calculation of added mass close to seabed

8/11/2019 Calculation of Added Mass Close to Seabed

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Calculation of Added Mass in the

Proximity of the Seabed for an Oscillating

Disc

Magnus Ingvard Rosvoll

Marine Technology

Supervisor: Hvard Holm, IMT

Co-supervisor: Tufan Arslan, IMT

Department of Marine Technology

Submission date: June 2012

Norwegian University of Science and Technology


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Abstract

This thesis presents the goodness of dynamic mesh in Ansys Fluent, through graphical comparison of

the Stokes second problem an oscillating disc. The results have been presented analytically and

numerically.

Simulations of an oscillating disc near the seabed have also been done, in 2D and in 3D. The data are

plotted against Keulegan-Carpenter number. The data have been compared against relevant

literature.


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Acknowledgements

First of all I would thank supervisor Hvard Holm for giving me a formulation of a problem, and

guiding me into the world of CFD. Valuable ideas, suggestions and comments are much obliged. He

has been helpful whenever asked for, and given valuable feedback.

I would also thank co-supervisor Tufan Arslan for all the assistance he has been providing me. He

have always had an open door, and given me relevant literature whenever needed. He also provided

me a basic set-up for a mesh in 3D with features such as sliding mesh and layering.

I would also thank Ragnhild Bjerke for making these two last semesters here in Trondheim

memorable.

Much appreciated are also the help and fruitful discussions given by Rune Bjrkli and Henrik Paulsenregarding Ansys Fluent and CFD.

I would also thank ystein Johannesen for writing a Matlab-script making post-processing for the

Stokes second problem straight-forward.

Last but not least, a big thank you goes out to all of the students at Tyholt giving me five fine years in

Trondheim.


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Contents

Abstract ....................................................................................................................................................ii

Acknowledgements ................................................................................................................................. iii

References ............................................................................................................................................... vi

List of figures ......................................................................................................................................... viii

List of formulas ......................................................................................................................................... x

1. Introduction ..................................................................................................................................... 1

1.1 Motivation and scope of thesis ............................................................................................... 1

1.2 Outline of thesis ...................................................................................................................... 1

2. Theory .............................................................................................................................................. 2

2.1 The Navier-Stokes equation .................................................................................................... 2

2.2 Flow regimes ........................................................................................................................... 2

2.3 Forces on a cylinder ................................................................................................................. 3

2.3.1 Added mass ..................................................................................................................... 4

2.4 Flow around a cylinder in oscillatory flows ............................................................................. 5

3. Pre-processing ................................................................................................................................. 9

3.1 Mesh theory ............................................................................................................................ 9

3.1.1 Sliding mesh ....................................................................................................................... 10

3.1.2 Smoothing methods ...................................................................................................... 11

3.1.2 Dynamic Layering .......................................................................................................... 12

3.1.3 Local remeshing ............................................................................................................. 13

3.2 Turbulence-modeling ............................................................................................................ 13

3.2.1 Reynolds Averaging ....................................................................................................... 14

4. Running .......................................................................................................................................... 16

4.1 Choosing the right solver, convergence criteria and time step. ........................................... 16

5. Post-processing ............................................................................................................................ 18

5.1 Extracting hydrodynamic force coefficients ................................................................................ 18

5.1.1 The power transfer method .......................................................................................... 18

5.1.2 The Least-square fit method...................................................................................... 19

6. Case 1 ............................................................................................................................................ 21

6.1 Results and discussion ........................................................................................................... 22

7. Case 2............................................................................................................................................ 25

7.1 2D-case, results and discussion ............................................................................................. 25

7.2 3D-case, results and discussion ............................................................................................. 29


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8. Discussion and recommendations for further work. .................................................................... 40

Appendix A .............................................................................................................................................. A

Appendix B ...............................................................................................................................................F


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References

(u.d.). Hentet fra http://www.iccbss.net/2002/pdf/February%206/Albert_Cecilia.pdf

Ansys Fluent, T. G. (2012). www.ansys.com.

Barton, I. (1998). Comparison of SIMPLE- and PISO-type algorithms for transient flows. Internationaljournal for numerical methods in fluids.

Bearman, P. (1985). Vortex trajectories in oscillatory flow. Proc. Separated Flow around Marine

Structures.

Chen, F. (2009). Coupled flow discrete element method application in granular porous media using

open source codes. University of Tennessee, Knoxville.

Chenu, B., Vu, K. H., & Thiagarajan, K. P. (u.d.). Hydrodynamic damping due to porous plates.

Faltinsen, O. M. (1990). Sealoads on ships and offshore structures.Cambride University Press.

Faltinsen, O. M., & Timokha, A. N. (2009). Sloshing.Cambridge University Press.

Ferziger, J., & Peric, M. (2002). Computational Methods for Fluid Dynamics.Springer-Verlag.

Hyem Aronsen, K. (2007).An experimental investigation of inline and combined inline and crossflow

vortex induced vibrations, PhD Thesis.Tapir Uttrykk.

Institute, C. S. (u.d.). Hentet fra http://www.iccbss.net/2002/pdf/February%206/Albert_Cecilia.pdf

Jasak, H. (u.d.). OpenFOAM Development: CFD in C++. Hentet fra

http://www.h.jasak.dial.pipex.com/development.html

Justesen, P. (1990). A numerical study of oscillating flow around a circular cylinder.J. Fluid

Mechanics.

Lafforgue, D. (u.d.). Sails: from experimental to numerical. Hentet fra

http://www.finot.com/ecrits/Damien%20Lafforgue/article_voiles_english.html

Lake, M., He, H., Troesch, A. W., & Perlin, M. (2000). Hydrodynamic coefficient estimation for TLP and

Spar structures.ASME.

Morrison, J., O'Brien, M., Johnson, J., & Schaaf, S. (1950). The force exerted by surface waves on piles.

Pettersen, B., & Faltinsen, O. (1987). Application of a vortex tracking method to separated flow

around marine structures.Journal of Fluids and Structures.

Pettersen, B., Ong, M. C., Utnes, T., Holmedal, L. E., & Myrhaug, D. (2010). Numerical simulation of

flow around a circular cylinder close to a flat seabed at high Reynolds number using a k-e

model. Elsevier, Coastal engineering.

Sarpkaya, T. (1986). Force on a circular cylinder in viscous oscillatory flow at low Keulegan-Carpenter

numbers.Journal of Fluid Mechanics, ss. 61-71.


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Sarpkaya, T., & Isaacson, M. (1981). Mechanics of wave forces on offshore structures.Litton

Educational Publishing, Inc.

Sumer, B. M., & Fredse, J. (1997). Hydrodynamics around cylindrical structures.World Scientific

Publishing.

