 Introduction

Hydrodynamic impact is a problem which is hard to solve analytically, because the solution

is highly dependent on the shape of the water close to the impacting object. Moreover,

boundary conditions must be applied to the water surface of which the position is not

known. This makes the slamming problem a highly non-linear phenomenon. In this

chapter a brief history of the research in the field of hydrodynamic impact is presented.

Both analytical as well as numerical models are discussed.

2.2 Analytical Models

2.2.1 Early Investigations

One of the first persons to investigate the slamming phenomenon was Theodore Von

Karman [8]. In 1932 he developed an analytical formula based on the conservation of

momentum, in order to estimate the forces on a seaplane when it lands on the water

surface. When an object hits the water it slows down; if momentum is to remain constant,

this means that the mass has to increase: MV = (m + M)v, with V the initial impact

velocity and m the added or virtual mass [9]. Newman [10] states that “The added mass

can be interpreted as a particular volume of fluid particles that are accelerated with the

body.” When differentiating the momentum equation with respect to time, the impact

force and pressure can be calculated. The accuracy of this method depends on the choice

of the value of the added mass. Von Karman assumed the added mass to be one half of

the inertia of a flat plat accelerated in a liquid, because the fluid underneath the plate is

accelerated, but the fluid above is not. With this assumptions, Von Karman obtained an

5

Chapter 2. Literature Study

expression for the maximum pressure at impact pmax (Equation 2.1) which is suitable for

a two-dimensional wedge on a flat water surface with an impact velocity V .

pmax =ρV 2

2πcot(α) (2.1)

α is the angle between the water surface and the wedge, called the deadrise angle (Figure

2.1), and ρ is the density of the water.

Figure 2.1: Definition of the deadrise angle α

The influence of the dynamic pressure due to the velocity, ρV 2/2 and the influence of

the shape, πcot(α), can clearly be distinguished. Furthermore, Von Karman neglects the

buoyancy, friction and the cushioning effect of air. Water elevation is also not considered,

which results in an underestimation of the pressure.

Wagner [11] expanded Von Karmans theory to two-dimensional wedges with small deadrise

angles, including the effect of the water pile-up and the spray thickness. He used an

incompressible potential flow to describe the problem. A lot of correction factors were

later added to Wagner formula’s, to include three-dimensional effects and larger deadrise

angles amongst other things.

The first person to experimentally investigate slamming is Watanabe [9]. In two papers

[12, 13] he describes experimental drop tests of cones with variable deadrise angle and

impact velocity and the resulting impact force. Such experimental results were extremely

important to validate the at that time recently developed analytical models.

2.2.2 Recent developments

Zhao et al. [1] derived an analytical expression for the pressure distribution on a body by

further developing Wagner’s theory, also using an incompressible and inviscid potential

6


flow. It is valid for small deadrise angles higher than 2, so the cushioning effect of air can

be neglected.

An expression for a slamming coefficient Cpmax is proposed:

Cpmax ≡pmax12ρV

2=

π2

4tan(α)2(2.2)

They also show that the global pressure distribution is not much influenced by the local

jet of water, which arises on the side of the impacting object.

Inspired by the work of Zhao [1], an analytical expression for the pressure on general two-

dimensional sections was developed by Mei et al. [14]. They generalize Wagners method

to a various range of body sections, and use the same initial-boundary-formulation as in

the work of Zhao [1] in which the pressure at the free surface is set to zero. Moreover,

the expression can be solved analytically for wedges and cylinders, in contrary to Zhao’s

expression, which can only be solved with numerical aid. Expressions for more general

ship sections are also derived. However, for three-dimensional objects, the expressions are

limited to a number of geometric shapes.

The pressure coefficient Cp(y, t) ≡ P (y, t)/(ρV2

2 ), y ≤ Y (t) is given by:

Cp(y, t) =2γ

A tan(α)

[(1− q2

q2

)θ ∫ 1

|q|

(w2

1− w2

)θdw − q

]−(

1− q2

q2

)2θ

− 2γ+ 1 (2.3)

In this equation:

• q is related to the position of the impacting object:

h(y)

V t= γA−1cos(β)

∫ q

0

(w2

1− w2

)θdw + γ (2.4)

• h(y) is the height on the impacting object

• vt is the submersion of the body, where v is the actual velocity and t is the time

after impact

• A, θ and γ are constants, depending on the deadrise angle.

Figure 2.2 gives an overview of the used definitions.

7


Figure 2.2: Slamming definitions

The results obtained by Mei [14] for wedges with different deadrise angles α are shown

in Figure 2.3, in which the influence of the deadrise angle is clearly visible: a smaller

angle gives a higher and shorter pressure peak, a large angle gives a lower and wider

pressure peak. This can be understood intuitively, because a sharper object cuts more

easily through the water. Furthermore, In Figure 2.3, a comparison is made with the

similarity solution obtained by Dobrovol’Ska [15], represented by the dotted line.

8


Figure 2.3: Solution for a wedge obtained by Mei for several deadrise angeles. Comparison with

the similarity solution obtained by Dobrovol’Skaya.

These expressions are derived based on the conservation of momentum. In 1951, Pierson

[16] noticed that when the concept of added mass is used in an energy-based analysis,

different results are obtained. However, Cointe et al. [17], proved that when the energy

flux going into the water jet is taken into account, the results are the same as when

9


conducting a momentum-based analysis.

2.3 Numerical models

The governing equations for fluid flow are the Navier-Stokes equations: conservation of

mass (Equation 2.5), momentum-equation (Equation 2.6) and energy-equation (Equation

2.7). These equations are not solvable in an analytical way.

∂ρ

∂t+∇ · (ρv) = 0 (2.5)

ρ∂v

∂t+ ρv ·∇(v) = ∇ · s + f (2.6)

ρ∂e

∂t+ ρv ·∇(e) = s : ∇(v) + f ·v (2.7)

The parameters in these equations are as follows:

• f is the external force density.

• v is the velocity vector

• The total Cauchy stress s is given by:

s = −p · I + µ(∇(v) +∇(v)T ) (2.8)

where p is the pressure and µ represents the dynamic viscosity

Together with the initial- and boundary conditions, these equations completely determine

the flow. For the slamming problem, two different regions on the boundary of the domain

are distinguished (Figure 2.4):

• The velocity is prescribed on ∂Ω1f : ·v(t) = g(t)

• The velocity isn’t prescribed, but the traction boundary condition is assumed to be

imposed on ∂Ω2f : · s ·n = h(t)

10


Figure 2.4: Fluid domain

When computers became powerful enough to solve large sets of equations, a lot of nume-

rical slamming models were developed. The Navier-Stokes equations were formulated in

a discrete form and solved numerically.

A lot of these numerical models are based on a mixed Euler-Lagrangian formulation.

Longuet-Higgings and Cokelet [18] first proposed this method in 1979. The flow velocity

potential is solved with Dirichlet boundary conditions on the free surface, and Neumann

boundary conditions on the solid body. The solution gives the velocity along the body

contour and, by deriving in the normal direction, the normal derivative on the free surface,

meaning the velocity components on the free surface are then known. By integrating the

velocity with respect to time, the position of the free surface is known. Applying the

Bernoulli-equation for an unsteady flow (∂v∂t 6= 0) results in the velocity potential along

the free surface. The problem with this method is that, after discretisation, the boundary

conditions on the intersection of the free surface and the solid body do not match, causing

a flow singularity, see Figure 2.5. A very fine jet appears, characterized by a large velocity

gradient. This results in an unreliable, very high pressures at first impact, located in the

spray root.

11


Figure 2.5: Boundary conditions in computational domain.

2.3.1 Recent developments

Battistin and Iafrati [3] developed a numerical model dealing with these issues, applied

to axisymmetric objects. The very fine jet is cut out of the computational domain, once

the distance between the first centroid on the free surface and the rigid body reaches a

fixed treshold value. This still leads to reliable results, because in spite of the high kinetic

energy of the jet, only a small changes of the pressure field occurs in this region[17]. Iafrati

et al. [19] validated that model for axisymmetric wedges. A similar model was suggested

by Zhao and Faltinsen [1].

Figure 2.6: Cut-off jet in computational domain

Peseux et. al [2] developed a model to solve the three-dimensional Wagner problem numer-

ically, both for rigid as well as for deformable structures. The potential flow assumptions

12


are valid: the fluid is inviscid, incompressible and irrotational. The domain is divided into

three different zones, as can be seen in Figure 2.7.

Figure 2.7: Definition of different domains in computational domain

In each of these domains an asymptotic expansion is performed. The first one is the

outer domain. Here, the solution resembles the flow obtained around a flat plate that is

immersed in a fluid, which is referred to as the outer problem. Next, the spray source

domain near the contact line, where the flow overturns to create a jet, is considered: this is

called the inner problem. The last zone, the jet domain, is not considered to influence the

solution and is not taken into account. A variational formulation of the Wagner problem is

developed both for two-dimensional and three-dimensional slamming problems. To solve

these equations, a finite element method was conceived. Moreover, a series of experimental

slamming tests with cones, both deformable and undeformable and with different deadrise

angles, were performed to compare with the obtained numerical results shown in Figure

2.8. Here, the pressure is shown for 4 different timesteps in the analysis, with t1 < t4. The

displacement of the pressure peak along the wedge in time can clearly be seen.

13


Figure 2.8: Comparison between cone and wig with same deadrise angle of 10at 4 different

timesteps. Impact velocity is 6m/s

2.3.2 Fluid-Structure Interactions (FSI)

The first models considered the impacting objects as rigid bodies. In real-life slamming

events, some deformation of the structure occurs. This deformation influences the flow

around the object and also the pressures. In general, the pressures are lower, because

air gets trapped in concave cavities. This is called the cushioning effect, as illustrated in

Figure 2.9.

14


Figure 2.9: Air gets trapped in concave cavities when the impacting object deforms. The air is

compressed and the object slows down.

Equation 2.9 governs the deformation of the structure. It expresses the conservation of

momentum.

ρdv

dt= ∇(s) + f (2.9)

Two boundary conditions are applied (Figure 2.10):

• The displacement boundary condition on ∂Ωs1 : x(X, t) = D(t). Here, x(t) is the

displacement, X represents the reference coordinates X1, X2 and X3.

• The traction boundary condition on ∂Ωs2 : s ·n = τ (t). n is the normal orientated

outward on ∂Ωs2.

Figure 2.10: Structural domain

When the deformation of the structure is also taking into account, contact algorithms

must be developed to calculate the contact forces applied from the fluid to the structure

and vice versa. An explicit method is used to update the nodal forces at the interface

15


each time step to calculate the contact forces. To stay in contact with the structure, fluid

nodes must follow the structure at the interface. Souli and Zolesio [20] and Rabier and

Medale [21] present a coupling algorithm where remeshing is not applied. Only small fluid

motions are allowed in order to prevent mesh distortion, as this would lead to incorrect

results. When modeling fluids with such a mesh, the mesh gets severely distorted due to

the large deformations of the water, as can be seen in Figure 2.11(a). A remeshing method

is needed, which is very CPU-time consuming.

An often-used algorithm that addresses these issues is the Euler-Langrange coupling. One

example of such an algorithm can be found in Aquelet et al.[22], in which the structure

is modeled with a Langrangian mesh. The mesh follows the deformation of the structure.

Because the deformation is relatively small, no severe mesh distortion occurs. The fluid is

modeled in an Eulerian mesh, in which the mesh is fixed in space. The fluid flows through

the mesh, and distortion does not occur.

