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VORTEX INDUCED VIBRATION OF A RIGID SUBSEA JUMPER 2011

OLUFIDIPE OYEYINKA TOPE i

VORTEX INDUCED VIBRATION OF A RIGID SUBSEA JUMPER

OLUFIDIPE, Oyeyinka Tope

MSc Subsea Engineering and Management

(109049739)

“This project is submitted in partial fulfilments of the requirements for the degree of Master

of Science in Subsea Engineering and Management at Newcastle University, Newcastle upon

Tyne.”


OLUFIDIPE OYEYINKA TOPE ii

DECLARATION

I, Olufidipe Oyeyinka Tope, a postgraduate student of the School of Marine Science and

Technology, Newcastle University hereby declare

That this dissertation is my own original work and sourced material have been

referenced therein;

That it has been prepared specifically as part of the requirements of the fulfilment of a

Master Degree of the University of Newcastle and has not been submitted for the

same purpose either in this university or any other.

……………………………………

Olufidipe Oyeyinka Tope

9th

August, 2011


OLUFIDIPE OYEYINKA TOPE iii

ABSTRACT

Vortex Induced Vibration (VIV) plays a very major role in the exploitation and production of

offshore oil and gas reserves. This is due to the interactions that take place between offshore

equipment and their environments. A rigid jumper is a typically a series of short sections of

pipes connected together that span between production equipments offshore.

These spans are exposed to ocean currents from which their interactions with cause them to

experience VIV. The present work seeks to validate the increasing awareness that rigid

subsea jumpers are in fact susceptible to the conditions of VIV.

The present work comprises of a study of the flow around a two – dimensional (2D) smooth

circular cylinder numerically using the 2D Unsteady Reynolds – Averaged Navier Stokes

(URANS) equations with a Shear Stress Transport (SST k-ω) turbulence model; structural

properties (natural frequencies) and a semi – empirical response model in order to obtain the

response of a rigid subsea jumper exposed to a steady current of Reynolds Number in the

sub-critical regime (1.0 × 104 – 1.3 ×10

5).

The amplitude of the in-line response of the jumper was found to be about 10% of the cross

flow response albeit with greater intensities. The numerical study of the flow past the

cylinder also agree remarkably well with experimental data as obtained from literature.


OLUFIDIPE OYEYINKA TOPE iv

ACKNOWLEDGEMENT

I would like to use this medium to acknowledge the support of numerous individuals who

have directly or indirectly contributed to the success of this project.

My sincere appreciation goes to Dr. I.M. Viola for his tutelage, guidance and support in his

capacity as my Project supervisor.

My strong appreciation goes to my parents, and my siblings. This project would not have

been as a huge a success it is was it not for the constant support of my friends. They without

mentioning a long list of names have been a fortress on which I relied on through this

difficult time of putting this together.

Finally, to my colleagues on the MSc Subsea Engineering and Management Programme at

the School of Marine Science and Technology, it has been a wonderful year.

For those too numerous to mention, I would like to say thank you all and God bless.


OLUFIDIPE OYEYINKA TOPE v

TABLE OF CONTENTS

VORTEX INDUCED VIBRATION OF A RIGID SUBSEA JUMPER.................................... i

DECLARATION ....................................................................................................................... ii

ABSTRACT ............................................................................................................................. iii

ACKNOWLEDGEMENT ........................................................................................................ iv

TABLE OF CONTENTS ........................................................................................................... v

LIST OF FIGURES .................................................................................................................. ix

LIST OF TABLES .................................................................................................................... xi

CHAPTER ONE ........................................................................................................................ 1

1.0 INTRODUCTION ....................................................................................................... 1

1.1 PROJECT CASE ......................................................................................................... 2

1.2 PROJECT OBJECTIVES ........................................................................................... 3

1.3 PROJECT STRUCTURE ............................................................................................ 4

CHAPTER TWO ....................................................................................................................... 5

2.0 BACKGROUND ......................................................................................................... 5

2.1 IMPORTANT PARAMETERS .................................................................................. 7

2.1.1 REYNOLDS NUMBER ...................................................................................... 7

2.1.2 REDUCED FLOW VELOCITY ......................................................................... 7

2.1.3 STROUHAL NUMBER ...................................................................................... 7

2.1.3 STABILITY PARAMETER ................................................................................ 8

2.1.4 DAMPING RATIO .............................................................................................. 8

2.1.4 MASS RATIO ..................................................................................................... 9

2.1.5 ADDED MASS .................................................................................................... 9

2.1.6 EIGEN FREQUENCY AND MODE .................................................................. 9

CHAPTER THREE ................................................................................................................. 11

3.0. FLOW PAST A BLUFF BODY ............................................................................... 11

3.1 VORTEX SHEDDING AND VORTEX INDUCED VIBRATIONS (VIV) ........... 12


OLUFIDIPE OYEYINKA TOPE vi

3.2 REYNOLDS NUMBER AND VORTEX SHEDDING ........................................... 13

3.3 FORCES DUE TO VORTEX INDUCED VIBRATIONS ....................................... 15

3.3.1 FLUCTUATING LIFT ...................................................................................... 16

3.4 RESPONSE MODELLING ...................................................................................... 18

3.5 FLOW AND FLOW FIELD MODELLING ............................................................ 19

3.5.1 GOVERNING EQUATIONS ........................................................................ 20

3.6 TURBULENCE ........................................................................................................ 22

3.7 SUBSEA JUMPERS AND VORTEX INDUCED VIBRATIONS .......................... 24

3.7.1 RESPONSE MODELLING OF RIGID JUMPERS .......................................... 25

3.7.2 COUPLED STRUCTURAL AND FLUID ANALYSIS OF A COMPLEX

SUBSEA JUMPER .......................................................................................................... 28

CHAPTER FOUR .................................................................................................................... 30

4.0 INTRODUCTION ..................................................................................................... 30

4.1 NUMERICAL SIMULATION OF THE FLOW PAST A CYLINDER .................. 30

4.2 TURBULENCE MODELLING ................................................................................ 31

4.3 GRID DESIGN ......................................................................................................... 31

4.3.1 NEAR WALL TREATMENT ........................................................................... 33

4.4 GOVERNING EQUATIONS ................................................................................... 35

4.5 BOUNDARY CONDITIONS ................................................................................... 37

4.6 COMPUTATIONAL DOMAIN ............................................................................... 37

4.6.1 DESCRIPTION OF THE FLOW CASE ........................................................... 37

4.6.2 FLUID PROPERTIES ....................................................................................... 38

4.6.3 FLOW PHYSICS ............................................................................................... 38

4.7 CASE SETUP ........................................................................................................... 39

4.8 SOLUTION METHODS........................................................................................... 39

4.9 TIME STEP SIZE CONSIDERATION .................................................................... 40

4.10 POWER SPECTRAL DENSITY .......................................................................... 41


OLUFIDIPE OYEYINKA TOPE vii

4.11 MODAL ANALYSIS ............................................................................................ 42

4.12 RESPONSE MODELLING .................................................................................. 45

4.12.1 IN-LINE RESPONSE MODEL ......................................................................... 47

4.12.2 CROSS FLOW RESPONSE MODEL .............................................................. 49

CHAPTER FIVE ..................................................................................................................... 52

5.0 VERIFICATION ....................................................................................................... 52

5.1 DISCRETIZATION UNCERTAINTY ................................................................. 53

5.2 GRID DEPENDENCE STUDY ............................................................................ 58

CHAPTER SIX ........................................................................................................................ 60

6.0 RESULTS AND DISCUSSION ............................................................................... 60

6.1 INTRODUCTION ..................................................................................................... 60

6.2 MODAL ANALYSIS ............................................................................................... 60

6.3 VORTEX SHEDDING FREQUENCY .................................................................... 62

6.4 EFFECTS OF REYNOLDS NUMBERS ON FLOW PROPERTIES ...................... 67

6.4.1 STROUHAL NUMBER .................................................................................... 67

6.4.2 DRAG ................................................................................................................ 68