Tao, L., & Cai, S. (2003). Heave motion supression of a Spar with a heave plate. Elsevier, Ocean

engineering.

Tao, L., & Dray, D. (2008). Hydrodynamic performance of solid and porous heave plate. Ocean

Engineering.

Wadhwa, H., Balaji, K., & Thiagarajan, K. P. (2010, June). Variation of heave added mass and damping

near seabed. Proceedings of the ASME 2010 29th International Conference on Ocean,

Offshore and Arctic Engineering.

White, F. M. (2006). Viscous fluid flow.McGraw-Hill.


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List of figures

Figure 1 Description of flow regimes. (Sumer & Fredse, 1997) ............................................................ 3

Figure 2 Various flow-regimes for a smooth circular cylinder in oscillatory flow, Re=10^3. (Sumer &

Fredse, 1997) ......................................................................................................................................... 7

Figure 3 Flow-regimes around a smooth, circular cylinder in oscillatory flow at various Reynolds-

numbers. (Sumer & Fredse, 1997) ........................................................................................................ 8

Figure 4 Sliding mesh with non-conformal mesh (Jasak) ...................................................................... 10

Figure 5 Cell layering to a moving boundary (Ansys Fluent, 2012) ....................................................... 12

Figure 6 Sliding and layering (Ansys Fluent, 2012) ................................................................................ 13

Figure 7 Reynolds-Averaged Navier-Stokes, (Lafforgue) ....................................................................... 14

Figure 8 Stokes' second problem with startup effects .......................................................................... 23

Figure 9 Mesh for Stokes second problem ............................................................................................ 23

Figure 10 Comparison between analytical and numerical solution for stokes second problem .......... 24Figure 11 Mesh for the 2D-case, where a disc is oscillating at a height 2*radius from the ground. .... 25

Figure 12 Close-up for the 2D-case. ...................................................................................................... 26

Figure 13 Screenshot from visualization of oscillating disc at height 2*radius and KC=1.2 at t=4.8125

............................................................................................................................................................... 27

Figure 14 Added mass in 2D for a oscillating disc at 2*radius depth with increasing KC ..................... 28

Figure 15 Screenshot of the velocity-magnitude around the edge of an oscillating edge at height

1*radius and KC=1 ................................................................................................................................. 29

Figure 16 Planar motion mechanism for the heave-motion, (Wadhwa, Balaji, & Thiagarajan, 2010) . 30

Figure 17 3D-case, some 1660000 cells, seen from above ................................................................... 31

Figure 18 3D-case, some 1660000 cells, seen from the side. ............................................................... 31Figure 19 Close up for the mesh with 1660000 cells ............................................................................ 32

Figure 20 3D-case, with some 2450000 cells, seen from above. .......................................................... 32

Figure 21 3D-case, with some 2450000 cells, seen from the side. ....................................................... 33

Figure 22 Mesh convergence-study for an oscillating disc at height 2*radius from the bottom, at

KC=1. ...................................................................................................................................................... 34

Figure 23 Added mass for a oscillating disc at KC=1 at depth 3*radius. (Tao & Dray, Hydrodynamic

performance of solid and porous heave plate, 2008) ........................................................................... 34

Figure 24 Added mass for an oscillating disc at height 2*radius from the bottom, at KC=1. (Wadhwa,

Balaji, & Thiagarajan, 2010) .................................................................................................................. 35

Figure 25 Added mass for an oscillating disc for various KC at 2*radius .............................................. 36

Figure 26 Y-star values for the mesh with 1660000 at height 1*radius from the bottom. .................. 37

Figure 27 3D-added mass for an oscillating disc at height 1*radius for various KC. ............................. 38

Figure 28 Added mass for various heights and various KC, (Wadhwa, Balaji, & Thiagarajan, 2010). ... 39

Figure 29 From The far side by Gary Larsson, (Institute)................................................................... 40

Figur 30 Forces and velocity against time, Forces and acceleration against time, Force against time,

3D-case height 1*radius from bottom at KC=1. 1660k mesh. ................................................................ B

Figur 31 Forces and velocity against time, force and acceleration against time, force against time, for

the 3D-case with 2450k mesh, height 2*radius and KC=1. ..................................................................... C

Figur 32 Forces and velocity against time, force and acceleration against time, force against time, forthe 3D-case with 1660k cells, height 1*radius and KC=1.2. .................................................................... D


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Figur 33 Forces and velocity against time, force and acceleration against time, force against time, for

the 3D-case with 1660k elements, height 2*radius and KC=1.5. ............................................................ E

Figur 34 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.01T*osc .........................................................F

Figur 35 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.1T*osc ...........................................................F

Figur 36 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.2*Tosc .......................................................... G

Figur 37 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.3*Tosc .......................................................... G

Figur 38 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.4*Tosc .......................................................... H

Figur 39 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.5*Tosc .......................................................... H

Figur 40 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.6*Tosc ........................................................... I

Figur 41 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.7*Tosc ........................................................... I

Figur 42 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.8*Tosc ........................................................... J

Figur 43 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.9*Tosc ........................................................... J

Figur 44 Oscillating disc at 1*radius, KC=1, 1660k cells, 1.0*Tosc .......................................................... K


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List of formulas

Formula 1 Navier-Stokes equation .......................................................................................................... 2

Formula 2 Mach number. ........................................................................................................................ 2

Formula 3 Reynolds number ................................................................................................................... 2

Formula 4 Morrisons equation............................................................................................................... 3

Formula 5 Newtons second law............................................................................................................. 4

Formula 6 Total kinetic energy ................................................................................................................ 5

Formula 7 Total kinetic energy ................................................................................................................ 5

Formula 8 Rate of increase of the kinetic energy of the fluid ................................................................. 5

Formula 9 Added mass coefficient .......................................................................................................... 5

Formula 10 Total force acting on a moving cylinder ............................................................................... 5

Formula 11 Keulegan-Carpenter number. .............................................................................................. 6

Formula 12 Sinusoidal velocity ................................................................................................................ 6

Formula 13 Maximum velocity ................................................................................................................ 6Formula 14 Conservation equation for a general scalar on an arbitrary control volume V, whoseboundary is moving. ................................................................................................................................ 9

Formula 15 Time derivative term. ........................................................................................................... 9

Formula 16 nth time level volume .......................................................................................................... 9

Formula 17 The volume time derivative of the control volume ........................................................... 10

Formula 18 The increase of volume by time. ........................................................................................ 10

Formula 19............................................................................................................................................. 10

Formula 20............................................................................................................................................. 11

Formula 21............................................................................................................................................. 11

Formula 22............................................................................................................................................. 11