(a) Lagrangian mesh (b) Eulerian mesh

To model the contact interface, two conditions are applied: an impenetrability condition

(Equation 2.10) and a traction condition (Equation 2.11) to make sure that momentum

is conserved across the interface.

ss ·ns + sf ·nf = 0 (2.10)

d = vs ·ns + vf ·nf = (vs − vf ) ·ns ≤ 0 (2.11)

In this equation:

• s denotes the structural domain, and f denotes the fluid domain

16


• d is the penetration vector

• s is the total Cauchy stress (Equation 2.8)

• n is the normal on the domain boundary

The interaction itself can be modeled in different ways. One surface (either fluid or

structural) is the master surface, the other one is the slave surface, containing the slave

nodes. Kinematic contact, where fluid and structure have te same velocity, makes sure

momentum is conserved, but violates the energy equation. Another possibility is to apply

the penalty based method. A penalty is a resisting force to the slave node, dependent

on its penetration through the master node. To satisfy equilibrium, an opposite force is

applied to the nodes of the master element. In Aquelet [22] an expression for the forces is

proposed:

Fs = −k · d (2.12)

F im = Ni · k · d (2.13)

Ni is a shape function at node i. This can be interpreted as springs with stiffness k placed

between the slave nodes and the contact surface, see Figure 2.11. The stiffness depends

on the bulk modulus K of the master material, the surface area A of the master element

and the volume V of the master element:

k = pfKA

V(2.14)

where pf is scale factor for the interface stiffness. It must satisfy 0 ≤ pf ≤ 1.

17


Figure 2.11: Contact algorithm

2.3.3 Volume Of Fluid Method (VOF)

When using an Euler-Lagrange coupling, the fluid is modeled with a fixed mesh. The

location of the free surface cannot be calculated by looking at the displacement of the

nodes. Therefore, a different approach is needed to determine the free surface; several

methods exist:

In the level-set free surface tracking method, the distance function φ(x, t) is introduced

to know the distance of the surface x from its initial position at t = 0. The interface

corresponds to φ(x, t) = 0. The problem with this model is that mass is not conserved.

The most popular model is the Volume of fluid model (VOF) [23]. The volume fraction F

is the fractional volume of a cell occupied by the fluid, and has a value between zero and

one. Every cell has its own VOF. Cells with F = 0 or F = 1 are called pure cells; when

0 ≤ F ≤ 1, the cell is called a mixed cell. The VOF-method was first introduced by Hirt

and Nichols [24]. It has two important characterizing properties: The way the interface

is reconstructed and the way it is propagated. As explained later on in this section,

several methods exist to reconstruct the interface. Based on the obtained interface and

the velocity fluxes at the different surfaces of each cell, fluid is moved from a donor cell

to an acceptor cell. Only the value of the VOF in surrounding cells is needed to calculate

the VOF in cell. This makes it easy to divide the calculations in parallel processes.

Rider [25] identifies 4 steps to calculate the VOF, as shown in Figure 2.12. In the first step,

18


the volume is divided in descrete parts using a mesh. Next, the free surface is discretised.

Now the material fluxes can be calculated. In the final step, the volumes are integrated

to a new time level.

Figure 2.12: Steps taken to calculate Volume Fraction F

Two ways of reconstructing the surface are much used nowadays. First, there is the Simple

Line Interface Calculation method (SLIC), where the interface is always parallel to one of

the coordinate axes. As can be seen in Figure 2.13(a), this method is not very accurate.

A more sophisticated method is the Piecewise Linear Interface Reconstruction method

(PLIC), as can be seen in Figure 2.13(b). To reconstruct the interface, it is divided by a

19


line in a certain number of discrete partitions equal to the number of phases present in

the cell. The line is a linear approximation of the curved interface. Also, a discontinuity

between the different lines is allowed. This method is very accurate when the curvature

of the interface is small, but even when the fluid surface has a large curvature it remains

robust. This is important especially when modelling events which can contain droplets,

because infinite curvatures may appear when these droplets reconnect.

(a) SLIC (b) PLIC

Figure 2.13: Comparison between the SLIC and PLIC surface reconstruction algorithms

The advection equation that governs F for an incompressible fluid can be written as follows

[25]:

∂F

∂t+ u ·∇F → ∂F

∂t+∇ · (uF ) = 0 (2.15)

where u is the flowfield. This equation shows that volume is conserved along a streamline.

The fluxes through a cell face can be calculated using:

∂F = u ·nA∂t. (2.16)

WithA the cell surface and ∂t the time step.

When these fluxes are calculated, the VOF is updated from level n to n+ 1:

Fn+1 = Fn +∂Fe + ∂Fn − ∂Fw − ∂Fs

∂x∂y. (2.17)

20


The indices n, e, s and w stand for north, east, south and west, representing the fluxes

through the different faces of a cell in two dimensions. Away from the free surface, the

net flux becomes zero. Around the free surface the calculation of the fluxes becomes more

complicated, a method to do so can be found in Hurt and Nichols [24].

There are however some major drawbacks to this method. First of all, conservation of mass

may be violated when rounding F when F ≤ 0 or F ≥ 1 [23]. Further, a lot of droplets

that disconnect from the surface may appear, which is called flotsam or jetsam. This is

especially the case for the lower-order methods like SLIC [25]. To address these problems,

a local height function was introduced. A detailed description is given by Gerrits [26]. The

local height function h is defined in each surface cell, and gives the height of the surface

in a column of three cells each. The direction of the local height function is defined as the

direction most normal to the free surface.

Figure 2.14: Local height function h

Normally, when the fluxes at the boundaries are calculated, the individual values of the

VOF are updated. Here the local height function h is updated instead. Afterwards, the

VOF is calculated from the height of the fluid in each column. No underflow or overflow

of the column can appear, so this method is strictly mass conserving. Also, the amount

of flotsam or jetsam is much smaller, which is physically more correct. As a conclusion,

it can be stated that the introduction of the local height function definitely improves the

performance of the VOF algorithm.

21

Chapter 3

Implementation of a CEL-model

in Abaqus™/CAE

3.1 Introduction

Simulia, the company that develops Abaqus™, gives a detailed description of what is

needed to build a Coupled Euler-Lagrang (CEL) model, but no explanation is given about

why it is necessary to adjust certain features or parameters to a certain value. The goal

of this chapter is to explain the principles behind the construction of a CEL-model.

3.2 Explicit dynamic analysis

3.2.1 Properties

In order to simulate slamming events, a dynamic analysis is needed. In Abaqus™, CEL-

calculations are performed using a dynamic explicit method, due to the highly non-linear

nature of the governing equations. This explicit method is characterized by two important

properties. First, an explicit integration rule is applied, and second, a lumped mass matrix

is used.

The equations of motion of an object are integrated using an explicit central difference

integration scheme:

u(i+1/2) = u(i−1/2) +∆t(i+1) + ∆t(i)

2u(i) (3.1)

u(i+1) = u(i) + ∆t(i+1)u(i) (3.2)

u is the velocity, and u is the acceleration. The kinematic state of the body is updated,

using the values of u at (i − 1/2), and u at (i), making it an explicit method. (i − 1/2)

and (i+ 1/2) are called intermediate states.

22

Chapter 3. Implementation of a CEL-model in Abaqus™/CAE

To calculate the acceleration u at the beginning of each increment, Newton’s second law

is applied:

u(i) = M−1 · (F (i) − I(i)) (3.3)

Here, M represents the lumped mass matrix, F is the applied load vector, and I is the

internal force vector. The inverse of the mass matrix is triaxial, which makes the explicit

method calculating fast. When compared with an implicit method, the explicit method

generally needs more increments to complete a calculation. But in an implicit method,

a global set of equations must be solved each increment, while here this is not the case

because of the triaxial inversed mass matrix [27], so no iterations nor a tangent stiffness

matrix are needed.

3.2.2 Stable increment time

The increment time is based solely on the highest natural frequency occurring in the model.

When using an implicit method, the time increment is based on the needed accuracy and

the convergence of the calculations. This means that the calculation cost of an extra

increment is cheaper using an explicit method.

The stable increment time for each element is determined using Equation 3.4

∆t =2

ωelementmax

(3.4)

To estimate the increment time for the whole model, the minimum of the increment time

for all elements is chosen. This implies a very conservative estimate. It can also be

formulated in terms of a characterstic element dimension Le and the dilatational wave

speed in the material cd:

∆t = min

(Lecd

)(3.5)

Le depends on the maximum eigenfrequency of the element, which means the stable in-

crement time is smaller when a stiffer material is used. Expressions for Le exist for every

element type available in Abaqus™/explicit, see [27]. To control high-frequency oscilla-

tions, an amount of damping is automatically introduced. The stable time increment

becomes:

∆t =2

ωelementmax

(√1 + ξ2 − ξ

)(3.6)

The fraction of critical damping in the highest mode ξ is determined by:

ξ = b1 − b22Lecdmin(0, εvol) (3.7)

23


Here εvol represents the volumetric strain rate, b1 and b2 are bulk viscosity scaling factors.

See Equations 3.13 and 3.14 for an explanation of their influence.

The damping actually reduces the stable time increment. Normally, an algorithm that

automatically estimates the highest eigenfrequency is implemented in Abaqus™/explicit,

the maximum value of the eigenfrequency is updated every timestep. However, when using

fluids in a model, Abaqus™ does not use a global estimation, but an element-by-element

one. From this data, the stable increment time is calculated automatically. If desired, a

smaller increment can be chosen, but this is not of use in the models that are discussed

here.

3.2.3 Application to CEL

The materials used in the model determine the stable time increment, and so the calcu-

lation time. To reduce calculation time, two explicit dynamic timesteps are implemented.

First, the lagrangian part is placed 1 mm above the water surface. In the first step, an

initial velocity is given to the impacting object. The length of this timestep is chosen so

that the body drops 0.5 mm. Now, the body finds itself 0.5 mm above the water surface.

In the next step, the initial velocity is disabled, and the object falls the remaining 0.5 mm

solely under the influence of gravity, after which the slamming takes place. By partitioning

the event in two timesteps and using an initial velocity in the first one, the calculation

time is reduced severly because the free fall which gives the impacting object its velocity

does not need to be simulated.

3.3 Material definitions

In the CEL-analysis, two types of material are used. First, there is the structural material

of the impacting body. This can be steel, wood or a composite material for example.

These materials are modelled as an elastic material, with given density ρ, stiffness E and

poisson ratio ν.

The fluids - water and air - are modelled based on the equation of state (EOS). This

means the fluid is considered to be in thermodynamic equilibrum at all time. There

are four thermodynamic variables in the governing fluid equations (see literature study):

the pressure p, the density ρ, the internal energy i and the temperature T . By using an

equation of state, the state of the fluid is described using only two of those parameters. The

chosen equation of state relates these parameters to the other thermodynamic variables.

24


An example of an EOS is the ideal-gas law. By giving the temparture T and the density ρ,

the pressure is known by using p = ρRT , and the internal energy is specified by i = CvT .

To define a fluid in Abaqus™, the density ρ, the wave velocity in the material c =√γRT

and the dynamic viscosity µ are specified using the Us − Up formulation for the EOS; the

fluid is considered incompressible. When the fluid is incompressible, no relation exists

between the energy equation on one hand and the mass conservation and momementum

equations on the other hand. The energy equation must only be solved when temparature

variations are taken into account [28], but this is not the case here.