6.4.3 RESPONSE MODELLING ............................................................................... 71

6.4.3.1 DNV SCREENING ................................................................................... 71

6.4.3.2 RESPONSE MODELLING ..................................................................... 72

CHAPTER SEVEN ................................................................................................................. 75

7.1 CONCLUSION ......................................................................................................... 75

7.2 RECOMMENDATIONS .......................................................................................... 75

REFERENCES ........................................................................................................................ 76

APPENDIX A – M-SCRIPTS ................................................................................................. 83

A-1 – M - Script for fixed point iteration solution of the order of accuracy ....................... 83

A – 2 – M- Script for Power Spectral Density ..................................................................... 83


OLUFIDIPE OYEYINKA TOPE viii

APPENDIX B – PLOT OF COEFFICIENT OF LIFT AND POWER SPECTRAL

DENSITIES ............................................................................................................................. 84

B-1- COEFFICIENT OF LIFT ............................................................................................ 84

B-2: POWER SPECTRAL DENSITIES ............................................................................. 86


OLUFIDIPE OYEYINKA TOPE ix

LIST OF FIGURES

Figure 1: Typical Rigid Subsea Jumper (Carruth and Cerkovnik, 2007) .................................. 2

Figure 2: Schematic showing the different structural responses to fluid flow .......................... 6

Figure 3: The boundary layer separation due to adverse pressure gradients ........................... 12

Figure 4: The discrete rolling of the vortex as the boundary layer is separated from the

structure (Chen, 1987) ............................................................................................................. 13

Figure 5: Regimes of flow around a smooth, circular cylinder in steady current (Lienhard,

1966, cited in Sumer and Fredsoe, 1999). ............................................................................... 14

Figure 6: Flow Regime Division. Zdravkovich (1990) ........................................................... 15

Figure 7: Variation of the Strouhal number with Reynolds number ....................................... 16

Figure 8: Cross-section of a rigid subsea jumper experiencing VIV ....................................... 19

Figure 9: Flowchart representing the Computational Fluid Dynamics process. ...................... 20

Figure 10: Out of Plane response of the Jumper Spool (translating to In-line VIV) (Carruth

and Cerkovnik, 2007)............................................................................................................... 27

Figure 11: In - Plane response of the Jumper Spool (translating to Cross-flow VIV) (Carruth

and Cerkovnik, 2007)............................................................................................................... 27

Figure 12: The subcritical flow past a cylindrical bluff body (Williamson, 1988, Edited) ..... 31

Figure 13: Fine Grid resolution around the cylinder wall........................................................ 32

Figure 14: Fully Meshed Computational Domain ................................................................... 32

Figure 15: Divisions of the near wall region plotted in semi-log coordinates ......................... 34

Figure 16: Graphical Representation of the Wall y-plus for on the developed grid ................ 35

Figure 17: Computational Domain .......................................................................................... 38

Figure 18: Cross-section of the rigid jumper ........................................................................... 42

Figure 19: Cross section of the ANSI B16:9 long radius elbow.............................................. 43

Figure 20: 3-Dimensional jumper in the ANSYS WORKBENCH environment .................... 44

Figure 21: Coordinate representation of the In-Line response model ..................................... 47

Figure 22: Reduction function with respect to turbulent intensity and flow angle .................. 49


OLUFIDIPE OYEYINKA TOPE x

Figure 23: Coordinate Representation of the Cross-Flow Model (DNV-RP-F105, 2006) ...... 50

Figure 24: Mode shape (1) out-of- plane translating to In-Line VIV ...................................... 61

Figure 25: Mode shape (2) in-plane- translating to Cross- flow VIV; shaded portion

represents the undeformed model ............................................................................................ 62

Figure 26: Coefficient of Lift against flow time (seconds) (Re = 58982) ............................... 63

Figure 27: Coefficient of Lift against flow time (seconds) (Re = 11796) ............................... 64

Figure 28: Power Spectral Density of the Coefficient of Lift (Re = 58982) ........................... 64

Figure 29: Vortex shedding frequencies for observed Reynolds Numbers ............................. 65

Figure 30: Contours of vorticity magnitude (Re = 23,593) at time instant 312 seconds clearly

depicting the 2S mode of vortex shedding- two single vortices are shed at each cylinder

oscillation cycle. ...................................................................................................................... 66

Figure 31: Contours of vorticity magnitude (Re = 106,167) at time instant 312 seconds

depicting the obliteration of the shed vortices due to the turbulent diffusion as described in

Roshko (1952). ......................................................................................................................... 67

Figure 32: Flow past a Cylinder: Variation of Reynolds Number against shedding frequency

.................................................................................................................................................. 68

Figure 33: Unsteady component of drag (Re = 23,593) .......................................................... 69

Figure 34: Present Simulation: Averaged Values of the Coefficient of Drag (Blue); Scatter

Diagram of the Coefficient of Drag of a Cylinder in atmospheric turbulence. (Yuji, 2004)

(Brown) .................................................................................................................................... 70

Figure 35: Pressure distribution across the transverse section of the cylinder in cross flow ... 70

Figure 36: Plot of In-Line Response using obtained Coordinates ........................................... 73

Figure 37: Plot of Cross-Flow Response using obtained Coordinates ................................... 74


OLUFIDIPE OYEYINKA TOPE xi

LIST OF TABLES

Table 1: Specifications of Rigid Jumpers ................................................................................ 25

Table 2: Fluid Properties at 4oC ............................................................................................... 38

Table 3: Describing the percentage occurrence of currents and current velocities in a selected

region offshore West Africa (Hariharan, Cerkovnik and Thompson, 2004). .......................... 39

Table 4: Dimensions of the single length sections of the rigid jumper ................................... 43

Table 5: Refinement factors; do is the distance of the first node to the cylinder wall and to is

the initial time step size as obtained in section 4.9. ................................................................. 52

Table 6: Mesh densities ........................................................................................................... 53

Table 7: Averaged control values as obtained from the simulation of flow past a cylinder (Re

= 11769) ................................................................................................................................... 54

Table 8: Averaged Drag Coefficients as obtained from the simulation of flow past a cylinder

with varying time steps (Re = 11769) ...................................................................................... 57

Table 9: Verification Results ................................................................................................... 58

Table 10: Grid dependence ...................................................................................................... 59

Table 11: Mode Shapes, Natural Frequencies and Effective masses of the Rigid Jumper

Spool (immersed in still water and dry). .................................................................................. 61

Table 12: DNV Screening Criterion ........................................................................................ 71

Table 13: Coordinates of the In-Line Response VIV .............................................................. 73


OLUFIDIPE OYEYINKA TOPE 1

CHAPTER ONE

1.0 INTRODUCTION

Economic development of offshore oil and gas fields require the drilling of oil or gas wells

around a central drill point. This point is often occupied by a subsea facility/equipment which

is used in gathering the produced fluids into a single comingled flow. Alternatively, a subsea

oilfield development may also require the connection between different subsea equipment in

order to improve the economic viability or for some other engineering design purposes.

A subsea jumper refers to that piece of subsea equipment which is used in connecting two

equipment/facilities on the seabed. Examples includes connections between two subsea

Christmas trees, subsea Christmas tree to subsea manifold, or pipeline end terminals just to

mention a few. The choice between a rigid (steel) and flexible (composite) subsea jumper

depends on one or a combination of the following: water depth; pressure; temperature; seabed

conditions and preference of the operator. The rigid subsea jumper is a combination of

cylindrical steel pipes which are welded together to form a singular piece of kit.

The study of the flow around cylindrical cross-section have gained and retained prominence

ever since Da Vinci first recorded the interactions between a fast flowing stream and the

boulders present therein. The exploitation of the offshore oil and gas resources constantly

employs the use of flexible and elongated cylindrical structures. They feature as risers,

offshore platform legs, pipelines, etc.