Formula 23............................................................................................................................................. 11

Formula 24............................................................................................................................................. 11

Formula 25............................................................................................................................................. 12

Formula 26............................................................................................................................................. 12

Formula 27............................................................................................................................................. 12

Formula 28............................................................................................................................................. 14

Formula 29............................................................................................................................................. 14

Formula 30............................................................................................................................................. 14

Formula 31............................................................................................................................................. 14Formula 32............................................................................................................................................. 18

Formula 33............................................................................................................................................. 18

Formula 34............................................................................................................................................. 18

Formula 35............................................................................................................................................. 18

Formula 36 Expression for average power. ........................................................................................... 19

Formula 37 Forces in phase with velocity. ............................................................................................ 19

Formula 38 Forces in phase with acceleration. ..................................................................................... 19

Formula 39 Forced oscillations .............................................................................................................. 19

Formula 40 In matrix form .................................................................................................................... 19

Formula 41 In matrix form .................................................................................................................... 19Formula 42 Minimizing the coefficients ................................................................................................ 20


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Formula 43 Harmonic oscillation........................................................................................................... 20

Formula 44............................................................................................................................................. 20

Formula 45 Forces in phase with velocity ............................................................................................. 20

Formula 46 Forces in phase with acceleration ...................................................................................... 20

Formula 47 Added mass coefficient ...................................................................................................... 20

Formula 48 Excitation coefficient .......................................................................................................... 20

Formula 49 Velocity for an oscillating plate .......................................................................................... 21

Formula 50 Velocity at far field ............................................................................................................. 21

Formula 51............................................................................................................................................. 21

Formula 52............................................................................................................................................. 21

Formula 53............................................................................................................................................. 21

Formula 54............................................................................................................................................. 21

Formula 55............................................................................................................................................. 21

Formula 56 Ordinary differential equation ........................................................................................... 21

Formula 57............................................................................................................................................. 22Formula 58 The final solution ................................................................................................................ 22

Formula 59............................................................................................................................................. 22


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1. Introduction

1.1 Motivation and scope of thesis

The subsea- and offshore industry are facing challenges continuously. As the industry is going

towards deeper waters, accurate knowledge of the forces that comes into play when deploying trees,

manifolds, suction anchors and other structures is imperative. Over prediction of the hydrodynamic

forces results in under-utilization of operating crane vessels, and under prediction can result in

damaging and or losing the structure.

The scope of this thesis is calculating the hydrodynamic forces in phase with acceleration through

CFD, and comparing it with relevant literature.

1.2 Outline of thesis

Chapter 1 is introduction

Chapter 2 is the theory chapter, which defines the most important parameters and flow regimes.

Chapter 3 describes pre-processing, with focus on mesh-generation and theory.

Chapter 4 describes the running of cases with CFD.

Chapter 5 describes the post-processing, how to extract the forces out of a time-series.

Chapter 6 describes case 1, which deals with Stokes second problem.

Chapter 7 describes case 2, an oscillating plate at various heights from the seabed and various

Keulegan-Carpenter numbers.

Chapter 8 is the discussion and further work.


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2. Theory

2.1 The Navier-Stokes equation

To describe the motion of fluid substances, one must resort to the Navier-Stokes equations. The

Navier-Stokes equations are the basic governing equations for a viscous, heat conducting fluid. It is a

vector equation obtained by applying Newtons law of motion to a fluid element, also referred to as

the momentum equation. Further it is supplemented by the mass conservation equation, and the

energy equation. The term Navier-Stokes equations is used to refer to all of these equations.

Formula 1 Navier-StokesequationWhere U is the fluid vector ([u v w]

T), is the differential operator, is the density of the fluid, p is

the pressure, is the kinematic viscosity and g is the acceleration of gravity.If we assume an incompressible flow of a Newtonian fluid this leads to simplifications. The

assumption of incompressibility rules out the possibility of sound or shock waves to occur. The

incompressible flow assumption holds if we are dealing with fluids at low Mach numbers, say M < 0.3

Formula 2 Mach number.

where M is the Mach number, V is velocity of a fluid, and a is the velocity of sound in that medium.

This assumption leads to simplifications to the Navier-Stokes equation, ruling out the first part on the

left hand side inFormula 1.

2.2 Flow regimes

Flow regimes have been classified in numerous manners, but they all resort to the non-

dimensionalised Reynolds-number. The Reynolds-number gives the relation between inertial forces

and viscous forces in the boundary layer. It is defined as

Formula 3 Reynolds number

Where the density for the fluid, U is is the velocity, D is the diameter of our object, is thekinematic viscosity.

A somewhat crude division is found in (Sumer & Fredse, 1997), but one should be aware of that the

divisions of the various regimes into Reynolds number ranges are not definite. Various imperfections

and disturbances may have a profound influence of the flow and change the Reynolds-ranges for

where the various regimes are seen. Such disturbances and imperfections may be surface roughness,

inflow turbulence and various quality of the cylinder, e.g. welding seams etc.


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Figure 1 Description of flow regimes. (Sumer & Fredse, 1997)

2.3 Forces on a cylinder

Because of its central importance in fluid mechanics, the flow about a cylinder and its resulting forces

will be described in the following chapter. The following theory is found in (Faltinsen, Sealoads on

ships and offshore structures, 1990).

If one would calculate the forces on a slender object, Morisons equation (Morrison, O'Brien,

Johnson, & Schaaf, 1950) is often used to calculate wave loads on circular cylindrical structural

members of fixed offshore structures when viscous forces matter.Formula 4 Morrisons equation

reads the horizontal force dF on a strip of length dz of a vertical rigid circular cylinder.

Formula 4 Morrisons equation


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The mass and drag coefficients Cm and Cd have to be empirically determined through the use of CFD

or experimentally, and are dependent of the parameters such as Reynolds number and Keulegan-

Carpenter number. What one must bear in mind is that Morisons equation not at all can predict the

forces orthogonal to the wave propagation direction and in the cross-sectional plane.

When viscous forces effects matter for offshore structures the flow will generally separate. A

consequence of separation is the time-varying pressure-forces due to viscous effects are more

important than shear forces. The lack of a precise definition of what is meant by separation in

unsteady flow (Faltinsen, Sealoads on ships and offshore structures, 1990) makes it somewhat

difficult to compare the different data sets and experiments made by various researchers. For

instance, P.W Bearman reported that nobody had seen vortices at KC-values less than 3. (Bearman,

1985). But one year later, Sarpkaya reported of separation occurring at KC=1.25 (Sarpkaya T. , 1986).

Clearly, vortex shedding is also a matter of Reynolds number, not only KC.