The equation of state gives the pressure as a function of the density and the internal

energy: p = f(ρ, i). It is possible to eliminate the internal energy i and eventually obtain

an pH = f(ρ) relationship, which is called the Hugoniot-curve. pH is called the Hugoniot-

pressure [27]. It can only be obtained experimentally.

Normally, the Us−Up formulation of EOS is used to simulate shocks in solid materials. Now

it is used to define fluid materials. The Hugoniot curve is approximated using following

expressions:

Us = c0 + sUp (3.8)

p =ρ0c

20µ

(1− sµ)2

(1− Γ0µ

2

)+ Γ0ρ0i (3.9)

Us is the linear shock velocity, Up the particle velocity. Γ0 and s are parameters of the

approximation. To model fluids, these parameters are set to zero: Γ0 = 0 and s = 0.

Eventually, we the following expression for the equation of state is obtained:

p = c20ρµ (3.10)

When defining a viscosity for the material, it is treated as a Newtionian fluid, meaning

the viscosity depends only on the temperature. This is the default setting in Abaqus™.

Since the temperature is constant throughout the analysis, the viscosity also is.

The velocity of air is not higher than 100 m/s, so the Mach-number does not exceed

0.3. When only low mach numbers occur, air can be treated as an incompressible gas.

Morever, when air is modelled using the ideal-gas EOS, the energy equation has to be

taken in account, resulting in an increase of calculation time.

To conclude, the parameters for each of the fluids used in the model are given in Table

3.1.

25


Table 3.1: Values of parameters used to model water and air

water air

ρ [kg/m3] 1000 1.2

c0 [m/s] 1481 343.2

µ [Pa · s] 0.001 1.78E-05

Γ0 0 0

s 0 0

3.4 Contact interactions

General contact is applied to account for the interaction between the lagrangian part and

the eulerian domain. The contact formulation is based on an immersed boundary method.

This means the lagrangian part occupies void in the eulerian domain. A rough friction

constraint is applied on the contact interaction. Mechanically, this implies that the friction

coefficient µfric =∞. Once the surfaces makes contact and undergoes rough friction, the

contact should remain closed. As a consequence, a layer of fluid remains attached to the

langrangian part. The contact interaction is implemented using a penalty based method

as described in Chapter 2. Abaqus™ calculates the contact interaction twice. The second

time the calculation is performed, the surface that previously was the master surface, now

becomes the slave surface and vice versa. Eventually a weighed average is taken in order

to calculate the contact for that particular step. Implementing a penalty based method

makes the model stiffer, because artificial springs are implemented between the slave and

master surface. This could implicate that the stable time increment decreases, raising the

calculation time needed. Abaqus™ automatically prevents this, thus limiting the stable

increment time.

Because a layer of fluid remains attached to the langrangian part, buoyancy is accounted

for. To test this, a drop test was performed using a wooden sphere (ρ = 500 kg/m3) as can

be seen in Figure 3.1. After impact, the sphere shows a short damped oscillating motion,

after which it stays afloat on the water surface.

26


(a) t = 0 s (b) t = 0.1 s

(c) t = 0.4 s (d) t = 1 s

Figure 3.1: Drop test with wooden sphere

Contact between different fluids in an eulerian domain does not need to be defined, as it

occurs by default. Because of the kinematic assumption that a single strain field is applied

to the materials present in one element, the interaction is rough.

3.5 Predefined fields

In the initial time step, the initial volume occupied by the water must be determined. This

can be done using a predefined field for the material assignment. The eulerian domain

must be divided in different sections to be able to do this. A volume fraction between 0 and

1 can be assigned to a chosen section of the domain. It is important to include a certain

region filled with void. When the langrangian part moves through the eulerian domain,

the eulerian elements occupied by the langrangian part become filled with void. Assigning

at least one layer of void results in the formation of a free surface. This is especially

important when the langrangian part is positioned outside of the eulerian domain in the

initial state of the model. But even when this is not the case, including void makes sure

27


that water can be pushed out of the eulerian element by the impacting body.

3.6 Boundary conditions and loads

Non-reflecting outflow eulerian boundaries are used in order to model an infinite domain,

and can be specified in the boundary condition editor. However, they cannot be imple-

mented in the initial timestep, but must be activated in an analysis timestep. They are

important because otherwise shockwaves are reflected on the eulerian boundaries, as a

result of which the pressure on the interface between the impacting object and the water

surface is influenced. Another advantage is that the eulerian domain can be made rela-

tively small, decreasing the computational time enormously. The boundary conditions are

implemented on the eulerian nodes, and they affect the fluid when it passes these nodes.

Rigid bodies are implemented by using a rigid body constraint on the deformable impacting

object. It is important that in this case the boundary conditions are applied on a chosen

reference point on the part. If this is not the case, the lagrangian part will not move.

3.7 Mesh

There is only one element availble in the Abaqus™ element library to mesh an eulerian

part, called EC3D8R. ’E’ stands for eulerian, ’C3D8’ means it’s a three-dimensional brick

with 8 nodes, and ’R’ indicates reduced integration is applied. It is based on a langrangian

element (C3D8R), but with added controls to be able to handle multiple materials in one

element, and to support the transport phase in the calculation of the volume fraction.

The default formulations for the element are applicable for a wide range of problems, both

dynamic and quasi-static, but are not very efficient [27]. To optimize the element control

for the high-rate dynamic event, hourglass control control is used, which is an artifcial

numerical stiffness applied to a single element. Otherwise, the linear element’s behaviour

is not stiff enough.

When applying reduced integration, only the linear varying part of the incremental dis-

placement is used to calculate the increment of physical strain. The remaining part of the

incremental displacement is called the hourglass field, and can be expressed in hourglass

modes. It may occur that these hourglass modes get excited, resulting in severe mesh

distortion without any resisting stress. To prevent this, there are several methods of hour-

glass control. In this model, viscous hourglass control is appropriate, because it the most

efficient approach for very dynamic simulations [27]. Three parameters can be adjusted:

28


the displacement hourglass scaling factor s, and the linear and quadratic bulk viscosity

scaling factors b1 and b2.

3.7.1 Displacement hourglass scaling factor s

A viscoelastic approach of the Kelvin-type is governed by the following Equation:

Q = s

[(1− α)Kq + αC

dq

dt

](3.11)

Here, q is an hourglass mode magnitude, and Q is the force or moment related to q. K

represents the hourglass stiffness, and is calculated automatically by Abaqus™/explicit.

C is the linear viscous coefficient. The first term on the righthand side, determines the

elastic behaviour, while the second term accounts for the viscous behaviour. α is a scaling

factor to model the balance between elastic and viscous control. Because a pure viscous

hourglass control is chosen, α equals one. Equation 3.11 becomes:

Q = sCdq

dt(3.12)

The eulerian elements do not move in space, so we can apply s = 0, resulting in Q = 0.

3.7.2 Bulk viscosity scaling factors b1 and b2

Bulk viscosity introduces damping in the model. It can be adjusted for the whole model

in each time-step. By doing this for the eulerian element in the section controls, the

adjustment is only valid in the eulerian domain. The bulk viscosity parameters for the

lagrangian part are set to a default value. For the eulerian domain, both b1 and b2 are set

to zero. This can be explained as follows:

The linear bulk viscosity scaling factor b1 is used to damp ringing of the highest element

frequency in the model, which would give oscillations in the stable time increment. The

damping introduces a bulk viscosity pressure given by the following expression:

pbv1 = b1ρcdLeεvol (3.13)

This pressure has a linear relation with the volumetric strain rate εvol. The quadratic

viscosity scaling factor b2 is related to a pressure that smears possible shock fronts across

different elements in order to prevent they should collapse under extremely high velocity

gradients.

pbv2 = ρ (b2Leεvol)2 (3.14)

29


Again, because the eulerain mesh is not moving, this damping is not necessery and can

be set to zero. This increases the stable time increment, reducing calculation time. The

pressure associated with the damping is purely a numerical aid, and makes no difference

in the obtained pressure at the nodes.

3.7.3 Mesh size

Of course, the mesh of the eulerian domain must be small enough to capture the impact

pressure and the boundary layer around the impacting object. Considering the advice of

Abaqus™ support and relying on the knowledge present in the department of Mechanics

of Materials and Structures at Ghent University, an eulerian mesh element size of 0.5 mm

was chosen. Moreover, the element size of the langrangian mesh a must have the same

order of magnitude in order to prevent numerical leaking, where the fluid leaks into the

langrangian part.

30

Chapter 4

Calculations for rigid bodies

4.1 Introduction

Before calculating slamming pressures on complex geometries, it is necessary to perform

a series of calculations on basic geometries such as wedges, cones, cylinders and spheres.

These are called testcases. Next, the results obtained for these testcases are examined,

and compared with results found in international literature, both numerical, analytical and

experimental. Doing so, the performance and accuracy of the Abaqus™-CEL calculations

can be examined. In this chapter, rigid bodies are considered. Gravity is always taken into

account, and the testcases are modelled as free-falling objects, meaning they can decelerate

and do not have a prediscribed velocity throughout the analysis. Surface tension cannot

be modelled in Abaqus™.

4.2 Wedge

4.2.1 Introduction

A series of calculations on wedges with different deadrise angles are performed. The

parameters of the model are as follows:

• initial velocity: 5.2 m/s

• deadrise angle α: 6, 10and 14

A rigid body constraint is applied to the wedge, with a reference point on the lowest point

of the wedge. The wedge is modelled as a solid, meshed with a linear element C3D8R

(Figure 4.1), because this will give a smaller calculation time than a quadratic element,

31

Chapter 4. Calculations for rigid bodies

and does not cause accuracy problems because the rigid body constraint is applied. For

all rigid bodies described in this thesis, this element type is used.

Figure 4.1: Meshed wedge, deadrise angle = 10

Because only three-dimensional eulerian elements exist, the parts are only one element

wide in depth. Moreover, several boundary conditions are applied:

• Initial velocity (Figure 4.2(a))

• Symmetry in the XY-plane (Figure 4.2(b)): Uz = Urx = Ury = 0, where U denotes

the displacement, and Ur is the rotational displacement. In this way, a wedge of

infinte length is simulated.

• Symmetry in the YZ-plane (Figure 4.2(c)): Ux = Ury = Urz = 0, so only one half of

the wedge needs to be modelled.

32


(a) Initial velocity (b) Symmetry in XY-plane

(c) Symmetry in YZ-plane

Figure 4.2: Boundary conditions for wedge

Boundary conditions also have to be applied to the eulerian domain:

• Non reflective outflow eulerian boundary (Figure 4.3(a)): They prevent shockwaves

from reflecting on the boundaries of the domain; in this way an infinite water basin

is modelled.

• Symmetry in the XY-plane (Figure 4.3(b)): Uz = Urx = Ury = 0; the wedge is

simulated to have an infinte length, and the same boundary condition must be

applied to the euler domain as well.

• Symmetry in the YZ-plane (Figure 4.3(c)): Ux = Ury = Urz = 0; because only half

of the wedge is modelled, the same symmetry condition must also be applied to the

eulerian domain.

33


(a) Non reflective (b) Symmetry in XY-plane


Figure 4.3: Boundary conditions for eulerian domain

The whole domain, together with the applied boundary conditions, can be seen in Figure

4.4.