One of the requirements for the efficient design of structures in the offshore oil and gas

industry is an understanding and predicting the interaction between the structures and its

environment (i.e. winds, waves and currents) all of which are fluids. In the study of the flow

past a cylindrical structure it is observed that the fluid impacts an alternating sequence of

forces to the structure. These forces are known as the drag and lift forces and they excite the

structure into an oscillatory motion. This phenomenon is commonly referred to as fluid

structure interaction.

Interdisciplinary studies are required in the proposition of solutions to the fluid structure

interaction problems experienced in the offshore industry and engineering in broader aspects.

This covers mostly structural mechanics and fluid dynamics. Computational (Numeric) and

experimental solutions to the dynamics of fluid structure interactions remain the best

alternative as analytical solutions are not readily available and are much more complicated. In

relation, the solutions through the experimental setup for oilfield components while



physically feasible in some respect are all together uneconomical. Hence, the most common

solutions employed in the offshore oil and gas industry remain prevalently computational or

semi-empirical through the use of standards which are based on empirical models.

Computational fluid dynamics is now broadly employed in different fields of engineering to

simulate the fluid structural interactions and the suite of ANSYS WORKBENCH which

consists as parts and coupled structural mechanics (MECHANICAL) and fluid dynamics

(FLUENT) was used extensively to study the interactions of fluid flow past the free span of a

rigid subsea jumper.

Subsea pipeline spans occurs due to uneven seabed or the scouring of the soil upon which the

pipe is laid due to its non-cohesive properties of the soil. However in most cases, the subsea

jumper with its free span is more often than not installed at a clearance above the seabed. The

figure below shows a typical subsea jumper as is installed between two subsea Christmas

trees.

Figure 1: Typical Rigid Subsea Jumper (Carruth and Cerkovnik, 2007)

This study is the first step in investigating the coupled fluid structure interaction and its

computation with an industrial computational fluid dynamic code which could then be used

on the complex structure of the subsea jumper.

1.1 PROJECT CASE

In recent times, the oil and gas industry has undertaken numerous researches into the

phenomenon of fluid structure interactions. It should however be pointed out that the

prediction of response amplitude in real world structures is best based on empirical data



(Hariharan, Cerkovnik and Thompson, 2004); the means of obtaining such data is more often

than not prohibitive especially for deep water structures. The numerous researches into this

phenomenon have however been exclusive of subsea rigid jumpers basically because they are

believed to be static structures and hence not susceptible to VIV.

This fact has with improvement of metocean data been rendered invalid due to the fact that in

offshore developments where rigid subsea jumpers are mostly employed there has become

more evidence to point to the existence of bottom ocean currents with speeds capable of

exciting the jumper structure.

However, the geometry of the rigid jumper which also allows it to accommodate large end

displacements from pressure and temperature load cycles makes them susceptible to

excitation by even low velocity currents (Hariharan, Cerkovnik and Thompson, 2004).

The combination of these facts and the non-availability of standards that provide safe

practices into the design and fabrication of subsea rigid jumpers present a viable research

topic into establishing the presence of VIV around subsea rigid jumpers.

1.2 PROJECT OBJECTIVES

The objectives of this dissertation include the following:

Demonstrate a good grasp of the underlying principles of vortex induced

vibrations, computational fluid dynamics and fluid flow around cylindrical cross

sectional structures;

Perform the three dimensional modal analysis of a given rigid jumper in order to

obtain the natural frequencies, modal shapes and corresponding participating

masses of the structure;

Perform the two dimensional flow around a cylinder at specific Reynolds

Numbers in order to determine the vortex shedding frequency of the cylinder and

other flow properties;

Finally, Seek to establish the potential evidence of vortex induced vibrations on a

subsea rigid jumper using semi-empirical models of vortex induced vibrations as

in an industry guideline;

The major advantages of computational fluid dynamics over empirical studies of the

phenomenon of the fluid flow around objects (fluid structure interaction) especially in the

offshore industry includes the ability to simulate flow conditions that not easily reproduced



experimentally; gives more detailed and comprehensive results and information; eliminates

human and instrumental errors in the measurement of characteristic hydrodynamic values;

more cost effective and reduces development time. However it must be noted that

computational fluid dynamics problems and solutions currently surpass most available

computer configurations and the need for validation of its results with empirical data and

these are its major hindrances.

1.3 PROJECT STRUCTURE

This chapter proposes an introduction into the subject matter, it defines the aims and

objectives of the author and briefly introduces the tools required to undertake this

dissertation. Chapter 2 introduces the reader into some of the important parameters and an

understanding of their various meanings. Chapter 3 presents a background study of the

concept of fluid – structure interactions using the circular cylinder as a case study, these

includes the hydrodynamic forces acting on a bluff body, semi-empirical models and their

usage and the inherent concepts that are experienced when a fluid flow past a bluff body is in

the subcritical regime. Chapter 4 introduces the methods of solving individually the

associated structural and fluid dynamics concepts and practical steps used in solving the

problem statement. Chapter 5 is a verification process used in measuring the uncertainty

inherent in the numerical (computational) methods that have been employed in this

dissertation. Chapter 6 presents an articulate report on the findings and validates the results

obtained with known experimental works. Chapter 7 presents a conclusion summarizing all

the findings and present an outlay for future study.



CHAPTER TWO

2.0 BACKGROUND

A condition which is experienced by an elastic structure due to the coupling of an excitation

force(s) imposed on the flow due to the relative motion of fluid flow with respect to the

structure is known as flow induced vibrations. This phenomenon is not restricted to only fluid

flow past a structure but might be experienced due to a pulsating flow through the structure.

Flow induced vibrations (FIV) have since been observed in ancient times; examples recorded

include the observation of the sound produced by King David’s kinnor – (stringed musical

equipment) as he hung it over his bed on windy nights. A sketch representing a row of

vortices as it flowed past a pile in a stream was produced by Leonardo da Vinci but the

systematic study of this condition did not commence until Strouhal, a physicist observed

through experiments with wires in 1878.

Flow-induced vibration is experienced in a number of fields which includes but not limited to

the following; aerospace, power generation and transmission, civil engineering and for the

purpose of this dissertation the oil and gas (subsea) industry. The study of the concept of

flow-induced vibrations is essential in submarine pipelines as it contributes adversely to the

fatigue life of the pipelines.

Flow induced vibration of an elastic structure, either immersed in a fluid or conveying the

fluid would experience a distributed force that is exerted on it by the fluid and the structure

might respond in any or a combination of the following ways:

Statically deflect

Instability caused by the divergence of the flow

Resonate due to the periodic excitation of the flow

Respond to fluid excitation and or

Be subjected to dynamic instability.

Four classes of FIV are commonly identified, they are: vortex induced vibrations, galloping,

flutter and buffeting (Kumar, Sohn and Gowda, 2008). A full schematic showing the

interactions of the fluid flow and corresponding structural response is shown below:



Figure 2: Schematic showing the different structural responses to fluid flow

(Steady and unsteady flow) (Blevins, 1974).

This dissertation is focused on the steady flow past a structure which implies from the above

that the structure would experience vibrations due to the fluid flow. The flow past a structure

can cause large amplitude variations in an elastic structure exposed to steady flow. If the

structure periodically sheds vortices at a frequency near a harmonic (multiples) of the natural

frequency of the structure, vortex shedding may be coupled with structural vibration and

generate a synchronous oscillating force on the structure (Bailey et al., 1999). Tsahalis,

(1987), states that the damage associated with the fatigue life of a pipe undergoing vibrations

is proportional to the product of the amplitude of vibration and the frequency of vibrations;

this is governed by the relationship A4f.

A dimensional analysis of the controlling parameter of VIV and its effects, yields according

to Vandiver, (1993), the following: density of the fluid “ρ”; dynamic viscosity of the fluid

“μ”; velocity of the incoming flow “Uo”; diameter and length of the cylinder “D, L”; spring

constant k; mean roughness height of the cylinder ks; structural damping factor ε; mass of the

cylinder (excluding the effects of added mass) “m”; mean shear “dU/dy”; taper “dD/dy”



characteristic turbulence intensity “εt” and integral length scales of the ambient flow “IILS”

and Schewe parameters “SP”. A brief background and/or effects into the important parameters

in which the dimensions make up of are described in the subsequent section(s).