The following section is found in (Sarpkaya & Isaacson, 1981).Given Newtons second law,Formula 5 Newtons second,two of the given three quantities must be

known in order to give the last one. In vacuum, this is straight forward as far as the mass of the

accelerated body is concerned. In a fluid with a given density, this takes a bit more consideration,

since the fluid of the motion as well as of the body itself influences the effective mass.

Formula 5 Newtons secondlaw

The induced mass varies with the instantaneous shape and volume of the wake or cavity as well aswith their rates of change. The magnitude of the added mass or added mass-moment of inertia at a

certain moment depends on the time history of the motion.

2.3.1 Added mass

One easy way of describing the added-mass, is given by (Sumer & Fredse, 1997); The

hydrodynamic mass is defined as the mass of the fluid around the body which is accelerated with the

movement of the body due to the action of pressure.

Another way of describing it, a bit more theoretical is given by (Sarpkaya & Isaacson, 1981).

Consider an unsteady motion of a cylinder normal to its axis in a fluid otherwise at rest. The potential

function may be represented by Formula 6 Potential function, where U is the velocity of the cylinder,

G(r) is of order r-n

(n=2,3,4 ..).

Formula 6.1 Potential function

Then the kinetic energy of the fluid becomes


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Formula 6 Total kinetic energy

Inserting Formula 6.1 Potential function intoFormula 6 one gets

Formula 7 Total kinetic energy

Where dS is an elemental area of the cross section and the integral is over the cylinder surface. The

work done per unit time, which makes power, must equal the time rate of increase of the kinetic

energy of the fluid. Therefore we getFormula 8.

Formula 8 Rate of increase of

the kinetic energy of the fluid

Which could be rewritten further, the added mass coefficient is the ratio of added mass over the

density of the fluid multiplied by the volume of the object,Formula 9.

) Formula 9 Added masscoefficient

The body has a mass Mb of its own and in real fluids it experiences a resistance primarily due to

separation of the flow or the resulting forces and a combination of these two. This is called the form

drag. If one does the assumption of adding the instantaneous values of the drag and inertial forces,

the total force acting on a cylinder moving unidirectional with a time-dependent velocity U(t) turns

out to become

Formula 10 Total force actingon a moving cylinder

In real life, the added mass coefficient and the drag coefficient depends on time, Reynolds number,

Keulegan-Carpenter-number (which will be investigated further in later section) and on the

parameters characterizing the history of motion. The consequences of this are that each data-series

is unique, and must be treated accordingly. As will be shown later, this leads to a time-varying lift-and drag-force.

2.4 Flow around a cylinder in oscillatory flows

Harmonically oscillating bodies with small amplitude have been used for a long time in order to

determine the added mass of bodies of various shapes.

When an object, say a cylinder, is subject to a harmonic flow normal to its axis, the cylinder

experiences a flow accelerating from and decelerate to zero, but the flow changes direction as well

during one cycle. This in turns leads to a reversal of the wake, complicating the inflow-conditionswhenever the velocity changes sign. The boundary layer over the cylinder may undergo changes


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varying from fully laminar to fully turbulent states, and the Reynolds-number may change from sub-

critical to post-supercritical over a given cycle (Sarpkaya & Isaacson, 1981).

The vortices which have been formed or have been shed during the first half of a cycle are being

reversed around the cylinder during the wake reversal, which in turns give rise to a transverse force

with or without vortex shedding. Particularly significant are the changes in the lift, drag and inertial

forces when the reversely-convected vortices are not symmetric.

The following chapter is found in (Sumer & Fredse, 1997).

Say that our cylinder with diameter D is situated in a harmonically oscillating fluid transverse to the

cylinders axis. After some time, when the start-up effects have decayedone can describe the flow

with another parameter than just the Reynolds number. The Keulegan-Carpenter number is defined

by

Formula 11 Keulegan-

Carpenter number.

Where the Um is the maximum velocity and Tw is the period of the oscillatory flow. Say that our flow is

sinusoidal with the velocity given by

Formula 12 Sinusoidal velocity

The maximum velocity then turns out to become

Formula 13 Maximum velocity

Where a is amplitude of the motion.

A physical meaning of the Keulegan-Carpenter number is that small KC-number means that the

orbital motion of the water particles is small relative to the total width of the cylinder. When KC is

very small, say smaller than 4, vortex shedding may not even occur. Any distinct answer to when

separation occurs is hard to give, as stated above, as this is connected to the Reynolds number.

Large Keulegan-Carpenter numbers means on the other hand that the water particles have to travel

large distances relative to the total width of the cylinder.

Figure 2 Various flow-regimes for a smooth circular cylinder in oscillatory flow, Re=10^3. Figure 2

gives the various regimes for flow around a smooth circular cylinder in an oscillatory flow. What one

must bear in mind, that this division is dependent on the Reynolds-number. This can be seen in

Figure 3.

The region divided by a has no separation, it is a creeping flow.The region a has no separation,

but the boundary-layer is turbulent. The region b has separation with Honji-vortices. The region c

has a pair of symmetric vortices. The region d has a pair of symmetric vortices, but turbulence over

the cylinder surface.


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Figure 2 Various flow-regimes for a smooth circular cylinder in oscillatory flow, Re=10^3. (Sumer & Fredse, 1997)


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Figure 3 Flow-regimes around a smooth, circular cylinder in oscillatory flow at various Reynolds-numbers. (Sumer &

Fredse, 1997)


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3. Pre-processing

3.1 Mesh theory

Each simulation with the aid of CFD has the same step one must go through in order to get a

satisfactory result. The steps are pre-processing, running and post-processing. The following chapter

will give a brief introduction into what is happening during pre-processing and simulation when using

dynamic mesh. The following sub-topics are chosen on the basis of what the author self has

encountered.

The dynamic mesh model in Ansys Fluent provides the opportunity to move the boundaries of a cell

zone relative to other boundaries of the zone, and to adjust the mesh accordingly. The motion of the

boundaries can be rigid, like a piston moving inside a cylinder, or a foil changing the angle of attack as

the time goes on, or deforming, such as the elastic wall of a balloon during inflation. Either way, the

nodes that define the cells in the domain must be updated as a function of time, and hence the

dynamic mesh solutions are inherently unsteady (Ansys Fluent, 2012).

The integral form of the conservation equation for a general scalar,, on an arbitrary control volumeV, whose boundary is moving, can be written asFormula 14.

( )

Formula 14 Conservation

equation for a general scalar on an arbitrary control volume

V, whose boundary is moving.

Where is the fluid density, is the flow velocity vector, is the mesh velocity of the moving mesh,is the diffusion coefficient and is the source term of. Now we must discretize this continuousequation into discrete equations.