34


Figure 4.4: Overview of model domain with applied boundary conditions

4.2.2 Results

Figure 4.6 shows the pressure on different paths in the water along the wedge for different

timesteps, where a path is a linear curve in the water domain, parallel and close to the

wedge (Figure 4.5, shown in red colour).

Figure 4.5: Path for visualisation of the pressure along the wedge

The blue curve represents the pressure at first impact. When time passes, the pressure

peak moves along the wedge. A maximum pressure of 10.9 bar is reached. At first

impact, the maximum pressure is low. The reason for this can be found in the fact that

the pressure was still building up at the time the output was generated. The obtained

maximum pressure is compared with an analytical formula proposed by Zhao and Faltinsen

[1]:

pmax =1

2ρV 2 π2

4tan(α)2(4.1)

For the wedge this gives pmax = 10.7 bar. This is lower than the pressure that is obtained

here, but the difference is small. As a result of a numerical study, Peseux [2] obtained a

35


maximum pressure of 14.1 bar for a wedge with an impact velocity of 6 m/s. When making

the pressures dimensionless, this results in a pressure coefficient Cpmax = pmax

1/2ρV 2 = 78.3.

For the results obtained here, Cpmax = 80.62. Again the difference is small, and the

calculated maximum pressure is on the conservative side.

0 0.005 0.01 0.015−2

0

2

4

6

8

10

12

Distance [m]

Pressure [bar]

t = 0 mst = 0.04 mst = 0.13 ms

Figure 4.6: Pressure along wedge with deadrise angle 10for different timesteps

Next, the pressure evolution in a single element can be seen in Figure 4.7, where x denotes

the distance of the chosen element away from the vertical axis of symmetry.

36


1 2 3 4 5 6

x 10−4

−20

−15

−10

−5

0

5

10

15

20

Time [s]

Pressure [bar]

x = 0 mm

x = 16 mm

x = 23 mm

Figure 4.7: Pressure in three different elements with respect to time. Deadrise angle=10

Again, the discplacement of the pressure peak away from the axis can be seen. A lot

of scatter is present for every element, once the pressure peak has passed. Also, large

negative pressures occur due to a shock wave that is propagated trough the water after

impact. This is not observed in literature. The maximum pressure obtained here is higher

than the maximum pressure in Figure 4.6, almost twice the value of the values found in

literature. This also proves the large amount of scatter. Normally, the pressure for each

element is averaged over different elements to obtain a continous contourplot. However,

when disabling the averaging of results between the different elements it can be seen

that there are a lot of single elements with a much higher pressure than the elements

surrounding them, without a clear pattern that emerges. This is especially the case in

the area surrounding the maximum pressure, see Figure 4.8.This is probably due to a

numerical instability.

37


Figure 4.8: Pressure for t = 0.595 ms after impact. The results are not averaged

Figures 4.9 and 4.10 show the evolution of the pressure troughout the calculations.

Several trends can be distinguished:

• As expected, the location of the maximum pressure can be found where the free

surface overturns to eventually form a jet. This is the location where at that moment

in time the body enters the water.

• Normally, when the pressure peak moves along the wedge, it should decrease because

the wedge slows down. Here, howerer, the maximum pressure both decreases and

increases from frame to frame.

• The source of the radial pressure wave is located on the same place as the maximum

pressure in every step. Because the wedge slows down, the root of the pressure front

should remain at the point of first impact. The fact that this is not the case here

can be contributed to the fact that the maximum pressure sometimes increases in

from one output frame to another.

38


(a) first impact

(b) t after impact = 0.025 ms

(c) t after impact = 0.125 ms

Figure 4.9: Evolution of pressure part 1

39


(a) t after impact = 0.225 ms



Figure 4.10: Evolution of pressure part 2

Effect of deadrise angle α

A higher deadrise angle should give a lower pressure, because the wedge cuts more easily

through the water. The pressure for t=0.15 ms after impact on a path along the wedges

40


with different deadrise angles can be seen in Figure 4.11 .

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−5

0

5

10

15

20

25

Distance [m]

Pressure [bar]

6°

10°

14°

Figure 4.11: Pressure on a path along the wedge for different deadrise angles. t after impact=0.15

ms

For every wedge, the chosen path has a different angle, equal to the deadrise angle. Because

the location of the pressure peak depends on the vertical submersion of the wedge, they

occur on a different location for every wedge. In Table 4.1, a comparison is made between

the calculated pressures and Zhao’s Equation 4.1:

Table 4.1: Comparison of calculated pressure and Equation 4.1

α pmax calculated [bar] pmax Zhao [bar]

6 20.96 30.20

10 8.76 10.73

14 4.91 5.73

The deviation from the analytical Equation 4.1 becomes larger for a smaller deadrise angle.

The calculated pressures are on the low side. The result depends highly on the time after

impact that is chosen. The evolution of the pressure in the element where the first impact

occurs, is given in Figure 4.12. To make the image more clear, the pressure peaks are

41


given a certain offset along the x-axis, and the moments of first impact are indicated. The

maximum pressures obtained here differ from the maximum pressures calculated along the

wedge. For α = 10 and α = 14, they are higher than the values predicted by Equation

4.1. For α = 6, however, it is much lower. Also, extremely large negative pressures

occur.

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

x 10−4

−35

−30

−25

−20

−15

−10

−5

0

5

10

15

20

Time [s]

Pressure [bar]

6°

10°

14°

Figure 4.12: Pressure evolution in element where first impact occurs for different deadrise angles

To illustrate that the pressures vary highly from element to element and from frame to

frame, the pressure history in an element is shown in Figure 4.13; the element is located

along the wedge, 3mm horizontal from the axis of symmetry. The pressure reaches 14.90

bar, when a maximum pressure of 5.73 bar is expected.

42


4 5 6 7 8 9

x 10−4

−10

−5

0

5

10

15

Time [s]

Pressure [bar]

Figure 4.13: Pressure history in element located 3 mm from axis of symmetry. α = 14

Influence of the implementation of air

A more realistic approach can be obtained by implementing a layer of air above the water

surface. However, for a rigid object the results should not differ much from the results

without air because cushioning cannot take place, and drag forces for the velocity used

are very small compared to gravitational force. First, the vertical velocity between the

models with and without air is compared (Figure 4.14). The start of the second step in the

analysis, where the initial velocity is disabled and the wedge moves only under the influence

of gravity, is indicated. When air is not implemented, the wedge accelerates due to gravity,

but with air, the wedge immediatly slows down, which can be explained as follows: In

Abaqus™ it is not possible to model a slamming event with two different eulerian domains,

one for the water and one for the air, because they cannot interact. In order to model the

interaction between the water and the object, a rough contact interaction is applied for

the whole eulerian domain, so the same contact interaction applies for the contact between

air and the impacting object. Because this interaction is rough, the wedge slows down

more than is expected physically, leading to a difference in velocity of the wedge between

the two models. A smaller maximum pressure in the case where air is implemented is

expected. However, because the difference in velocity is small, the maximum pressures

should not differ that much.

43


0 1 2 3 4 5 6 7x 10−4

5.14

5.15

5.16

5.17

5.18

5.19

5.2

5.21

Time [s]

Vertical velocity [m/s]

airno air

Figure 4.14: Vertical velocity for model with and without air. α = 10

The pressure on a path along the wedge is given in Figure 4.15. A maximum pressure of

8.50 bar is obtained, which is indeed slightly smaller than when air is not implemented.

The second peak pressure (8.50 bar) is higher than the first peak pressure (8.37). Altough

Figure 4.14 clearly shows that the wedge slows down in time, the pressure increases in

time. However, this increase is very small and can be considered insignificant.

44


0 0.005 0.01 0.015 0.02−1

0

1

2

3

4

5

6

7

8

9

Distance [m]

Pressure [bar]

t = 0.235 ms

t = 0.400 ms

Figure 4.15: Pressure along wedge for two different timesteps. α = 10

The squeeze-velocity of the air, which is the horizontal velocity, is shown in Figure 4.16,

and is much higher than the vertical velocity in this case. A maximum value of 75.5 m/s

is obtained, but the air mach-number does not exceed M = 0.3. As such, this clearly

justifies the modelling of air as in incompressible gas, since up to mach-numbers of 0.3,

the divergence of the governing equations between an incompressible and a compressible

fluid is only about 1% [29].

45


(a) t after impact = 0 ms


Figure 4.16: Horizontal velocity of air for two different timesteps. α = 10

4.3 Cone

4.3.1 Introduction

The cone is a three-dimensional version of the wedge, rotated around its axis of symmetry.

Steel (ρ = 7800 kg, E = 210GPa, ν = 0.3) is used as material. The conus has a diameter

of 300 mm, and the deadrise angle α is 10. To reduce calculation time, only one fourth

of the cone is modeled. Calculations on one eight of a cone were also performed, but

the results were not accurate in the area around the axis of symmetry, where the brick

elements had a corner of 45. Because a rigid body constraint is applied, linear brick

elements are used to model the cone, in order to reduce calculation time.

The implemented boundary conditions for the cone are as follows:


46


• Symmetry in the XY-plane: Uz = Urx = Ury = 0 (Figure 4.17(b))

• Symmetry in the YZ-plane: Ux = Ury = Urz = 0 (Figure 4.17(c))

(a) Initial Velocity (b) Symmetry in XY-plane


Figure 4.17: Boundary conditions for cone

For the eulerian domain, following boundary conditions were implemented:

• Non reflective outflow eulerian boundary (Figure 4.18(a)):

• Symmetry in the XY-plane (Figure 4.18(b)): Uz = Urx = Ury = 0;

• Symmetry in the YZ-plane (Figure 4.18(c)): Ux = Ury = Urz = 0;

47


(a) Non reflective (b) Symmetry in XY-plane


Figure 4.18: Boundary conditions for eulerian domain

The explanation of the boundary conditions, both for the cone and the eulerian domain,

is the same as for the wedge. Furthermore, a general view of the model with applied

boundary conditions can be seen in Figure 4.19.

Figure 4.19: General view of the cone model with applied boundary conditions.

48


4.3.2 Results

Pressure

First, the same impact velocity V = 5.2m/s as for the wedge is used, but to see the

influence of the impact velocity, a simulation with V = 2.6m/s is also performed. The

influence of the impact velocity is very clear from Figure 4.20, where the pressure is shown

for the element where the first contact between the cone and the water surface takes

place. Since pmax ∼ V 2 (see Equation 4.1), the maximum pressure should be 4 times

higher when the impact velocity is doubled. For V = 5.2m/s, a maximum pressure of

11.1 bar is reached, resulting in a pressure coefficient Cp = 82.10. For V = 2.6m/s,

the maximum pressure is 3.69 bar, with a pressure coefficient Cp = 109.17. Altough the

impact pressure is much lower when the impact speed is halved, it is still higher than

expected.

1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

x 10−4

−15

−10

−5

0

5

10

15

Time [s]

Pressure [bar]

v = 5.2 m/sv = 2.6 m/s

Figure 4.20: Pressure evolution in element were first impact takes place for two different impact

velocities. α = 10

The pressure on a path along the cone is visualised in Figure 4.21, where the path is

defined parallel to the cone, just as for the wedge, and has an angle of 45with respect

to the XY-plane. The time after impact is 0.165 ms. Again, there is clearly a difference

when the impact velocities are different. For V = 5.2m/s, the maximum pressure is 10.30

bar, while for V = 2.5m/s, the pressure reaches 3.28 bar.