2.1 IMPORTANT PARAMETERS

2.1.1 REYNOLDS NUMBER

The Reynolds number is a dimensionless number that is significant in the design of a model

of any system in which the effect of velocity is important in controlling the velocities or the

flow pattern of a fluid (Chen, 1987). It is given by the equation below:

(2.1)

Where Re is the Reynolds number, ρ is the fluid density; Uo is the flow velocity; L is the

characteristic length which in cylindrical cross sections is the diameter; and μ is the absolute

viscosity.

Alternatively the Reynolds number has been proved to be the ratio of the inertia force to the

viscous force in the fluid.

2.1.2 REDUCED FLOW VELOCITY

Chen (1987) defines the reduced flow velocity as the representative ratio between the fluid

kinetic energy to strain energy in the structure. The frequency of the vortex shedding relative

to the frequency of the structural motions can be described by the parameter of reduced

velocity; this property is where it gains its relevance.

(2.2)

VR in the above expression is defined as the reduced flow velocity; while f s is the vortex

shedding frequency.

2.1.3 STROUHAL NUMBER

The Strouhal number is a dimensionless number used in examining the oscillations of a body

where a fluid is flowing past it. It relates the frequency of vortex shedding to the velocity of

the flow and a characteristic dimension of the body (in cylindrical structures this dimension is

the diameter). Mathematically it is shown as below:

(2.3)



S is defined as the Strouhal’s number in the expression above.

The frequency of vortex shedding of a body at rest is determined by this dimensionless

parameter. It is uniquely related to the velocity of the flow and the characteristic size of the

body (Holmes, Oakley, and Constantindes, 2006). The Strouhal number is a function of

Reynolds number (Chen, 1987) and as investigated for the vortex shedding of a circular

cylinder, this parameter is found to be constant over a range of Reynolds numbers between

102 ~ 10

5.

The Strouhal number albeit constant over a range of Reynolds number fluctuates as the flow

passes through laminar (low Reynolds number); transitional and finally turbulent flow.

2.1.3 STABILITY PARAMETER

The stability parameter, Ks, determines the magnitude of the vibrations experienced by a

structure experiencing FIV. It is a function of the structure’s characteristic damping and the

mass ratio between the structure and the fluid displaced and is expressed in the relationship

below:

(2.4)

Where the total modal damping ratio; and me is is the effective mass per unit length (this

includes the structural mass, added mass and the mass of any fluid contained within the pipe).

This is a non-dimensional damping parameter and may also be referred to as the Scruton’s

number.

2.1.4 DAMPING RATIO

Chen (1987) and Blevins (1974) define damping as the dissipation of energy with time. The

damping ratio is a measure of the amount of damping in a structure which can effectively

reduce structural vibration at resonance (Ji and Bell, 2008). It is a dimensionless quantity that

measures how oscillations in a system decay after it has been disturbed. In essence it is often

represented as a percentage. This is because the damping ratio is the ratio of the actual

damping to the critical damping where the critical damping is the minimum amount of

damping which prevents system oscillation and is regarded as unity. The damping inherent in

the structure is related to the material constituents of the structure, while the fluid damping is

a result of the viscous dissipation and fluid drag due to the interaction at the boundary of the

bluff body.



2.1.4 MASS RATIO

The mass ratio is defined as the ratio of the mass of a structure submerged in a fluid to the

mass of the fluid displaced. In analysing flow induced vibrations of cylindrical structures, the

mass ratio is given by Chen (1987) as:

(2.5)

Where m* is the mass ratio; and m is the mass per unit length

The mass per unit length implied includes the mass of the cylindrical structure and the mass

of the fluid it contains while excluding the hydrodynamic mass of the structure in the external

fluid (added mass). It is a measure of the relative importance of buoyancy and added mass

effects (Blevins, 1974). The mass ratio parameter is therefore the ratio of the oscillating

structural mass to the displaced fluid mass (Stappenbelt and O’Neil, 2007).

2.1.5 ADDED MASS

Added mass is the pressure force per unit acceleration acting on an oscillating floating body,

due to the acceleration field set up in the surrounding fluid (Lata and Thiagarajan, 2007).

This parameter represents the force associated with the acceleration imparted to the fluid

particles due to the disturbance of the flow caused by the body in accordance with the added

mass effect. The added mass effect as detailed in Chen, 1987 describes the resistance which a

structure experiences when moving through a fluid at a variable velocity. This causes the

structure to behave as though an added mass of fluid were firmly attached to and moving with

it. Mathematically, it is represented as:

(2.6)

Where Ca is the added mass coefficient.

2.1.6 EIGEN FREQUENCY AND MODE

A structure is characterized by a natural mode and this mode is associated with a mode shape

and frequency that characterizes the form of free vibrations that the structure might

experience. The mode shapes of simple structures can be found by solving the linear

equations of motion governing the system and for each degree-of-freedom of motion for a

structure there is a corresponding vibration mode (Liu and Holmes, 1995).



The Eigen- mode refers to the characteristic mode of a vibrating structure. A mode in this

case is referred to as the summation of travelling waves which have culminated into a single

standing wave after the reflections caused by the system’s boundary conditions. It is the

sequence of sinusoidal structural motions which all of the individual parts of the system

experiences with the same frequency and phase. The mode determines the form of VIV (in-

line or cross-flow) a body experiences at any given instant.



CHAPTER THREE

3.0. FLOW PAST A BLUFF BODY

There are no agreed existing definitions of the term bluff and hence as observed in literature

the definition of the term lie in the exclusive preserve of the investigator. The definitions that

follow describe which is most agreeable to this author.

Blevins (1974), Kumar, Sohn and Gowda (2008), simply allude to the fact that a bluff body is

one in which the fluid flow separates from a large section of the external cross section of the

structure. Otherwise, it refers to an elastic or elastically mounted fore-and-aft body of proper

mass, material damping, and shape whose cross-section facing the ambient flow at high-

enough Reynolds numbers gives rise to separated flow and hence to two shear layers, which

interact with each other and bound an unsteady wake (Sarpkaya, 2004). The bluff body

considered in this dissertation is the circular cylinder and any reference to a bluff body would

infer as such from here henceforth.

Most structures are bluff bodies, which stems from the fact that they are not streamlined and

one of the most common form of bluff body which is profoundly covered in research is the

circular cylindrical structure. Previous research have been able to classify bluff bodies

according to some certain parameters, some of which are either smooth or rough; rigid

(structure with infinite structural stiffness) and elastic (structure with finite structural

stiffness). In the context of this dissertation a subsea jumper is an elastic structure on fixed

supports and can be related to most literature in lieu of the fact that it is a coupling of

cylindrical sections.

The steady flow past a bluff body is characterized by the Reynolds number and at certain

value which is approximately 40; there would be the existence of vortices which have been

alternately shed from either side of the downstream part of the body. The Reynolds number at

which this occurs is referred to as the critical Reynolds number (Recr ≤ 40) and this process is

referred to as vortex shedding.

The trailing alternate shedding of vortices is what is popularly referred to as the Von Karman

vortex street and is defined as “a repeating pattern of swirling vortices caused by the

unsteady separation of flow over bluff bodies” (Chaudhury, 2011). The undisturbed region of

flow behind the cylinder is referred to as the wake; this region bears a lot of engineering

importance as would be observed in the subsequent sections.



3.1 VORTEX SHEDDING AND VORTEX INDUCED VIBRATIONS (VIV)

In the viscous steady flow around a bluff body, the flow at a point very close to the solid

surface slows down due to the effects of fluid viscosity and forms a thin slow moving fluid

layer called the boundary layer. As flow progresses past the surface, the boundary layer

separates from the body due to the adverse pressure gradients the fluid particles experience

due to the increasing pressure in the direction of the flow (low pressure gradients generated

on the downward side of the body). This is referred to as the boundary layer separation and it

occurs at different angular positions depending on the turbulence of the flow (categorized by

the Reynolds number). The separated boundary layer forms a free shear layer which

eventually rolls into a single vortex. The vortices are generated on both sides of the body and

this phenomenon is referred to as vortex shedding. As observed by Bearman (Bearman,

1984), the presence of this two shear layers are fundamentally responsible for the vortices

been shed, while the presence of the bluff body modifies the shedding process by allowing

for a feedback between the wake and the recirculation of shedding at the separation points.