If we use a backward-facing difference formula, the time derivative term inFormula 14,writes out

like this;

Formula 15 Time derivative

term.

The n+1th time level, VN+1

, is computed from

Formula 16 nth time level

volume

Where is the volume time derivative of the control volume. We must satisfy the mesh

conservation law, so the volume time derivative of the control volume is computed from following

relationship


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Formula 17 The volume time

derivative of the control

volume

Where is the number of faces on the control volume and is the j area face vector.

Formula 18 The increase of

volume by time.

Where is the volume swept over by the control volume face j over the time step .

3.1.1 Sliding mesh

The sliding mesh feature gives you the possibility to set up a problem where separate zones move,

rotationally or translational, relative to each other. The nodes move rigidly in a prescribed dynamic

mesh zone given by the user.

Additionally, multiple cells zones are connected with each other through non-conformal interfaces.

Since the mesh motion is updated in time, the non-conformal interfaces are likewise updated to

reflect the new positions each zone. SeeFigure 4

Figure 4 Sliding mesh with non-conformal mesh (Jasak)

The general conservation equation formulation given for dynamic meshes above, seeFormula 14

Conservation equation for a general scalar on an arbitrary control volume V, whose boundary ismoving.Formula 14,is also valid for sliding meshes. Since the mesh motion in our dynamic sliding

mesh formulation is rigid, all cells have their original shape and volume. This gives us, that the time

rate of change of the cell volume is zero, andFormula 16 is simplified to the following;

Formula 19


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AndFormula 15 leads to the following simplification;

Formula 20

Finally,Formula 20,leads to the following simplification

Formula 21

A consequence to this dynamic mesh is that the solutions forFormula 14 for sliding meshes are

inherently unsteady, as all dynamic meshes are.

3.1.2 Smoothing methods

The following part is found in the User Manual for Ansys Fluent.

If one chooses the spring-based smoothing, one can look upon the edges between any two mesh

nodes as idealized network of interconnected springs, where the initial spacings of the edges before

any mesh motion is the equilibrium state of the mesh. Hooks law is applied in order to find the

forces proportional to the displacement along all the springs connected to the node.

Hooks law can be written like

Formula 22

Where and are the displacements of node i and its neighbor j, is the number ofneighboring nodes connected to node i, and is the spring stiffness between node i and theneighbor j.

The spring stiffness for the edge connecting nodes i and j is defined asFormula 23,

| |Formula 23

When equilibrium occurs, the net force on a node due to all the springs which are connected to the

node must equal zero. This condition gives an iterative equation,

Formula 24


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Since we already know the displacements at the boundaries,Formula 24 is solved with a Jacobi-

sweep on all the interior nodes. When convergence occurs, the positions are updated giving us

Formula 25

Where n+1 and n are used to denote the positions at the next and current time step, respectively.

In the GUI in Ansys Fluent, you can prescribe the spring constant factor.

3.1.2 Dynamic Layering

If the mesh consists of hexahedral and/or wedge mesh zones, one can use dynamic layering to add

and remove cell-layers based on a preset height of the layer adjacent to the moving surface. The

layers which exceed this preset height merge or collapses into the next cell layer, seeFigure 5 Cell

layering to a moving boundary

Figure 5 Cell layering to a moving boundary (Ansys Fluent, 2012)

If the cells in the layer j are expanding, the cell height is allowed to expand to increase a certain limit

is met,Formula 26.

Formula 26

Where is the minimum cell height of the cell layer j, is the ideal cell height, and is thelayer split factor.

On the other side, if the cells in the layer j are being compressed, they can be compressed until a

certain level is met,

Formula 27Where is the layer collapse factor. SeeFigure 6 to get an understanding of what is happening.In most cases, mesh sliding and layering is superior to other dynamic meshing, when it comes to less


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numerical diffusion, time consumed updating the mesh. And the fact that if the mesh consists of

quads and hexes, a smaller number of faces or cells are needed, compared to tris and tets.

Figure 6 Sliding and layering (Ansys Fluent, 2012)

3.1.3 Local remeshing

On zones with a triangular or tetrahedral mesh, the spring-based smoothing method is often used.

When the boundary displacements are large, at least if one compares it with the local cell sizes, the

quality of the cell tends to deteriorate. This will eventually invalidate the mesh, and give the user a

non-satisfactory result. In order to solve this problem, Fluent marks the cell which violates the preset

skewness or size-criteria. Next Fluent locally remeshes the agglomerated cells or faces, given that the

new cells or faces satisfy the skewness criteria. Otherwise the new cells or faces are discarded.

Fluent evaluates each and every cell out of the following criteria; maximum skewness, smaller than a

specified length, larger than a specified length and that its height does not meet the specified length

scale at moving face zones.

3.2 Turbulence-modeling

Turbulent fluid motion is an irregular behavior of a flow where the various quantities such as

velocity, pressure, concentration, temperature etc., show a random variation with time and space

coordinates. The most accurate approach to turbulence simulation is to solve the Navier-Stokes

equations without averaging or approximation other than numerical discretizations whose errors can

be estimated and controlled (Ferziger & Peric, 2002).

In order to assure that all of the significant structures of the turbulence have been captured, the


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domain must be at least the size of the physical domain or the largest turbulent eddy. A valid

simulation must capture all of the kinetic energy dissipation.

This is however not feasible for the nearest future, given the huge amount of time needed. As of

now, the largest DNS-simulation done used 40963

mesh points, which isnt very much. This

introduces the necessity for modeling turbulence.

3.2.1 Reynolds Averaging

When one chooses a RANS-turbulence model, say k-or k-, the solution variables in the

instantaneous Navier-Stokes equation are decomposed into the mean and fluctuating components

(Ansys Fluent, 2012), seeFigure 7.

Figure 7 Reynolds-Averaged Navier-Stokes, (Lafforgue)

For the velocity components, this gives usFormula 28,

Formula 28Where and are the mean and fluctuating velocity components, where i=1,2,3 indicatingdirection.

The same principle counts for pressure and other scalars,Formula 29.

Formula 29Where denotes a scalar such as pressure, energy or a concentration of a species.If we substitute the expression given inFormula 28 into the instantaneous continuity and

momentum equations, and taking the time-averaged, this yields averaged momentum equations.

Formula 30

Formula 31


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The equations given inFormula 30 andFormula 31 are called the Reynolds-Averaged Navier-Stokes

equations. Additional terms do now appear compared to the traditional Navier-Stokes equation

given inFormula 1 Navier-Stokes equation.These terms, the Reynolds stresses , must bemodeled in order to closeFormula 31.