49


0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02−1

0

1

2

3

4

5

6

Distance [m]

Pressure [bar]

v = 5.2 m/sv = 2.6 m/s

Figure 4.21: Pressure on a path along the cone for two different impact speeds. α = 10

The cone is relatively heavy since it is made of steel, and only the first stages of the impact

is studied. This means there is a strong gravitational force on the cone, while only a little

wetted surface is in contact with the water. The upward force on the cone is small. As

a result, the cone does not immediatly slow down when it hits the water. Only when the

wetted surface has become large enough, it slows down. The vertical velocity of the cone

throughout the simulation is shown in Figure 4.22; the point of impact is denoted by the

dotted line. However, the variation in velocity is rather small. The calculation can be

considered as performed with a constant velocity.

50


1 2 3 4 5 6 7

x 10−4

5.195

5.196

5.197

5.198

5.199

5.2

5.201

5.202

5.203

5.204

5.205

Time [s]


Figure 4.22: Vertical velocity of rigid cone. α = 10

Comparison with experimental results

The calculated results are compared with experimental results obtained by Peseux et al.[2],

where tests on both rigid and deformable cones were performed, and compared with their

own numerical model, which was discussed in chapter 2. In the experimental setup, the

cone is equiped with two pressure sensors. The first one is placed along the cone 40 mm

from the tip, the second one is placed 90 mm from the tip. The location of both sensors can

be seen in Figure 4.23(a). To obtain a constant velocity throughout the experiment, the

velocity is measured at the point of impact with a cable sensor connected to the support,

and is controlled and kept constant from that moment on. The experimental setup can

be seen in Figure 4.23(b).

51


(a) Location of the pressure sensors (b) Experimental setup

Figure 4.23: Experiments by Peseux et al.[2]

Their results for a rigid cone with impact velocity V = 5.2m/s and deadrise angle α are

shown in Table 4.2. From these results it can be concluded that the pressures calculated

with Abaqus™ are highly conservative, since they almost double the experimental values,

and are 50% larger than the value obtained with the numerical model.

Table 4.2: Experimental and numerical results by Peseux [2]

Experimental [bar] P1 3.9

P2 5.5

Numerical [bar] 7

4.4 Cylinder

4.4.1 Introduction

A series of slamming tests is performed for a rigid cylinder. This is a more difficult object

to perform a calculation on than a wedge. The wedge has a first contact with only one

singular element, after which there is a linear submersion. For the cylinder, this is not the

case. After first contact, a large part of the cylinder surface quickly slams on the water

surface.

The model parameters are chosen as follows:

52


• cylinder diameter: 300 mm

• thickness: 3 mm

• material: steel (ρ=7800 kg/m3, E=290 GPa, ν=0.3)

• initial velocity: 5 m/s

The cylinder is modelled using shell elements S4R, and a rigid body constraint is applied,

with a reference point in the center of the cylinder. The same type of boundary conditions

as for the wedge are implemented for the cylinder:


• Symmetry in the XY-plane: Uz = Urx = Ury = 0 (Figure 4.24(b))

• Symmetry in the YZ-plane: Ux = Ury = Urz = 0 (Figure 4.24(c))

(a) Initial velocity (b) Symmetry in XY-plane (c) Symmetry in YZ-plane

Figure 4.24: Boundary conditions for cylinder

The boundary conditions for the eulerian domain are exactly the same as for the eulerian

domain used in the model for the wedge, because the domains are equal. Also, the same

mesh element size is used.

53


4.4.2 Results

The pressure in the water on a circular path along the cylinder can be seen in Figure 4.26,

where the path is defined in the same way as for the wedge, but is now circular (Figure

4.25).

Figure 4.25: Circular path to visualise pressure along cylinder

At first impact, represented by the blue curve, the pressure reaches 45.8 bar. 0.065 ms

after impact, the maximum pressure is even 51.8 bar. Afterwards the pressure decreases,

as is expected because the cylinder slows down. The pressure history for two elements

in time is shown in Figure 4.27. A maximum pressure of 44.2 bar is obtained. There

is stil a lot of noise after the pressure peak has passed. For the element that is located

23 mm away from the axis of symmetry, there is some influence of the flowfield before

the maximum pressure is obtained. The difference in width of the pressure peak between

Figure 4.26 and Figure 4.27 shows that the pressure peak occurs very shortly in time, but

is more spread out in space.

54


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035−10

0

10

20

30

40

50

60

Distance [m]

Pressure [bar]

t = 0 mst = 0.065 mst = 0.265 ms

Figure 4.26: Pressure on a path along the cylinder for three different timeframes

2 3 4 5 6 7 8

x 10−4

−10

0

10

20

30

40

50

Time [s]

Pressure [bar]

x = 23 mmx = 0 mm

Figure 4.27: Pressure history for two different elements

An overview of the pressure in the water can be seen in Figure 4.28 and 4.29:

Several trends can be seen:

• The maximum pressure occurs at the point were, at that time, impact takes places.

This is the point where the flow eventually overturns to form a jet.

55


• A strong shock wave with large negative pressures occurs. This is not seen in liter-

ature.

• The value of the maximum pressures vary highly from frame to frame, but eventually

the pressure reaches a smaller value (Figure 4.29(c)) than the initial values. This

shows again that the value of the maximum pressure is dependent of the time frame

that is chosen. The reason for this is that the pressure can become very high in

a single isolated element, due to numerical instabilities, which Figure 4.30 clearly

demonstrates. A maximum pressure of 85.8 bar is reached in that timeframe, which

is much higher than in most of the other timeframes.

56


(a) first impact



Figure 4.28: Evolution of the pressure in the water domain for the impact of a 2D rigid cylinder,

part 1

57


(a) t after impact = 0.09 ms



Figure 4.29: Evolution of the pressure in the water domain for the impact of a 2D rigid cylinder,

part 2

58


Figure 4.30: Very high pressure in a single element caused by a numerical instability. Non-

averaged values.

Implementation of air

When a layer of air is implemented above the water surface, the following results are

obtained for the pressure along the cylinder on two different timesteps:

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04−5

0

5

10

15

20

25

Distance [m]

Pressure [bar]

t = 0 mst = 0.3 ms

Figure 4.31: Pressure along cylinder for two different output frames when air is implemented

59


At first impact, a maximum pressure of 22.3 bar is reached, which is lower than compared

to the results calculated without air; the air damps the impact because the velocity de-

creases rapidly. Due to the shape of the cylinder, this effect is stronger than in the case of

a wedge. Therefore, the maximum pressure is lower, but this is mainly due to the decrease

in velocity and the rough fricition between the cylinder and the air. The pressure peak

also decreases with time, as is expected. However, when looking at the pressure in time

for a single element (Figure 4.32), it can be seen that the pressure can reach very high

values in a single element, and that there is a lot of fluctuation in the pressure. When

extracting the results along a path, the pressures are averaged along the surrounding ele-

ments. However, for the evolution of the pressure in a single element, this is not the case,

hence the difference in pressure between Figure 4.31 and Figure 4.32. This proves again,

that very high pressure occur in a single element, and that the results must be interpreted

with caution.

2 3 4 5 6 7 8 9 10

x 10−4

−10

−5

0

5

10

15

20

25

30

35

Time [s]

Pressure [bar]

x = 0 mmx = 23 mm

Figure 4.32: Pressure history in time for two different elements when air is implemented.

Comparison with experimental results

In order to assess the correctness and accuracy of the calculations, they are compared

with the experiment results obtained by Lin and Shieh [4], who performed tests on acrylic

cylinders with diameter 0.2 m and 0.3 m. Their results are given in dimensionless form

in Figure 4.33. A comparison with Wagners theory is made, and it can be seen that the

deviation is larger for smaller angles.

60


Figure 4.33: Results of Lin and Shieh [4]. Dimensionless pressure

A summary of the results calculated with Abaqus™can be found in Table 4.3:

Table 4.3: Maximum pressure coeffecient for different locations on the cylinder

sin(θ) = x/R Cpmax

Without air 0 414.4

0.1533 403.6

With air 0 280.0

0.1533 228.3

The actual velocities were used to calculate the maximum pressure coefficient. Altough it

can not be stated that the experimental results are the real maximum pressure coefficient,

it is clear that the pressures that are calculated using Abaqus™ are very high. For x/R = 0,

Lin and Shieh obtain a Cpmax of 75. This is more than 5 times smaller than the calculations

done without air, and more than 3 times smaller than the calculations with air.

61


Comparison with numerical model

A comparison is made with the numerical results of Battistin and Iafrati [3] (see Figure

4.34) who developed a numerical model for symmetric and axisymmetric bodies, in which

the effects of gravity and surface tension are neglected. Moreover, the water is considered

incompressible, and air is not implemented. Their results are also compared with the

asymptotic solution by Zhao [1]. The results are given in dimensionless form for different

values of z/R, where z is the penetration depth, and R is the radius of the cylinder. The

dimensionless form of the arc length on the cilinder, y/R, is found on the x-axis, where y

denotes the arc lenght. The pressure is written in dimensionless form using the pressure

coefficient Cp, in function of the dimensionless arc length.

In the initial stage of the water impact, their results are in good agreement with the asymp-

totic theory. For larger values of y/R, the difference increases, because the asymptotic

theory is not accurate for larger deadrise angles.

Figure 4.34: Results by Battistin and Iafrati [3]. Left: Free surface for cylinder and sphere

for z = 0.01 and z = 0.05. Right: Pressure coefficient Cp corresponding to the

penetration depth on the left.

As can be seen in Figure 4.35 the shape of the curve corresponds quite well with the nu-

merical results by Battistin and Iafrati. The value of the pressure coefficient Cp, however,

is much higher than in Figure 4.34, for both values of z/R.

62


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−20

0

20

40

60

80

100

120

140

160

180

y/R

Cp

z/R = 0.01z/R = 0.05

Figure 4.35: Dimensionless pressure for different dimensionless penetration depths z/R

Formation of the jet

Lin and Shieh [4] used high speed cameras to capture the appearance of the jet and spray

that forms when the cylinder slams on the water. A comparison is made between their

experimental results, and the calculations performed with Abaqus™. The diameter D of

the cylinder is 100 mm, and the initial velocity is 1 m/s. Their experiment is shown in

Figure 4.36, in which can be seen that in the first stage of impact, after the pressure

has occured, droplets form. Next, the water rises up along the cylinder, and eventually

separates for the cylinder. This happens in a timeframe of about 0.1 s. The water surface

calculated with Abaqus™ can be seen in Figure 4.37. Figure 4.37(b) shows that the amount

of droplets that is formed is small when compared with the experimental results. This

can be atributed to the fact that the eulerian elements have a width and length of 0.5

mm. Droplets that are much smaller than these dimensions are not formed. At 0.006 s

after impact, the flow starts to overturn to form a jet. The splashing starts somewhat

later than in the experimental setup. Except for the amount of droplets, the free surfaces

match quiet well. The elevation of the free surface in the outer region is the same for both

the experiments and the calculations. The direction of the spray also corresponds with

the experiments.

63


(a) t = 0 − 0.0057s (pres-

sure occurs)

(b) t = 0.0057 − 0.0114s

(splashing)

(c) t = 0.0114 − 0.017s

(splashing)

(d) t = 0.017 − 0.0288s

(splashing)

(e) t = 0.0228 − 0.0285s

(rising up)

(f) t = 0.0285 − 0.0342s

(rising up)

(g) t = 0.0341 − 0.0398s

(rising up)

(h) t = 0.0398 − 0.0455s

(separation)

(i) t = 0.0512 − 0.0.0569s

(separation)

(j) t = 0.0626 − 0.0683s

(separation)

(k) t = 0.0740 − 0.0796s

(separation)

(l) t = 0.0853 − 0.0910s

(separation)

Figure 4.36: Visualisation of the flow field for a cylinder slamming on a flat water surface obtained

by Lin and Shieh[4]. D = 100mm and v = 1m/s.