The entrainment of fluid from the internal formation region and its renewal due to the flow

reversal determines the frequency of vortex shedding (fs). The frequency of the shedding of

vortices around the external surface of the cylinder is determined by the Strouhal number and

as the flow velocity past the fixed bluff body increases, an almost linear increase of frequency

of the vortices shed is observed.

Figure 3: The boundary layer separation due to adverse pressure gradients

(Blevins, 1974). Edited)



Figure 4: The discrete rolling of the vortex as the boundary layer is separated from the

structure (Chen, 1987)

The alternate shedding of vortices in the near wake, in the classical vortex street

configuration, leads to large fluctuating pressure forces in a direction transverse to the flow

and may cause structural vibrations, acoustic noise, or resonance, which in some cases can

trigger failure (Williamson, 1996). The interactions of the alternately shed vortices with the

structure causes an elastic deformation of the body due to the generation of an uneven

pressure distribution which develops between the alternate sides of the body and this

phenomenon which is a subset of FIV is what is referred to as Vortex induced vibrations

(VIV). A phenomenon observed with vortex induced vibrations known as the “lock-in”

where the structural motion of the cylindrical bluff body dominates the shedding process. The

frequency of the vortex shedding in this instant does not obey the relationship as described

earlier with the Strouhal’s number but rather locks in to the natural frequency of the structure

causing increased vibration amplitudes (resonant).

The occurrence of this phenomenon thus implies the amalgamation of the shedding

frequency, structural oscillations and the natural frequency of the structure. The most

common example of adverse effect of this phenomenon is the collapse of the Tacoma Bridge

in USA. A few pipelines have failed by this specific reason only; such as the Ping Hu 10” oil

pipeline which failed due to the occurrence of strong near seabed flows which was generated

by a typhoon in the region and the Shell pipeline which failed at 4 different locations almost

immediately after construction due to the strong currents and fine sand which made it

susceptible to scouring and hence generated spans that were longer than the critical span

length.

3.2 REYNOLDS NUMBER AND VORTEX SHEDDING

The VIV response of a bluff body is largely dependent on the Reynolds number. It defines

the flow regime and serves as a foundational base for the non-dimensional quantities that



describe the flow around a smooth cylindrical structure. The flow pattern around a circular

cylinder can generally be characterized by the Reynolds number of the incident flow and by

the location of points at which the flow separates from the cylinder surface, which in turn

depends on the state of the boundary layer (laminar or turbulent) (Pantazopoulous, 1994).

The major Reynolds regimes and their characteristic vortex shedding are as shown below:

Figure 5: Regimes of flow around a smooth, circular cylinder in steady current

(Lienhard, 1966, cited in Sumer and Fredsoe, 1999).

In the context of this dissertation, the subcritical range (300 ≤ Re ≤ 3 × 105) is a function of

the cylindrical diameter where the vortex shedding is strong and periodic.

Based on existing literature, there have been several divisions of this region with its attendant

implications on the wake of the cylinder. In (Raghavan and Bernitas, 2010), in the range of (1



x 103 ≤ Re ≤ 2 x 10

5) which is called the “shear-layer transition regime” it was asserted that

in this region, the Strouhal number gradually decreases, which is due to the instability of the

separating shear layers from the sides of the body.

This instability is as observed by (Zdravkovich, 1997) as the formation of a three dimensional

formation of the flow in the wake of the cylinder. The division in Raghavan and Bernitas,

(2010) is predated by the first classifications of Zdravkovich (1990); which was based mainly

on the transition in shear layers as the fluid flow past a bluff body. In the generalized sub-

critical range and as relating to the current work, the division of Zdravkovich (1990) exists as

thus:

TrSL2 - transition vortices in free shear layers 1,000 to 2,000 < Re < 20,000 to 40,000

TrSL3 - fully turbulent shear layers 20,000 to 40,000 < Re < l00,000 to 200,000

TrSO - onset of transition on separation 100,000 to 200,000 < Re < 320,000 to

340,000

Achenbach and Heinecke, (1981), present a division of the subcritical regime, which is

defined between 150 – 1.4 × 105. This classification is mainly based on the characteristics of

the Strouhal number in this regime which is constantly assumed to be at 0.2. Zdravkovich

(1990) presents a more stronger argument concerning the flow regime division as the

variation experienced within these regimes influences significant changes in the fluctuating

and time-averaged forces (lift and drag) exerted on the cylinder.

Figure 6: Flow Regime Division. Zdravkovich (1990)

3.3 FORCES DUE TO VORTEX INDUCED VIBRATIONS

A periodic force is exerted on the bluff body due to an asymmetric flow pattern (vortex

shedding) which alters the pressure distribution, the component of this periodic force in the



perpendicular direction i.e., lift force) has the same frequency as the vortex shedding, while

the component in the in-line direction has a frequency equal to twice the shedding frequency

(Yang et al., 2009). The force component of the in-line direction is regarded to as the drag

force and possesses the lower amplitudes compared to the lift forces (Posdziech and

Grundmann, 2007); this force (i.e. the drag force) is dominated by fluctuating pressures

which are in-phase between the upper and lower side of the cylinder (Norberg, 2003).

VIV is also a frictional process (movement between boundary layers and fluid particles)

which in effect causes a torsional force to be experienced by the cylinder even in low laminar

regimes, however there is a consensus that the effects of this force is insignificant compared

to the already discussed forces (Lecointe and Piquet, 1989).

3.3.1 FLUCTUATING LIFT

The conceptual study into the phenomenon of the fluctuating lift in a continuous stream of

flow is credited to Drescher, H. (Norberg, 2003). The lift forces are referred to as fluctuating

lift due to the fluctuating pressures acting on the bluff body (Norberg, 2003) and except for

the aft portion of the body, the energy induced by the pressure fluctuations is concentrated

around the vortex shedding frequency. The shedding frequency as defined earlier is closely

related to the Reynolds number of the flow and as shown below is the graphical

representation showing its variation.

Figure 7: Variation of the Strouhal number with Reynolds number

(Sumer and Fredsoe, 1999)

The vortex shedding process is dominated by the fluctuating lift forces under normal

conditions and is regarded as the main source of cross stream FIV. The fluctuating lift around

the bluff body follows a sinusoidal path and is harmonic with the shedding frequency and can

be numerically modelled as follows



( )

( ) (3.1)

The vibration caused by the action of the force induced by the action of the alternating forces

due to the effect of the fluctuating lift forces can be obtained by equating the equation of

motion of the bluff body to the above equation. This is as derived in the equation below

( )

( ) (3.2)

The time derivative solution of the above equation yields the steady state amplitude of the

alternating transverse forces which is normalized with the cylinder diameter.

( )

√[( (

) ) ( (

) ) ]

(3.3)

Where m is the structural mass; ωn is the cylinder natural frequency; ωs is the vortex shedding

frequency; A/D is the normalized steady amplitude of vibration; β is the damping ratio; y is

the direction of motion (transverse lift force); the dotted symbols represent its derivatives and

K is the structural stiffness. The phase angle is given as;

(3.4)

The expressions above refer to the elementary derivation of the single degree of freedom

response of a structure experiencing VIV.

The term CL is commonly referred to as the lift coefficient and is as defined as

(3.5)

However; considering the alternating periodicity of the phenomenon the magnitude of the

coefficient of lift can be represented in terms of its root mean square (RMS) value. This is as

shown below:

'2 2/Ud

LC

c

L

(3.6)

Where L’ is the RMS of the lift fluctuations acting on a span wise segment of the length

where lift is assumed to have a zero mean value.