3.2.1.1 Choosing the proper turbulence-model

After discussion with co-supervisor Tufan Arslan, the choice fell on k-with Shear-Stress Transport(SST) for turbulence modeling regarding case two. A brief description found in the User Guide for

Ansys Fluent is given below.

The SST-version of the k-was developed by Menter. The target were to effectively blend the robustand accurate formulation of the k-model in the region near wall, with the free-streamindependence found in k- model in the far field. The features implemented by Menter makes the k-

SST-model more accurate and reliable for a wider specter of flows, such as for airfoils, transonic

shock waves and adverse pressure gradient flows. Further elaboration into the turbulence modeling

is not presented in this thesis, as this is out of this thesis scope.


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4. Running

4.1 Choosing the right solver, convergence criteria and time step.

The fundamental concept of the different schemes like PISO and SIMPLE (pressure-implicit with

splitting of operators and semi-implicit method for pressure-linked equations, respectively), is to

derive a pressure correction equation by enforcing mass continuity over each cell, (Barton, 1998).

According to the User-Guide for Ansys Fluent, steady-state calculation uses in general SIMPLE or

SIMPLEC, while PISO is recommended for transient calculations. PISO is in a larger number of cases

supreme to its competitors in transient simulations, since it has the possibility of having a larger time

step than the other schemes and a stable calculation, (Ansys Fluent, 2012). PISO may also be useful

for steady-state and transient calculations on highly skewed meshes.

Below is a work-chart for the PISO- and SIMPLE-schemes, in pseudo-code (Chen, 2009).

SIMPLE-scheme;

1. Set the initial velocity and pressure field Uo andpo.

2. Solve the under-relaxed discretized momentum equation to compute the tentative velocity

field U*.

3. Solve the pressure correction equation to update the pressure field using under-relaxation

factor p.

4. Correct the velocity field

5. Return to step (1) until convergence.

PISO-scheme

1. Set the initial or old velocity and pressure field Uo andpo.

2. Solve the discretized momentum equation to compute the tentative velocity field.

3. Corrector step 1:

a) Solve the pressure correction equation to update the pressure field.

b) Correct the velocity field

4. Corrector step 2:a) Repeat corrector step 1 with for the pressure correction equation

b) Correct the velocity field using

5. Return to step 1 until convergence.

For both cases, case 1 and case 2, the PISO-solver is the chosen one.

When it comes to choosing convergence-criteria, the default setting in Fluent is 10^-3. This is in

some transient cases, if the mesh is fine enough, not enough. This leads to an erroneous solution,

which ultimately can diverge. For the running of case 1, this default setting is good enough. For

case 2, lowering the convergence criteria to 10^-4 was a necessity.


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One way of judging if a time step is small enough, is to have it at least one order of magnitude

smaller than the smallest time-constant being modeled. Another way, which were the choice the

author fell on, were to have 15-20 iterations for each time-step. This choice were supported by

co-supervisor Tufan Arslan and the User-guide for Ansys Fluent.

Choosing the number of time-steps is a tradeoff between time and having a sufficient amount ofdata to describe the case accurately. For case one, Stokessecond problem, the amount of full

cycles were chosen to be three, in order to avoid the startup-effects. For case two, the optimum

choice would be to have more than 20 full oscillations. This turned out to be not feasible, with

the given computer power available. The author fell on two complete oscillations, which of

course isnt enough to conclude on anything. Nevertheless it gives a trend of what one could

expect of the added mass in the proximity of the seabed.


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5. Post-processing

In order to efficiently process the information gained through simulations done in Ansys Fluent, a

program was written in MATLAB. ystein Johannesen, a fellow student, helped me write a program

which made the post-processing easy and fast for case one. The program reads in informationregarding velocities at various y-coordinates at a certain time-step. This was then organized into two

matrices. The results will be presented in a later chapter. The code will be given in the appendix.

Regarding the post-processing the information retrieved for case number two, this has been done

with basis of what is outlined in (Hyem Aronsen, 2007).

5.1 Extracting hydrodynamic force coefficients

In this section two different methods capable of extracting the hydrodynamic forces in phase with

acceleration and velocity are presented.

5.1.1 The power transfer method

This method considers the power which is transferred between the fluid and the cylinder, i.e. in

other words energy per unit time.

Power transfer, P(t), is given by

Formula 32

Where x(t) is the cylinder motion. is hence the cylinders velocity, and the average powertransfer, , can be written as an integral, summing up all the components,

Formula 33

If we assume that a forced harmonic oscillation results in a harmonic hydrodynamic force, a second

expression for the average power transfer is given below,

Formula 34

Which can be further elaborated into the following,

Formula 35


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This reveals that the first integral equals 0 and the second integral = .This leads us ultimately to the expression for average power,

[ ] Formula 36 Expression for

average power.

FromFormula 33 and the force in phase with the velocity, , is given likethis;

Formula 37 Forces in phase

with velocity.

Using the same idea, but without linking it to a physical quantity as power, we can deduce an

expression for the forces in phase with the acceleration ,

Formula 38 Forces in phase

with acceleration.

5.1.2 The Least-square fit method

This method is also proposed by (Sarpkaya & Isaacson, 1981), but the one presented by (Hyem

Aronsen, 2007) is the one which will be outlined.

The method is based on Matlabs capability to solve matrices fast and efficient. The assumption is still

that the hydrodynamic force may be given as a linear combination of the acceleration and velocity

signal,

Formula 39 Forced oscillationsThis can be reorganized into a matrix C consisting of

and

,

Formula 40 In matrix form

And

Formula 41 In matrix form

The coefficients are then estimated by the minimization of


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Formula 42 Minimizing thecoefficients

The forced oscillations are given as a sinus-equation,

Formula 43 Harmonic

oscillation

Putting this intoFormula 39 Forced oscillations,this leads to

Formula 44

The force components can then be identified as

Formula 45 Forces in phasewith velocity

Formula 46 Forces in phasewith acceleration

Now that the time-varying forces have been figured out, it is time to put them into coefficients.

The added mass coefficient is defined as

Formula 47 Added mass

coefficient

And the dynamic excitation coefficient is defined as

Formula 48 Excitation

coefficient


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6. Case 1

As proposed by supervisor Hvard Holm, a case where the moving mesh-feature in Ansys Fluent

would be validated against a text-book problem was needed. Hvard Holm suggested solving the

Stokes second problem, which have an analytical solution.

Several text-books present this problem, like (Faltinsen & Timokha, Sloshing, 2009), but the one

chosen here is the (White, 2006).