64


(a) t = 0.003s (pressure oc-

curs)

(b) t = 0.006s (rising up) (c) t = 0.015s (splashing)

(d) t = 0.02s (rising up) (e) t = 0.025s (rising up) (f) t = 0.03s (rising up)

(g) t = 0.035s (separation) (h) t = 0.045s (separation) (i) t = 0.06s (separation)

(j) t = 0.065s (separation) (k) t = 0.08s (separation) (l) t = 0.095s (separation)

Figure 4.37: Volume fractions at different timesteps for an impacting cilinder calculated with

Abaqus™65


Conclusion

4.5 Sphere

To see the influence of using symmetry in the models, a slamming test was performed on a

sphere, without the use of symmetry, in which the sphere is made of steel (ρ=7800 kg/m3,

E = 210GPa, ν = 0.3), with a thickness of 3 mm. The impact velocity V is 5 m/s .

Figure 4.38: Overview of model with boundary conditions

First, the results are made dimensionless to compare with the results of Battistin and

Iafrati [3] seen in Figure 4.34. The pressure coefficient Cp is given in function of the

dimensionless arc length y/R in Figure 4.39. Due to three-dimensional effects in case of

the sphere, the wetted portion of the body is smaller. This results in a lower pressure

when compared to the cylinder. A maximum pressure coefficient Cp of 140 is obtained

(Figure 4.39), for z = 0.01, while Battistin and Iafrati obtain a maximum value of 80.

The dimensionless pressure calculated here is indeed lower than for the cylinder, but is

still much higher than the values found in literature.

66


Figure 4.39: Pressure coefficient for rigid sphere. z/R = 0.01

Next, the influence of the symmetry is studied. The pressure at the water surface at 0.215

ms after impact is shown in Figure 4.40. The pressure is not uniform on a circumpheral

path around the center, which is the point of first impact. This is due to the fact that

the eulerian elements are not all equal for points on this circumpheral path. Because a

brick element is the only eulerian element that is available in the Abaqus™element library,

it is not possible to obtain a mesh with ’pie-shaped’ elements, as if a two-dimensional

planar element was rotated around the central vertical axis because this would give highly

inaccurate results. Normally, the angle in a brick element is 90, but if this angle is made

smaller, what would be the case if pie-shaped elements were implemented, there would be

a high level of distortion in this smaller angle.

67


Figure 4.40: Water surface for impact of rigid sphere. Time after impact=0.215 ms

4.6 Conclusion

For the wedge, the obtained maximum pressures correspond quite well with Zhao’s ex-

pression for the maximum pressure. However, due to numerical instabilities, the pressure

can reach abnormally high values in some elements. A lot of noise is present in the pres-

sure history plots, and because large negative pressures occur, the flow pattern does not

correspond with patterns found in literature. The influence of the deadrise angle α can

clearly be seen: the pressure decreases when α increases, as expected, but the deviation

from Equation 4.1 increases when α decreases.

The results for the cone were compared with numerical and experimental results, from

which it appears that the calculated pressures are too high. The influence of the impact

velocity V can clearly be seen: the pressure is lower when V decreases.

For the cylinder, the pressures calculated with Abaqus™ are much higher than the

values found in literature, and large negative pressures occur. Moreover, in some elements

the pressure can also become very high due to numerical instabilities. However, the

formation of the jet and the water surface elevation in the Abaqus™simulation match

quite well with experimental results.

Somewhat lower pressures are obtained for both the cylinder and the wedge, when air

is implemented, altough the difference should not be great for rigid bodies. The lower

pressure can be explained by the fact that a rough interaction between the air and the

68


falling object is applied, meaning that the object immediately slows down, with a lower

pressure as a result. From this, it can be concluded that the drag caused by the air is too

large, and that the air has a rather stiff behaviour.

In general, it can also be noted that the resulting pressure depends highly on the

element and timestep chosen.

69

Chapter 5

Calculations on deformable bodies

5.1 Introduction

In this chapter, Fluid-Structure Interaction (FSI) calculations are performed. The local

pressure peak will deform the impacting object, which in return the will influence the

pressure distribution. This is a more realistic approach of real life slamming events. Dif-

ferent testcases are studied and the influence of the material properties are examined for

a cylinder, by implementing both steel and composite materials. In all cases, the same

boundary conditions as for the rigid bodies are implemented.

5.2 Wedge

5.2.1 Introduction

The wedge is made of steel plate with thickness 1 mm, to be able to study the influence of

the deformation of te structure, because a massive steel wedge would not deform enough

in the first stage of the impact. In Abaqus™, the wedge is modelled as a solid, and Figure

5.1 shows the model together with the applied boundary conditions, which are the same

as for the rigid wedge described in Paragraph 4.2.1, but without a rigid body constraint.

As for the rigid wedge, the impact velocity is 5.2 m/s. Because the thickness is small,

quadratic tetrahedral elements (C3D10M) were used to mesh the wedge [27], with a size

of 0.5 mm for the smallest element edges.

70

Chapter 5. Calculations on deformable bodies

Figure 5.1: Domain with boundary conditions for calculation on deformable wedge

5.2.2 Results

Deformation

The deformation of the wedge througout the analysis is shown in Figure 5.2, with a scaling

factor 20. From Figure 5.2(c) it appears that the left bottom part of the wedge is not the

lowest point, which is not the case, but a result of the use of a scaling factor to make the

deformations more visible. The maximal occuring Von Mises stress σv is 177.7 MPa, and

is derived from the stress tensor using Equation 5.1. If the Von Mises stress is higher than

the material yield strength, plastic deformation will occur. For typical steels, the yield

stress is 250 MPa, so the wedge will not yield during the analysis.

σv =

√(σ11 − σ22)2 + (σ22 − σ33)2 + (σ33 − σ11)2 + 6σ212 + 6σ223 + 6σ223

2(5.1)

71


(a) t after impact=0.25 ms

(b) t after impact=0.50 ms

(c) t after impact=0.75 ms

(d) t after impact=1.00 ms

Figure 5.2: Deformation of a steel wedge. α = 10

Pressure

The pressure is expected to be somewhat lower due to deformation [2]. Figure 5.3 shows

the pressure on a path along the wedge at two different times after impact. For the first

timestep, a maximum pressure of 8.64 bar is obtained, while for the second output frame

the maximum pressure is 8.62 bar. This is indeed lower than the maximum pressures

obtained for a rigid wedge. However, sometimes the pressure reaches an extremely high

value in a single element due to a numerical instability, as can be seen in Figure 5.4.

72


0 0.005 0.01 0.015 0.02 0.025−6

−4

−2

0

2

4

6

8

10

Distance [m]

Pressure [bar]

t = 0.3 mst = 0.47 ms

Figure 5.3: Pressure on a path along the wedge for two different output frames. α = 10

Figure 5.4: Numerical instability for the impact of a deformable wedge, 0.025 ms after impact.

α = 10

For a wedge with deadrise angle α = 10, the implementation of air should not cause a

big difference in pressure, because the deadrise angle is too large to see a large amount of

cushioning, and the deformations are relatively small. However, as seen for rigid bodies,

the air does have an influence on the velocity of the wedge, due to the rough friction

73


between the wedge and the air. Here, this is also the case, as can be seen in Figure

5.5. Moreover, it must be noted that the diffference in velocity is larger than for rigid

wedges due to deformation (see also Figure 4.14), so the pressure should be lower when

air is implemented. In Figure 5.6 the pressure on a path along the wedge is shown, for

t after impact = 0.47 ms, just as in Figure 5.3. With air, the maximum pressure is 5.27

bar, against 8.62 bar when air is not implemented. The difference between the maximum

pressures with and without air is larger when deformation is taken into account, meaning

that the deformation does not allow the air to squeeze as easily as for rigid bodies. This

shows that the air is modelled rather stiff.

0 0.2 0.4 0.6 0.8 1 1.2 1.4

x 10−3

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

Time [s]


No airAir

Figure 5.5: Vertical velocity for a deformable wedge

74


0 0.005 0.01 0.015 0.02 0.025 0.03−1

0

1

2

3

4

5

6

Distance [m]

Pressure [bar]

Figure 5.6: Pressure on path along wedge when air is implemented. t after impact = 0.47 ms

5.3 Cone

5.3.1 Introduction

The cone is made of steel plate with thickness=1 mm for the same reason as for the

wedge, and the domain and boundary conditions can be seen in Figure 5.7. The boundary

conditions are the same as for the rigid cone described in Paragraph 4.3.1, and the same

element type as for the wedge is used (see Paragraph 5.2.1). To be able to compare the

obtained results with the work of Peseux et al. [2], the impact velocity V is set to 2.5

m/s.

75


Figure 5.7: Domain with boundary conditions for calculation on deformable cone

5.3.2 Results

Pressure

Figure 5.8 shows the pressure evolution in the element where the first impact takes place

for three different values of the deadrise angle α, where the dotted line represents the

moment of impact. To make the image more orderly, the pressure curves were given a

certain offset along the x-axis. The resulting maximum pressures are given in Table 5.1,

where they are compared with experimental and numerical results obtained by Peseux

[2]. For their experiments, they also used a deformable cone with thickness=1 mm. The

maximum pressures obtained with Abaqus™ are very high compared with these results.

However, it must be noted that the highest pressure which can be seen in Figure 5.8 are

not appearing at first impact, but somewhat later, and are reached in a single timestep

only. Furthermore, for α = 6, there are two peaks visible, which could be caused by a

numerical instability. When looking at the pressures at first impact, they are lower and

correspond more with the results of Peseux, but are still on the high side.

76


0 1 2 3 4 5 6 7

x 10−4

−8

−6

−4

−2

0

2

4

6

8

Time [s]

Pressure [bar]

! = 6°! = 10°! = 14°

Figure 5.8: Pressure evolution in element where first impact of a deformable cone occurs, for

different values of α

Table 5.1: Comparison of results by Peseux [2] and calculated results for deformable cone

α Peseux Experimental [bar] Peseux Numerical [bar] Abaqus™[bar]

6 1.4 2.2 6.67

10 1.2 1.3 4.83

14 0.6 0.6 3.66

77


5.4 Cylinder

5.4.1 Introduction

The same cylinder is used as in Chapter 4, but is now modelled as a deformable solid.

Calculations with a deformable shell (element type S4R [27], element size 0.5 mm) were

also performed, but no significant difference in the results arose. The impact velocity of

the cylinder is 5 m/s, the outer diameter D=300 mm, and only half of the cylinder is

modelled, using symmetry boundary conditions. 4 layers of linear brick elements with

reduced integration (C3D8R) were used to model the cylinder [27]. The shortest edge of

the element is 0.5 mm.

5.4.2 Results

Deformation

First, the deformation of the cylinder is studied. The cylinder is made out of steel,

modelled as a linear elastic material (ρ = 7800 kg/m3, E = 210GPa, ν = 0.3). The

deformation througout the analysis can be seen in Figure 5.9. Note that the deformation

is scaled with factor 20 to make it more visible. As expected, the deformation is mostly

concentrated in the area where slamming takes place, because only a small part of the

cylinder experiences the opposing force of the water.