3.4 RESPONSE MODELLING

The cross flow motion of the cylinder can be obtained not only by the single degree of

freedom method shown above. The other two are referred to as the wake-body (wake-

oscillator) coupled models and the force decomposition model. These three combined present

the most common semi-empirical response models of VIV. They are phenomenological in

nature (that is mainly used on the explanation and simulation of experimental results). The

wake oscillator model considers the fluid around the cylinder as an oscillator, where the lift

coefficient is linked to the acceleration, velocity and displacement of the structure under

oscillation through the Van Der Pol equations. In the force decomposition models such as

proposed by Sarpkaya (1977), the hydrodynamic lift force acting on the cylinder are

decomposed into a drag and inertia force. An example is shown below:

( ( )

( ))

Where y is the displacement and its derivatives, ω is the ratio of the cylinder oscillation

frequency to its natural frequency (ωn); is the ratio of the fluid density to the density of the

cylinder and = ωnt. The values Cml and Cdl are the inertia and drag coefficients determined

from experiments at perfect synchronization.

The major disadvantage of these response models is that they are mainly tweaked to suit

experimental data and the models have not been able to predict with a great deal of accuracy

resonant vibration experiments in open literature (Choudhury, 2011).

The basic ideologies of VIV response modelling are based on either or a combination of i)

self-limiting and self-exciting models ii) assumption of resonance between the fluid and

structure (the model employed in this dissertation is based on this) and iii) a forcing term is

used in relating the cylinder to the fluid oscillator.



Figure 8: Cross-section of a rigid subsea jumper experiencing VIV

(Carruth and Cerkovnik, 2007)

3.5 FLOW AND FLOW FIELD MODELLING

Solutions or approach to fluid flow/dynamics problems can be either of i)

practical/experiments; ii) Theoretical; iii) Computational fluid dynamics. Computational fluid

dynamics constitutes a new “third approach” in the philosophical study and development of

the whole discipline of fluid dynamics (Vandiver, 1987). Computational fluid dynamics is

basically an iterative/direct numeric solution to the partial differential equations of fluid

mechanics.

The solution to fluid flow problems using computational fluid dynamics follow a sequence of

steps as to achieving satisfactory results and this is independent of the specific application

which is been considered. They are Pre-processing; solving; and Post-processing; below is a

flow chart as to the representation of this sequence.



Figure 9: Flowchart representing the Computational Fluid Dynamics process.

Aremu, (2009)

3.5.1 GOVERNING EQUATIONS

The set of governing equations of CFD are as obtained by the Navier – Stokes derivation of

the Euler propositions for an inviscid flow in 1822. They are partial differential equations

which consist of four independent variables (i.e. spatial coordinates; x, y, and z; and time t);

six dependent variables density, pressure P; temperature T and the components of vector

velocity; u, v, and w.

These equations represent the mathematical statements of the conservation laws of physics.

They are as obtained from Anderson (1995):

Continuity equation: mass is conserved for the fluid;



Momentum equation: This is based on the Newton’s second law of motion which

states that the rate of change of momentum of a body is equivalent to the sum of

forces acting on the body. In relation to CFD, the rate change of momentum equals

the sum of forces acting on the fluid; and

Energy equation: this is based on the first law of thermodynamics which simply states

that energy cannot be created nor destroyed but be converted to another when a

change in process occurs. Anderson (1995) states it as the rate of change of energy

been equal to the sum rate of change of heat addition to and the rate of work done on

the fluid.

The 3D- Unsteady Navier-Stokes equations are:

Continuity

( )

( )

( )

(3.7)

Momentum in the x-direction

( )

( )

( )

( )

( )

[

] (3.8)

Momentum in the y- direction

( )

( )

( )

( )

( )

[

] (3.9)

Momentum in the z-direction

( )

( )

( )

( )

( )

[

] (3.10)

Energy equation

( )

( )

( )

( )

( )

( )

( )

[

]

[

( )

( )

( )] (3.11)

In computational fluid dynamics, these equations are discretized over the flow field and the

flow thus modelling the fluid flow simultaneously over a given time. The discretization of the

flow field is what is most commonly referred to as meshing (gridding) and is one of the most

important steps required in obtaining valid results from a CFD simulation. In this dissertation,



a quadrilateral structured mesh has been employed; this choice is solely arbitrary as there are

no evidences in literature to suggest that any other choice would produce more valid results.

In the discretization of the flow; three most common methods are referred to in literature they

are: i) Finite Difference method; ii) Finite element method and iii) Finite Volume method.

The finite volume method is the most popular amongst the commercially available CFD

codes and is what is employed in the ANSYS WORKBENCH FLUENT. It discretizes the

equation representing the fluid flow integrally directly over the computational domain. The

advantage of the finite volume method includes its conservativeness of the governing

equation and the elimination of the need to transform the equations as experienced with the

finite difference methods.

3.6 TURBULENCE

In figure 3.2, it is observed that the region behind the cylinder in the subcritical regime is

fully turbulent.

A major challenge of CFD is to predict accurately complex flows, especially the ones with

large separations. The objective from an engineering point of view is for the CFD code to be

able calculate such flow characteristics such as pressure, skin friction and velocity accurately.

Turbulence is irregular or random; causes rapid mixing and flow resistance and is of a major

concern in most engineering applications. Turbulence dominates greatly in any instance

where it is observed and its successful modelling in turn improves the quality of numerical

simulations of fluid flow.

The complexities of modelling turbulent flows first; stems from the equations that define

fluid flow (i.e. Navier-Stokes Equations) which are intrinsically time dependent, non-linear

and exhibits random spatial variations (3D). In order to model this phenomenon, there exists

variant avenues, these include:

The direct numerical solutions of the governing equations popularly referred to as the

(DNS);

The large-eddy simulations (LES) and;

The Reynolds Averaged Navier Stokes (RANS) approaches.

The most popular approach still remains the RANS because it is fairly simple, robust and

inherently capable to model industrial turbulent flows as witnessed today. The other two

whilst more accurate than the RANS approach are exceptionally prohibitive in terms of



computational requirements. An example is the DNS approach which would need nodes in

the region of about 107 and a time step size of about 10

-5 seconds for flows in Reynolds

numbers of about 800.

Reynolds (1895) proposed decomposing the flow into a mean motion, defined as an

ensemble-average (and in most cases, a time average), and turbulent fluctuations. This

approach eliminates the rapidly fluctuating elements of the Navier Stokes equations by

substituting it with new terms referred to as the Reynolds stresses. The equation is as shown

below and popularly regarded to as the Reynolds Averaged Navier Stokes Equations.

For continuity:

( ) (3.12)

For momentum:

( )

[ (

)]

(

) (3.13)

The accent in the equations means the averaged values of density, pressure and velocities; μ

is the molecular viscosity

In comparison with equations (3.7 - 3.11), a new term is observed. The new term

(Reynolds Stresses) presents a fundamentally new problem which is referred to as the closure

problem. The Boussinesq hypothesis is employed in order to accurately account for

turbulence and effectively model the Reynolds stresses and achieve closure of the equation.

This is accomplished by relating the Reynolds stresses to the mean velocity within the flow.

The resulting solution is given as

(

)

(

) (3.14)

Where the turbulent (or eddy) viscosity and k is is the turbulent kinetic energy. The

implication of the solution is that the shear stresses are divided into two individual

components which are the turbulent and viscous shear stresses. In a mathematical sense this

makes no significant difference however physically the molecular viscosity is a fluid property

while the turbulent viscosity is a flow property.

There are various forms of the eddy viscosity models; and are mostly classified based on the

number of terms required in solving the closure problem. The most popular forms of the eddy



viscosity models are shown below which are widely accepted in the CFD codes however not

exhaustive.