An oscillating stream or wall is often referred to as Stokes second problem. If we look at an

oscillating wall with the velocity

Formula 49 Velocity for anoscillating plate

Where as usual is the circular frequency. The fluid in the far field is at rest,

Formula 50 Velocity at far fieldThe solution for a laminar flow with moving boundaries, can be found if one makes a parallel-flow

assumption of

Formula 51This makes the momentum equation turn out like

Formula 52

The pressure gradient can only be a function of time for this flow, and hence can be absorbed into

the velocity by a change of variables. If we define the velocity like

Formula 53

Then

Formula 54

The steadily solution ofFormula 54,must be of the form

Formula 55Where i= SubstitutingFormula 55 intoFormula 52,gives the ordinary differential equation

Formula 56 Ordinary

differential equation

Where the solution takes the form of


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Formula 57

Now still holding on to the oscillating wall, the final result is

Formula 58 The final solutionWhere

Formula 59

6.1 Results and discussion

The dynamic mesh consists entirely of tris, seeFigure 9,and has a total of approximately 29 000

elements. It is locked onto the oscillating plate. The domain is 10x3 meters. Smoothing and

remeshing were applied. Spring-constant factor equal to 0.3, and the remeshing criteria were

minimum length scale equal to the minimum length found in the mesh, and the maximum length

scale 50% larger than the shortest length scale. Maximum cell skewness were floored to the one

found in the mesh.

The Reynolds-number is kept low; below Re


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Figure 8 Stokes' second problem with startup effects

The second attempt, with some knowledge acquired from run number one, gave some hints of what

the next mesh should have.

The mesh was made in Gambit with scaling functions on the vertical sides, and on the oscillating disc.

A line were created in Ansys Fluent in the center of the mesh, at x=0, with height equal to 1. The

velocities were measured along this line. The boundary-conditions were all symmetry-plane, except

for the disk itself which was wall. The total domain is 10x3 meters, which is the standard unit for

Gambit.

Figure 9 Mesh for Stokes second problem


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The amplitude of the oscillation of the plane was 0.1, and the period was 9 seconds. One can argue

that the plate is oscillating a bit too close to the wall, which would result in fluid streaming over the

disc, but if plotting the velocities measured numerically versus the analytical ones, it isnt that far off,

seeFigure 10.

A suggestion for refinement would be to put more elements higher above the oscillating plate, asthe profiles are more off up there than on the plate itself, which has more elements on it. Another

suggestion for refinement would be to make the oscillating plate shorter, in order to avoid effects

from the symmetry-plane on the sides.

But being happy with the results, bigger challenges awaits.

Figure 10 Comparison between analytical and numerical solution for stokes second problem


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7. Case 2

The second case is where all the features are brought together, like dynamic meshing with sliding

and layering, and eventually extracting the forces in phase with acceleration and velocity. The

objective was to see if the use of CFD with RANS-modeling were capable of calculating the

hydrodynamic forces for an oscillating disc near the seabed is possible.

7.1

2D-case, results and discussion

Starting off with a 2D-case, the sliding and layering functionality were to be tested. Thereafter the

functionality of the post-processing was tested. The mesh is shown inFigure 11.The total domain is

4x1 meters, whereas the disc is 0.4x0.02 meters.

Figure 11 Mesh for the 2D-case, where a disc is oscillating at a height 2*radius from the ground.

The boundary conditions are symmetry plane at the boundaries to the left, right and top, whereas

the oscillating disc and the bottom is wall. Total number of faces is 315620. The distribution of faces

close to the discs ends is shown inFigure 12.


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Figure 12 Close-up for the 2D-case.

Co-supervisor Tufan Arslan gave me the advice to have at least 15-20 nodes at the discs edge. As

shown inFigure 12 this is the case here. The motion of the disc was controlled by a UDFuser

defined function written in C, which is loaded and compiled into Fluent and finally builds an adequate

library. The code can be seen in the appendix.

The turbulence-model was chosen to be the k-with SST. The coefficients were set as default givenby Fluent. The fluid was set to be regular water, which resulted in a Reynolds-number equal to 95448

at KC=1.2. This is in the sub-critical region, as given inFigure 1.

The solver was chosen to be PISO, with second order accuracy for the spatial discretization, but only

first-order implicit transient formulation. This is due to the dynamic mesh which limits the user to

first-order implicit. The residuals-limit were chosen to be absolute and 10^-4 for the continuity,

velocities in x- and y-direction, and for k and omega.

To the authors big disappointment, any comparable data at the given ratio between the discs

diameter and thickness, and the equal Keulegan-Carpenter number, were hard to find.

Recommendations for further work are here obvious. A screenshot were taken at t=4.8125 seconds,

which equals 0.8125*T, where T is the period of one oscillation, seeFigure 13.


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Figure 13 Screenshot from visualization of oscillating disc at height 2*radius and KC=1.2 at t=4.8125

The added mass for the 2D-case is given as

Formula 60

SeeFigure 14 for the results. As stated above, some simulations with a more standard geometry like

a square or a cylinder would be highly beneficial, in order to validate simulation themselves and the

post-processing.


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Figure 14 Added mass in 2D for a oscillating disc at 2*radius depth with increasing KC

One simulation with 5 complete oscillations took 53 hours on a computer equipped with an Intel i7-

2600 processor running at 3.40 GHz and 8 GB of RAM.

Another picture of this 2D-case, although with a finer mesh with 452800 faces, is given inFigure 15.It

would of course be interesting to see some oscillations for this case with that mesh, but due to

convergence-trouble, the time step which would give small enough oscillating residuals to be

somewhat converging within a reasonable number of iteration, were 2.5*10^-4. One complete

oscillation would then take 58 hours, but the simulation was aborted, due to the need of computer-

time for other simulations.


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Figure 15 Screenshot of the velocity-magnitude around the edge of an oscillating edge at height 1*radius and KC=1

7.2

3D-case, results and discussion

Seeing that the sliding and layering features worked out fine, it was time to go 3D.

The simulations have been done with the intention to reproduce the results found in the following

article (Wadhwa, Balaji, & Thiagarajan, 2010). This article investigates the variation of added mass

and damping in heave direction near the seabed as a function of Keulegan-Carpenter number.

The motion is carried out by a planar motion mechanism, as shown inFigure 16.The experiments

were carried out in the Keulegan-Carpenter range of 0.26-1.77. The distance to the seabed ranges

from 0.2a-2a, where a equals radius for the oscillating disc. The total physical domain for the

experiment is 1x1x (0.76-0.84), length, breadth and depth respectively. Why the depth is varying is

not described in the article.


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Figure 16 Planar motion mechanism for the heave-motion, (Wadhwa, Balaji, & Thiagarajan, 2010)

Tufan Arslan helped me to set-up the basics of the mesh, but in the end some tweaking has been

done.