78


(a) before impact (b) first impact (c) t after impact=0.25 ms

(d) t after impact=0.75 ms (e) t after impact=1.25 ms (f) t after impact=1.75 ms

Figure 5.9: Deformation of a steel cylinder when slamming on a water surface. Scaling factor=20

The distance the cylinder has travelled is shown in Figure 5.10. Because the cylinder

is deformable, both the bottom and top displacements are shown. Before the impact,

both move at the same vertical velocity because the cylinder is in a free fall. At impact,

79


the bottom is slowed down first, and the top part still moves further down. The relative

deviation of the diameter (DOD) is shown in Figure 5.11, and is calculated with Equation

5.2. When the impact occurs, the diameter changes in a linear way. At the end of the

calculation, 2.2 ms after impact, the relative deformation is 1.29%.

DOD =original diameter − deformed vertical diameter

original diameter· 100 (5.2)

0 0.5 1 1.5 2 2.5

x 10−3

0

0.002

0.004

0.006

0.008

0.01

0.012

Time [s]

Distance [m]

Top

Bottom

Figure 5.10: Displacement for both the top and the bottom of the cylinder

80


0 0.5 1 1.5 2 2.5

x 10−3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [s]

DOD [%]

Figure 5.11: Procentual deviation of the original diameter for an impacting cylinder

Pressure

For the pressures, a comparison is made with the work of Battistian and Iafrati [3], of

which the results can be seen in Figure 4.34. The dimensionless pressures obtained with

Abaqus™ are shown in Figure 5.12.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−40

−20

0

20

40

60

80

100

120

140

160

y/R

Cp

z/R = 0.01z/R = 0.05

Figure 5.12: Dimensionless pressure for deformable cylinder

81


For z/R=0.01, a maximum Cp of 141.51 is obtained, which is rather high, but still lower

than for a rigid cylinder. The explanation for this can be found when looking at the vertical

velocity of the cylinder (Figure 5.13) and the actual pressure (Figure 5.14); to calculate

Cp, the velocity of the bottom of the cyilinder was used. As can be seen in the velocity

plot (Figure 5.13), the velocity drops suddenly at impact due to the deformation of the

cylinder, resulting in v = 3.67m/s for z/R = 0.01. This velocity is lower when compared

to the velocity of the rigid cylinder for the same submersion, and as a result the pressure

is much lower after the first impact. When looking at the actual pressure (Figure 5.14), it

can be seen that the actual pressure is indeed lower than in the case of a rigid impact. For

both values of the dimensionless submersion (z/R = 0.01 and z/R = 0.05, Figure 5.12),

the shape of the curve does not correspond as wel with the results in Figure 4.34. There

is a lot of scatter in the pressure, and two pressure peaks are visible. For z/R = 0.05, the

high value of Cp can be explained by the same considerations as for z/R = 0.01, and it

can be concluded that the maximum pressure is lower than for a rigid cylinder, but still

are very conservative, just as for rigid bodies.

0 0.5 1 1.5 2 2.5

x 10−3

2.5

3

3.5

4

4.5

5

5.5

Time [s]


Figure 5.13: Vertical velocity for deformable cylinder

82


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−2

0

2

4

6

8

10

Distance [m]

Pressure [bar]

Figure 5.14: Pressure on a path along a deformable cylinder for z/R = 0.01.

t after impact=0.37 ms

Calculations with air are also performed, with maximum pressures that are somewhat

lower than the pressures obtained with air, but as shown in Chapter 4, this is due to the

lower velocity resulting from the rough interaction between the air and the cylinder. Still,

the pressures are in the same order of magnitute as without air, and are still higher than

the results found in literature.

5.4.3 Composite cylinder

Modelling of the composite lay-up

The point absorbers that will be used in the SEEWEC-project are made out of a composite

material. For that reason, a slamming test is performed on a cilinder made of this material.

The lay-up scheme of the composite (Table 5.3) is the same as used in experimental tests

by Baro and Dervaux [30], performed at the departement of Mechanics of Materials and

Structures. A cylinder with diameter D = 300mm was used in the experiments. The

shell is composed out of 3 layers, and is 3 mm thick in total. The bottom layer is a mat of

short glass fibres, and can be considered isotropic. The next two layers are made of cross

windings of continuous glass fibre, wound under an angle of 70with respect to the the

axis of the cylinder. The mass of the used glass fibre per unit length is 600 TEX, where

1 TEX= 10−6 kg/m. The density of the material ρ = 1900 kg/m3. The properties of the

83


fibre are given in Table 5.2.

Table 5.2: Properties of glass fibre

E1 Elastic modulus parallel to the fibre direction 25.770 GPa

E2 Elastic modulus perpendicular to the fibre direction 6.2519 GPa

ν12 Poisson ratio 0.38

G12 Shear modulus 4.2 GPa



In Abaqus™, this material is modeled as a linear elastic lamina. Only planar stresses are

considered (σ33 = 0). The shear moduli G13 and G23 are required however because they

may be needed for modeling transverse shear deformation in the shell. The poisson ratio

ν21 is simply determined as (E2/E1)ν12. The material properties for the isotropic fibre

mat are not known, but are attributed to the cylinder as a whole. Because the windings

are crossed, each layer has to be defined twice in Abaqus™, under symmetrical angles. As

can be seen in Figure 5.15, a local cylindrical coordinate system was introduced to define

the material orientations, and eventually the lay-up scheme is given in Table 5.3.

Table 5.3: Modelling of composite lay-up

layer thickness [m] material orientation

1 0.0006 0

2 0.0006 20

3 0.0006 -20

4 0.0006 20

5 0.0006 -20

84


(a) Layer 1 (b) Layer 2 (c) Layer 3 (d) Layer 4 (e) Layer 5

Figure 5.15: Lay-up scheme with fibre directions in local coordinate system

5.4.4 Results

Deformation

The impact velocity V is 5 m/s, just as for the steel cylinder. Figure 5.17 shows the

deformations, with scaling factor 20 for the composite part. Due to the difference in

stiffness of both materials (represented by the elastic modulus E), the deformation is more

pronounced for the composite cylinder. The use of a scaling factor is necessary to display

the deformations, because otherwise the deformations would not be notable. However,

from Figure 5.17 it appears that the most left bottom point is not the lowest point on the

cilinder, which would not be a deformation that is expected physically. When looking at

the deformation without the scaling factor, it is clear that this is not the case, and the

shape of the cylinder after impact is elliptical. The relative deviation of the diameter is

shown in Figure 5.16, where at the end of the simulation it is 2.05%, which is higher than

for the steel cylinder.

85


0 0.5 1 1.5 2 2.5

x 10−3

0

0.5

1

1.5

2

2.5

Time [s]

DOD [%]

Figure 5.16: Relative deviation of the diameter for a composite cylinder

86


(a) before impact (b) first impact (c) t after impact=0.25 ms

(d) t after impact=0.75 ms (e) t after impact=1.25 ms (f) t after impact=1.75 ms

Figure 5.17: Deformation of composity cylinder when slamming on a water surface. Scaling

factor=20

87


Pressure

Figure 5.18 shows Cp for different values of the dimensionless arc length y/R. The max-

imum pressure coefficient Cp is 411.48, which is very high, but the shape of the curve

resembles quite well with the results by Battistin and Iafrati [3] (Figure 4.34). The expla-

nation for this result is the same as in previous for the steel cylinder: when the dimension-

less submersion is z/R = 0.01, the vertical velocity v is only 1.83 m/s, while the actual

pressure is 6.89 bar. Due to the higher level of deformation, this velocity is lower than for

the steel cylinder. This is the cause of the high pressure coefficient (Cp ∼ 1/v2), rather

than a high value for the maximum pressure itself.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−50

0

50

100

150

200

250

300

350

400

450

y/R

Cp

Figure 5.18: Dimensionless pressure for the impact of a composite cylinder. z/R = 0.01

Failure of the material

To determine if the material will fail under impact, the Tsai-Wu failure criterion, which

is a stress based failure criterion, is implemented. A combination of stresses in different

directions may not exceed 1 in order no to fail. The expression for the Tsai-Wu criterion is

shown in Equation 5.3, and can be interpreted as a surface in three dimensions around the

origin defined by σ11, σ22 and σ12. When a certain stress state σ11, σ22, σ12 is outside of

this surface, failure occurs.

IF = F1σ11 + F2σ22 + F11σ211 + F22σ

222 + F66σ

212 + 2F12σ11σ22 < 1 (5.3)

F1, F2, F11, F22, F66 and F12 are determined by the following expressions:

88


F1 =1

Xt+

1

Xc(5.4a)

F2 =1

Yt+

1

Yt(5.4b)

F11 = − 1

XtXc(5.4c)

F22 = − 1

YtYc(5.4d)

F66 =1

S2(5.4e)

F12 =1

2σ2biax

[1−

(1

Xt+

1

Xc+

1

Yt+

1

Yc

)σbiax +

(1

XtXc+

1

YtYc

)σ2biax

](5.4f)

(5.4g)

or

F12 = f√F11F22 (5.4h)

Xt and Yt are the tensile strength limits in the 1- and 2-direction, Xc and Yc are the

compressive stress limits in the 1- and 2-direction and S is the maximum shear tension in

the 1-2 plane. If the equibiaxial stress at failure σbiax is known, Equation 5.4f is used to

calculate the coupling term F12. When it is not known, Equation 5.4h can be used, where

f is a coefficient with values between -1 and 1. However, it is generally accepted that the

influence of F12 can be neglected [31]. In Abaqus™, both σbiax and f are set to zero. The

values of the stress limits used here are found in [30]: Xt and Xc are 600 MPa, Yt and Yc

are 60 MPa, and S is 50 MPa.

The stresses for the top and bottom layer are shown in Figure 5.19 and 5.20, both in the

direction of (σ11) as well as perpendicular to the fibre orientation (σ22). The shear stress

(σ12) for both layers can be seen in Figure 5.21. In order to visualize the stresses, the

cylinder was stretched in the z-direction, with a scaling factor 500. The time after impact

is 2 ms. The top and bottom layer are shown because these are most crucial, since the

cylinder experiences bending. Figure 5.22 shows the values of IF for the Tsai-Wu failure

criterion. The bottom region of the cylinder that is exposed to the slamming force appears

to most crucial, but the value for IF remains well below 1, meaning that failure will not

occur because the stresses are too low to cause any damage.

89


(a) bottom layer (b) top layer

Figure 5.19: Stress in the fibre direction (σ11) for the bottom and top layer


Figure 5.20: Stress perpendicular to the fibre direction (σ22) for the bottom and top layer

90



Figure 5.21: Shear stress (σ12) for the bottom and top layer


Figure 5.22: Tsai-Wu failure criterion

5.5 Conclusion

The maximum pressure for a deformable wedge is slighty lower when compared to a rigid

wedge. However, the influence of air is larger altough the deformation is small, which can

91


be atributed to the fact that the air is modelled too stiff, and that the drag caused by the

air is higher than expected physically.

For the cone, the maximum pressure is higher when compared to experimental results,

but this may be caused by numerical instabilities causing a high pressure in a single

timestep, because the pressure at first impact is lower. However, it is still higher than the

experimental results found in literature.