Algebraic turbulence models

Cebeci-Smith Model (Smith and Cebecci, 1967)

Baldwin-Lomax Model (Baldwin and Lomax, 1978)

One equation turbulence models

Spalart – Allmaras (Spalart and Allmaras, 1992)

Prandtl Mixing Length Model (Wilcox, 2004)

Baldwin – Barth (Baldwin and Barth, 1990)

Two equation turbulence models

K - Epsilon model

Standard k – Epsilon model (Wilcox, 2004)

Renormalized (RNG) k – Epsilon model (Yakhot et al, 1992)

Realisable k – Epsilon model (Wilcox, 2004)

K - Omega model

Standard k – Omega model (Wilcox, 2004)

SST k – Omega model (Menter, 1994)

k-tau model (Kato and Launder, 1993)

Reynolds stress model (Launder, Reece and Rodi, 1975)

3.7 SUBSEA JUMPERS AND VORTEX INDUCED VIBRATIONS

The availability of literature concerning this topic has just in recent years become available

and only few have been related to numeric simulation of the fluid structure interactions that

affects rigid jumpers; however of the few literature available concerning this subject, three

have been presented and the scope and dimensions which were observed are detailed in the

table below while their corresponding methods and results are presented in the adjoining

sections.



Jumper 1 Jumper 2 Jumper 3

Dimensions

(mm)

Outer diameter 406.4 168.28 167

Wall thickness 22.2 18.3 Na

Anti-corrosion

thickness 0.4064 Na Na

Corrosion allowance

(buoyancy section) 0 Na (600)

Horizontal distance

between jumper ends 30626 29570 30000

Vertical dimension

(highest point to

lowest point along

jumper

26304 5790 5000

Fluid

properties

Internal fluid Gas Gas Na

Internal fluid density

(kg/m3) 181 318.8 Na

External fluid Water Water Water

External fluid

density (kg/m3) 1025 1025 Na

Design

conditions

Design pressure

(MPa) 20.68 Na Na

Design temperature 85 Na Na

Sour service

expected Yes No Na

Water depth (m) 396.24 2133.6 Na

Material

properties

Pipe material grade API 5L X60 API 5L X65 Structural

steel

Coating material

(buoyancy material)

Fusion bonded

epoxy

Fusion bonded

epoxy

Syntactic

foam

Buoyancy section

density Na Na 509

Location

OFFSHORE

WEST AFRICA

GULF OF

MEXICO Na

Table 1: Specifications of Rigid Jumpers

3.7.1 RESPONSE MODELLING OF RIGID JUMPERS

In the case represented in Jumper 1 and 2, the response of a rigid jumper was estimated using

an amplitude response model as obtained from the recommend practice DNV-RP-F105,

which provides guidelines as pertaining to free spanning pipelines.

A rigid jumper is created as a model in the ANSYS software (PIPE59 component) with

proper care given to achieve the accurate bend radius. The PIPE59 is a uniaxial element with

tension-compression, torsion, and bending capabilities, with member forces simulating ocean



waves and current [Behr et al., 1995]. It is an efficient tool used in modelling the complex

inter-relationship between the hydrodynamics, soil mechanics and structural analysis

involved in free spanning pipelines on the seabed.

The dimensions representing the jumper geometries as described for case 1 and 2 in the table

above is statically analysed (modal analysis) in three dimensions also using the ANSYS

software; this is done to obtain the natural frequencies of the structure as constructed.

The natural frequencies ranged from 0.38Hz to 5.30Hz in 9 consecutive modes (1-9) for case

1 and ranged from 0.45Hz to 2.48Hz in 8 consecutive modes (1-8). The modes in both cases

varied either as in-plane or out of plane to the direction of the flow and the frequencies

obtained have built into them the effects of the submerged weight of the rigid jumper as it is

immersed in still water. This represented cross-flow and in-line VIV respectively. The flow

across the span of the jumper is assumed to be steady and in transverse direction to the plane

of the cross section of the rigid jumper.

A screening criterion for the onset of either cross-flow or in-line VIV in the recommended

practice DNV-RP-F105 was then applied to the natural frequencies obtained earlier. The

comparison of the natural frequencies of the jumper with the vortex shedding frequency was

obtained using the below equation:

(3.9)

Where

f n Natural frequency (in-line or cross-flow) of the jumper

U Maximum flow velocity

VR In-line or cross-flow onset value for reduced velocity

D hydrodynamic diameter of the jumper

γf factor of safety on the natural frequency for in-line or cross-flow VIV

γ screening factor for in-line or cross-flow VIV (Note: In-line screening factor

takes into account the length of the jumper

The analysis then follows a set of guidelines in the recommended practice showed for Jumper

1, a response to both forms of expected VIV, while for Jumper 2, shows that the rigid jumper

would experience in-line VIV but not cross-flow VIV.



Figure 10: Out of Plane response of the Jumper Spool (translating to In-line VIV)

(Carruth and Cerkovnik, 2007)

Figure 11: In - Plane response of the Jumper Spool (translating to Cross-flow VIV)

(Carruth and Cerkovnik, 2007).

This analysis is assumed to be adequate for predicting the VIV response of the rigid jumper;

this is because as in free spanning pipelines, the pipe diameters, current velocities and desired

minimum margins of safety are similar in both cases.

There are, however differences which would significantly impact the VIV response of the

free spanning pipeline as compared to a rigid jumper. The first is the end conditions; while in

the rigid jumpers the ends are typically fixed, the ends of the spanning pipeline are typically

supported. The rigid jumpers also experiences torsion with its out-of-plane motions with

complex mode shapes, this is not applicable to free-spanning pipelines. Finally the

interactions of the free-spanning pipelines with the supporting soil have been incorporated

into the VIV response models as prescribed in DNV-RP-F105.



3.7.2 COUPLED STRUCTURAL AND FLUID ANALYSIS OF A COMPLEX

SUBSEA JUMPER

In this case, a fully coupled fluid – structural interaction model was performed which

involves the combination of a numerical fluid model describing the fluid motion with a linear

structural vibration model

The structural analysis was performed based on eigenvalue (mode and frequency) analysis

from ABAQUS structural analysis codes and the fluid flow solver AcuSolveTM

using

unstructured grids and Detached Eddy Simulation (DES) turbulence model.

The structural analysis characterized responses totalling 20 modes with natural frequencies

ranging from 0.36Hz – 15.771Hz in air (i.e. without including the effects of the surrounding

fluid and added mass).

The fluid structure interaction analysis involved the simultaneous solution of the structural

response and the fluid response. An iterative scheme was used in each time step to solve for

this and convergence between the two results was obtained after five iterations. The fluid

structure interaction analysis was considered in four different cases which either involved

variation of current speed and/or geometry. Case 1 and 2 (variation of Eigen-modes and

jumper included buoyancy element in the centre of its free span); Case 3 and 4 (variation of

the current speed (0.5m/s, 0.454m/s) with bare jumper)

The flow domain around the jumper which was employed in modelling the fluid flow is a

rectangular block with width of 33.48m and height of 10m and a length in the flow direction

of 10m. The mesh used in the discretization of the fluid space were unstructured tetrahedral

elements (17.6 million and 19.3 million for case 1, 2 and 3, 4 respectively), while the jumper

was meshed with 4.1 million nodes and 2.4 million wedges for the boundary layer for case 1,

2 and 4.5 million nodes and 2.37 million wedges for the boundary layers respectively. The

boundary layers mesh was 7 elements thick with a first element thickness of 0.004m and a

stretch of 1.2.

A moving mesh scheme was used so as to be able to capture the coupled fluid imposed

vibration on the structure. The methodology of the moving mesh for the 3-dimensional

simulation of fluid structure interaction (VIV) is described in Spalart (2006) and Holmes

(2006).



The results found that for all the cases considered under the influence of the constant current,

there were no observed large VIV amplitudes and even for the worst case translated to

normalized Amplitude to Diameter ratio of 15cm (Case 4).