The first mesh has a total of some 1660000 cells; seeFigure 17 andFigure 18.The total domain

equals 10*radius meter of the disc in radial direction, and height equals 0.8 meters. The boundary

conditions are symmetry plane for everything except the oscillating disc and the bottom, which is

wall. The extent of the domain is the minimum of what is acceptable. The dimension of the disc is as

following, radius equals 0.1 meters, and thickness equals 0.002. Any domain-study has not beenperformed.


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The first mesh can be seen from above, seeFigure 17.

Figure 17 3D-case, some 1660000 cells, seen from above

The same mesh as inFigure 17,seen from the side, is given inFigure 18.

Figure 18 3D-case, some 1660000 cells, seen from the side.


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A close up picture of the cell-distribution close to the disc is given inFigure 19.A cut-plane gives the

distribution of cells along this plane. The distribution is equal all the way round.

Figure 19 Close up for the mesh with 1660000 cells

On the other side on the fine-mesh scale, a mesh with 2450000 cells is shown below, seeFigure 20.

One run with 2 complete oscillations, took 56 hours, where the time step were 10^-3.

Figure 20 3D-case, with some 2450000 cells, seen from above.


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The same mesh, seen from the side, is given inFigure 21.

Figure 21 3D-case, with some 2450000 cells, seen from the side.

The mesh convergence-study were done with a Keulegan-Carpenter-number equal to 1, at a height

2*radius from the bottom. SeeFigure 22.As can be seen, the nodes are clustered towards were the

action is, in the area of the oscillating disc. One run on the finest mesh took close to 84.5 hours for 2

complete oscillations with time step equal to 10^-3.

Some nice screenshots can be seen in the appendix for the oscillating disc at height 1*radius from

the seabottom at KC=1, seeFigur 34 toFigur 44.


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Figure 22 Mesh convergence-study for an oscillating disc at height 2*radius from the bottom, at KC=1. The x-axis gives

number of elements in thousands.

The mesh-convergence study reveals that the finest mesh with 2450000 cells gives the solution

closest to the experimental data, seeFigure 23 andFigure 24.

Figure 23 Added mass for a oscillating disc at KC=1 at depth 3*radius. (Tao & Dray, Hydrodynamic performance of solid

and porous heave plate, 2008)

The mesh-convergence study have been performed at KC=1, and with a frequency equal to 1. Thedata represented by the blue line, seeFigure 23,is pretty close to what is given by the finest mesh


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with some 2450000 cells. What one must bear in mind is that data achieved in theFigure 23 have

been performed at a depth equal to 3*radius. This could give the difference in the added mass

coefficient.

Some experimental data have also been produced at a different height, namely 2*radius from the

bottom, seeFigure 24.

Figure 24 Added mass for an oscillating disc at height 2*radius from the bottom, at KC=1. (Wadhwa, Balaji, &

Thiagarajan, 2010)

Unfortunately, simulations have been performed with the coarsest mesh. The reason for this is that

simulations with the finest mesh would spend too much time. Even though 84.5 hours sounds

reasonable for a CFD-simulation of a certain size, this amount of time is if everything runs smoothly.

That isnt always the case, which has been learned the hard way. Another recommendation for

further work is here given. Below is given the added-mass for an oscillating disc at height 2*radius

oscillating at different KC, seeFigure 25.


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Figure 25 Added mass for an oscillating disc for various KC at 2*radius

A substantial increase in the added mass can be seen in the range between KC equal 1 and 1.5 is

seen. If one compare it withFigure 24,over prediction is seen. Reason for this might be, except thelower cell count, the y-star values. Y-star is defined below.

Formula 61

As can be seen inFigure 26,the mesh has quite high y-star values at the bottom of the mesh, y-star

equal to 12, and at the disc the y-star is approximately 3.

According to the User-Guide for Fluent, and discussion with Tufan Arslan, the y-star values should be

in the region 30


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Figure 26 Y-star values for the mesh with 1660000 at height 1*radius from the bottom.


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Added mass for the same oscillating disc at 1*radius depth is given for various KC inFigure 27.

Figure 27 3D-added mass for an oscillating disc at height 1*radius for various KC.

The reason for why this graph only has two points and a line, instead of three points and a curve, is

because of trouble with convergence and lack of time to generate and run a proper simulation.

Another recommendation for further work is her given.

Comparing the data found inFigure 27 andFigure 28,shows a significant difference for the

calculated added mass. The reason for this might be as for height equal to 2*radius, lower quality of

mesh and not good enough y-star.


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Figure 28 Added mass for various heights and various KC, (Wadhwa, Balaji, & Thiagarajan, 2010).


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8. Discussion and recommendations for further work.

Working with CFD isnt always smooth sailing. Before you reach a certain level and gain a certain

amount of experience, various errors and misconceptions may occur from time to time.

Including a comic strip in a Master thesis may seem inappropriate,Figure 29,but I think it sums up

working with CFD; it isnt just something you do right away. The user must be systematic and do

careful planning to succeed. If one doesnt pay attention to the details, you would end up with an

answer which isnt much of an answer.

Figure 29 From The far side byGary Larsson, (Institute)

However, this thesis shows that is feasible to predict the added mass with CFD and Reynolds-

Averaged Navier-Stokes turbulence modeling. But in order to produce satisfactory results there are

some important things to pay special attention to.

First of all sufficient quality of the mesh is priority number one.

Number two is a time step that is small enough to ensure convergence, but large enough to ensure a

simulation as quick as possible.

Number three is to have more than 2 oscillations per simulation. This is clearly not enough to give a

representation of the forces in play. At least 20 simulations would yield a somewhat representative

dataset for the real world behavior.

Number four is to have a small enough Keulegan-Carpenter number, say lower than 1. Any

simulations with a higher Keulegan-Carpenter number than 1, one should investigate the possibility


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to use LES-modeling.


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Appendix A

Following figures gives the velocity, hydrodynamic forces and acceleration as a function of time.


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Figur 30 Forces and velocity against time, Forces and acceleration against time, Force against time, 3D-case height

1*radius from bottom at KC=1. 1660k mesh.


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Figur 31 Forces and velocity against time, force and acceleration against time, force against time, for the 3D-case with

2450k mesh, height 2*radius and KC=1.


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1660k cells, height 1*radius and KC=1.2.


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1660k elements, height 2*radius and KC=1.5.


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Appendix B

Screenshots for the oscillating disc at height 1*radius at KC=1. The mesh is the one with 1660k cells.

Figur 34 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.01T*osc

Figur 35 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.1T*osc


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Figur 36 Oscillating disc at 1*radius, KC=1, 1660k cells, 0.2*Tosc



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calculation of added mass close to seabed

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