Both steel and a composite material were used to model the cylinder. In both cases, the

pressure is lower than for rigid bodies, but is still much higher than numerical results found

in literature. The deformation is rather small, but larger for the composite cilinder, caused

by a smaller stiffnes of the material. Moreover, the deformation has a large influence on

the velocity of the cylinder. Both materials, however, will not fail under impact.

92

Chapter 6

Conclusion

In this thesis the slamming of an object, both rigid and deformable, on a flat water surface

was studied using an Euler-Lagrange coupling in Abaqus™. This feature is rather new in

Abaqus™, and it’s performance for slamming simulations was not yet known.

First, a theoretical explanation of the model parameters was given. With only one way

to implement fluids, the number of options that Abaqus™ has to offer to model fluids is

limited. Also, only one element is available in the element library to mesh an eulerian

domain.

Next, the results of the analysis were presented for both rigid and deformable bodies.

Pressures, water flow, deformations and stresses were studied, and compared with results

found in international literature.

The wedge is the most fundamental geometric shape to perform a calculation with. The

results match quite well with results found in literature, but the difference becomes larger

as the deadrise angle decreases. For a cone, the pressures were overestimated in the

calculations. Moreover, a large amount of oscillation is present in the pressure results, and

sometimes the pressure reaches a very high value in a single element, which is probably

caused by a numerical instability. Large negative values of the pressure occur, because

a shockwave is propagated through the fluid, which is not seen in literature. The origin

of this shockfront is located in the point where the object is entering the water at that

moment.

In case of a cylinder, which is more complex because the deadrise angle starts at 0and

eventually becomes larger, the pressures are largely overestimated. This is also seen for a

sphere, altough the pressure here is lower due to threedimensional effects. The influence of

using symmetry is studied, and it can be concluded that symmetry is useful, because it is

not always possible to create mesh made of eulerian brick that is completely symmetrical as

93

Chapter 6. Conclusion

well. The formation of the jet along the cylinder corresponds quite well with experimental

results, but the amount of spray is lower, because very small droplets are not resolved in

the mesh.

The same conclusions can be made for the flowfield when a deformable object slams on the

water, altough the actual pressures are somewhat lower than in the case of rigid objects,

which is physically correct. The deformation was studied for both steel and a composite

material. In both cases, the deformation is rather small, and no damage in the material

occurs.

Interaction between different eulerian domains is not possible in Abaqus™, so when air is

implement, both water and air must be defined in the same eulerian domain. Because the

interaction between the impacting object and the water must be rough, interaction between

air and the object is also rough. As a result, the drag force of the air is overestimated during

free fall, leading to a lower impact velocity and a lower maximum pressure. Moreover, air is

modelled too stiff, which is notable when the impact of a deformable object is considered.

The maximum pressures, however, are still higher than values found in literature.

In general, it can be concluded that the pressures are overestimated, and that the results

are hard to interpret due to the appearance of numerical instabilities in a single element.

94

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97

List of Figures

1 Overzicht van rekendomein voor wig . . . . . . . . . . . . . . . . . . . . . . ix

2 Dimensieloze druk voor cilinder bij verschillende waarden van z/R . . . . . x

3 Dimensieloze druk voor bol bij z/R = 0.01 . . . . . . . . . . . . . . . . . . . xi

4 Zeer hoge druk in element door numerieke instabiliteit. Niet-uitgemiddelde

waarden. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

5 Druk in element waar eerste impact plaats heeft, voor verschillende waarden

van α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

6 Vervorming voor stalen en composiet cilinder. t na impact is 1.25 ms . . . . xv

7 Procentuele afwijking van originele diameter . . . . . . . . . . . . . . . . . . xv

8 Dimensionless pressure for deformable cylinder . . . . . . . . . . . . . . . . xvi

9 Tsai-Wu failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1.1 WEC platform concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Slamming of composite hull during the 2008 Volvo Ocean Race . . . . . . . 3

1.3 Pressure distribution on an impacting wedge [6] . . . . . . . . . . . . . . . . 4

2.1 Definition of the deadrise angle α . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Slamming definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Solution for a wedge obtained by Mei for several deadrise angeles. Com-

parison with the similarity solution obtained by Dobrovol’Skaya. . . . . . . 9

2.4 Fluid domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.5 Boundary conditions in computational domain. . . . . . . . . . . . . . . . . 12

2.6 Cut-off jet in computational domain . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Definition of different domains in computational domain . . . . . . . . . . . 13

2.8 Comparison between cone and wig with same deadrise angle of 10at 4

different timesteps. Impact velocity is 6m/s . . . . . . . . . . . . . . . . . . 14

2.9 Air gets trapped in concave cavities when the impacting object deforms.

The air is compressed and the object slows down. . . . . . . . . . . . . . . . 15

2.10 Structural domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

98

List of Figures

2.11 Contact algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.12 Steps taken to calculate Volume Fraction F . . . . . . . . . . . . . . . . . . 19

2.13 Comparison between the SLIC and PLIC surface reconstruction algorithms 20

2.14 Local height function h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.1 Drop test with wooden sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1 Meshed wedge, deadrise angle = 10 . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Boundary conditions for wedge . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Boundary conditions for eulerian domain . . . . . . . . . . . . . . . . . . . . 34

4.4 Overview of model domain with applied boundary conditions . . . . . . . . 35

4.5 Path for visualisation of the pressure along the wedge . . . . . . . . . . . . 35

4.6 Pressure along wedge with deadrise angle 10for different timesteps . . . . . 36

4.7 Pressure in three different elements with respect to time. Deadrise angle=10 37

4.8 Pressure for t = 0.595 ms after impact. The results are not averaged . . . . 38

4.9 Evolution of pressure part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.10 Evolution of pressure part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.11 Pressure on a path along the wedge for different deadrise angles. t after

impact=0.15 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.12 Pressure evolution in element where first impact occurs for different deadrise

angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.13 Pressure history in element located 3 mm from axis of symmetry. α = 14 . 43

4.14 Vertical velocity for model with and without air. α = 10 . . . . . . . . . . 44

4.15 Pressure along wedge for two different timesteps. α = 10 . . . . . . . . . . 45

4.16 Horizontal velocity of air for two different timesteps. α = 10 . . . . . . . . 46

4.17 Boundary conditions for cone . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.18 Boundary conditions for eulerian domain . . . . . . . . . . . . . . . . . . . . 48

4.19 General view of the cone model with applied boundary conditions. . . . . . 48

4.20 Pressure evolution in element were first impact takes place for two different

impact velocities. α = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.21 Pressure on a path along the cone for two different impact speeds. α = 10 50

4.22 Vertical velocity of rigid cone. α = 10 . . . . . . . . . . . . . . . . . . . . 51

4.23 Experiments by Peseux et al.[2] . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.24 Boundary conditions for cylinder . . . . . . . . . . . . . . . . . . . . . . . . 53

4.25 Circular path to visualise pressure along cylinder . . . . . . . . . . . . . . . 54

4.26 Pressure on a path along the cylinder for three different timeframes . . . . . 55

4.27 Pressure history for two different elements . . . . . . . . . . . . . . . . . . . 55

99

List of Figures

4.28 Evolution of the pressure in the water domain for the impact of a 2D rigid

cylinder, part 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.29 Evolution of the pressure in the water domain for the impact of a 2D rigid

cylinder, part 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.30 Very high pressure in a single element caused by a numerical instability.

Non-averaged values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.31 Pressure along cylinder for two different output frames when air is imple-

mented . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.32 Pressure history in time for two different elements when air is implemented. 60

4.33 Results of Lin and Shieh [4]. Dimensionless pressure . . . . . . . . . . . . . 61

4.34 Results by Battistin and Iafrati [3]. Left: Free surface for cylinder and

sphere for z = 0.01 and z = 0.05. Right: Pressure coefficient Cp corre-

sponding to the penetration depth on the left. . . . . . . . . . . . . . . . . . 62

4.35 Dimensionless pressure for different dimensionless penetration depths z/R . 63

4.36 Visualisation of the flow field for a cylinder slamming on a flat water surface

obtained by Lin and Shieh[4]. D = 100mm and v = 1m/s. . . . . . . . . . 64

4.37 Volume fractions at different timesteps for an impacting cilinder calculated

with Abaqus™ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.38 Overview of model with boundary conditions . . . . . . . . . . . . . . . . . 66

4.39 Pressure coefficient for rigid sphere. z/R = 0.01 . . . . . . . . . . . . . . . . 67

4.40 Water surface for impact of rigid sphere. Time after impact=0.215 ms . . . 68

5.1 Domain with boundary conditions for calculation on deformable wedge . . . 71

5.2 Deformation of a steel wedge. α = 10 . . . . . . . . . . . . . . . . . . . . . 72

5.3 Pressure on a path along the wedge for two different output frames. α = 10 73

5.4 Numerical instability for the impact of a deformable wedge, 0.025 ms after

impact. α = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.5 Vertical velocity for a deformable wedge . . . . . . . . . . . . . . . . . . . . 74

5.6 Pressure on path along wedge when air is implemented. t after impact =

0.47 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.7 Domain with boundary conditions for calculation on deformable cone . . . . 76

5.8 Pressure evolution in element where first impact of a deformable cone oc-

curs, for different values of α . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.9 Deformation of a steel cylinder when slamming on a water surface. Scaling

factor=20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.10 Displacement for both the top and the bottom of the cylinder . . . . . . . . 80

5.11 Procentual deviation of the original diameter for an impacting cylinder . . . 81

100

List of Figures

5.12 Dimensionless pressure for deformable cylinder . . . . . . . . . . . . . . . . 81

5.13 Vertical velocity for deformable cylinder . . . . . . . . . . . . . . . . . . . . 82

5.14 Pressure on a path along a deformable cylinder for z/R = 0.01.

t after impact=0.37 ms . . . . . . . . . . . . . . . . . . . . . 83

5.15 Lay-up scheme with fibre directions in local coordinate system . . . . . . . 85

5.16 Relative deviation of the diameter for a composite cylinder . . . . . . . . . 86

5.17 Deformation of composity cylinder when slamming on a water surface. Scal-

ing factor=20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.18 Dimensionless pressure for the impact of a composite cylinder. z/R = 0.01 . 88

5.19 Stress in the fibre direction (σ11) for the bottom and top layer . . . . . . . 90

5.20 Stress perpendicular to the fibre direction (σ22) for the bottom and top layer 90

5.21 Shear stress (σ12) for the bottom and top layer . . . . . . . . . . . . . . . . 91

5.22 Tsai-Wu failure criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

101

List of Tables

1 Vergelijking tussen de berekende maximum druk en Vergelijking 1 . . . . . ix

2 Vergelijking van resultaten van Peseux [2] met berekende resultaten voor

vervormbare conus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

3.1 Values of parameters used to model water and air . . . . . . . . . . . . . . . 26

4.1 Comparison of calculated pressure and Equation 4.1 . . . . . . . . . . . . . 41

4.2 Experimental and numerical results by Peseux [2] . . . . . . . . . . . . . . . 52

4.3 Maximum pressure coeffecient for different locations on the cylinder . . . . 61

5.1 Comparison of results by Peseux [2] and calculated results for deformable

cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 Properties of glass fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.3 Modelling of composite lay-up . . . . . . . . . . . . . . . . . . . . . . . . . . 84

102

Documents

Study of simpli ed simulation models for slamming wave ...lib.ugent.be/fulltxt/RUG01/001/418/544/RUG01-001418544_2010_0001... · Slamming, Fluid-Structure Interaction (FSI), Numerical,