Holmes (2010) came to the following conclusions as to the low amplitude response of the

four cases observed. Firstly, “the strake buoyancy section acts as a sea anchor and tends to

damp out motions at the jumper centre”. The strakes act to disturb the flow pattern around the

jumper centre (preventing correlated vortex shedding). Secondly, the vortex shedding

frequencies experienced due to the fluid flow are small compared to the natural frequency of

the jumper structure. Also the jumper is stiff and heavy so that lock-in is limited to a narrow

range of frequencies further limiting the response. Finally, the complex structure of the

jumper also militates against the VIV response, this is because the modal shapes of the

jumper do not always align with the direction of the vortex shedding, hence reducing the

power (pressure fluctuation) experienced by the structure. It is however important to note that

there was an increase in the amplitudes of VIV response between Cases 1, 2 and Cases 3, 4.

This was largely due to the absence of the strake section at the jumper centre.

Comparisons can be made directly between Jumper 2 and Jumper 3, as they have almost

identical hydrodynamic properties (diameter and mass), however the difference between the

response systems applied for each of these, is the effect of added mass and the differences in

their submerged weights.



CHAPTER FOUR

4.0 INTRODUCTION

The steady and unsteady flow (laminar and turbulent) behind a cylinder has received wide

considerations as observed through the numerous experimental and computational studies.

The subsequent sections would present how the flow around a 2-Dimensional circular

cylinder cross section in Reynolds numbers representing fluid flow velocities across the

external cross section of a rigid subsea jumper. It would also state how the modal analysis of

the rigid jumper was carried out.

The modal analysis of a structure is used in determining the natural frequencies of a structure

and its corresponding mode shapes. The natural frequencies of structures can be obtained

empirically and formulae exist to obtain such, however the rigid jumper is a continuous

serpentine structure albeit with a circular cross-section making it difficult to obtain its natural

frequencies; mode shapes and corresponding effective weights empirically. The ensuing

sections would highlight how the 3-dimensional modal analysis was performed in the

ANSYS WORKBENCH environment.

4.1 NUMERICAL SIMULATION OF THE FLOW PAST A CYLINDER

The numerical solution to the unsteady incompressible Reynolds Averaged Navier Stokes

equations coupled with an appropriate turbulence equation are solved using the finite-volume

discretization method with collocated grid system as described by (Menter, 1994) and a

pressure-correction technique as incorporated in the commercially available computational

fluid dynamics code FLUENT. In this method, the spatial discretization of the governing and

model transport equations is performed by the integration of the equations over each control-

volume defined by the grid.

The flow pattern around a circular cylinder in cross flow is very dynamic consisting of

separation; adverse pressure gradients; vortex shedding; recirculation and reattachment all of

which introduces additional complications in numerical analyses. In the subcritical regime of

Reynolds numbers been considered in this case, there is an onset of turbulence in the wake of

the cylinder just as shown below:



Figure 12: The subcritical flow past a cylindrical bluff body (Williamson, 1988, Edited)

Highlighting (a) separation point; (b) Laminar boundary layer and (c) the turbulent wake

4.2 TURBULENCE MODELLING

In the earlier sections, it has been clarified that the turbulence models are required to close

the Reynolds stress terms in the Reynolds Averaged Navier Stokes (RANS) equations in

order to solve the Reynolds averaged equations.

The accuracy of prediction amongst the eddy viscosity models for flow characteristics such

as Strouhal numbers; detailed local distributions of pressure especially the two equation from

various studies (Bardina, Huang and Coakley, 1997; Atlar, Unal and Goren, 2010), accede to

the fact that the turbulence model as prescribed by Menter (1994) i.e. shear stress transport k-

ω (SST) models, which is a modified form of the k-ω model is very much superior. The

reasons are that it combines the strengths of the k- ω model in the near wall region with the

strengths of the k-e in the free-stream prediction of flow characteristics; also it is less

sensitive to the inlet conditions. This is why this choice was implemented in this dissertation.

4.3 GRID DESIGN

The computational domain is discretized using quadrilateral cells, with a fine mesh around

the cylinder in order to capture the effects of the turbulent separation behind the cylinder

cross-section and coarser grids farther away from the cylinder wall. The grid as generated

around the cylinder is as shown below:



Figure 13: Fine Grid resolution around the cylinder wall

Figure 14: Fully Meshed Computational Domain

The numbers of intervals along the edges that extend to the cylinder wall are determined

using a formula which defines the sum of infinite terms of a geometric progression. This is

given as

(

( )

)

(4.1)

Where S is the edge length; r is the growth ratio; n is the number of intervals and ao is the first

edge length.



A bias factor which ensures the geometric progression between two intersecting edges is

given by:

(4.2)

z is the bias factor and the equation represents a ratio between the largest and smallest node

distances along an edge.

In an instance where the first cell height from the cylinder wall was determined to be 1.0472

× 10-4

m, due to the necessitation of resolving the fine mesh needed as described earlier for

the cylinder wall. The number of intervals along the edge was found to be 197 using a growth

ratio of 1.05 while applying a bias factor of 14227.

4.3.1 NEAR WALL TREATMENT

Near wall turbulence can be divided into an active vortical component arising in the inner

layer which contributes to shear stresses; an inactive component arising from pressure

fluctuations and large scale vortical structures in the outer layer which does not contribute

(Townsend, 1961).

The presence of walls meaningfully affects turbulent fluid flows. The near wall modelling

significantly impacts the fidelity of numerical solutions, in as much as walls are the main

source of the mean vorticity and turbulence (George, 2007). The particular attention paid to

the wall in turbulent flows is due to the large gradients of the solution variables; the effects of

the no-slip condition that has to be satisfied at the wall (cylinder) on the mean velocity field

and predominance of the viscous effects over the turbulence which occurs in this region and

hence a precise exemplification of the flow in the near wall zone is required in order to

adequately predict wall bounded turbulent flows.

The turbulent flow been considered (in the wake of the cylinder) generally requires a fine

grid resolution around the cylinder wall, and although coarse grids can be employed it is

difficult to obtain the accuracy needed in predicting the high degree of variation of the flow

properties past a bluff body in turbulent regimes.

The near-wall region can be divided into three layers; viscous sub-layer: where the flow

behaves close to a laminar; buffer layer: where both the laminar and turbulent properties are

both important; and the fully turbulent layer; here the turbulent properties of the flow play the

major role.



The yardstick for measuring which of the division in which an analysis is been carried out is

the non-dimensional distance “wall y+” from the wall which is similar to the local Reynolds

number and the wall shear parameters, and is often used in CFD to describe how coarse or

fine a mesh is for a given flow condition. It is referred to as the ratio between the turbulent

and laminar influences in a cell.

The following figure represents the divisions of the near wall region;

Figure 15: Divisions of the near wall region plotted in semi-log coordinates

(George, 2007).

There are two approaches to the solution of the near the wall problems: wall function

approach and near-wall model approach; the wall function approach does not achieve the

desired accuracy when applied to fluid flows that involve flow separation such as vortex

shedding as is been considered in this case. This is because it predicts the logarithmic

velocity distribution outside the viscous sub-layer whereas the effects of turbulence are

observed mostly in this region (i.e. the accuracy is largely dependent on the position of the

first grid point in the computational domain as this must be in the logarithmic region).

The SST k-ω model is by design applicable throughout the boundary layer, provided near

wall mesh resolution is sufficient. This has prompted the use of the near-wall modelling

(Low-Re resolved boundary layers which require a refined boundary layer mesh with the first

set of grid points at the wall).



The user manual of FLUENT reports that an average value between 4 and 5 for the y+ when

solving through the viscous sub-layer of the boundary layer but Menter (1994) reports that for

the first grid point, a value of y+ = 2 is appropriate; however in all of the simulations

performed in this thesis it didn’t exceed the value of 1. This is has evidenced in wall y+ chart

around the cylinder wall shown below.

Figure 16: Graphical Representation of the Wall y-plus for on the developed grid

The correlation observed along the upper and lower bounds of the cylinder wall corresponds

to an even flow around the cylinder, however it can be observed that this symmetry

disappears at the rear edges of the cylinder wall where the effects of vortices cause

unsteadiness in the flow

4.4 GOVERNING EQUATIONS

The Reynolds averaged equations for the conservation of mass and momentum respectively

in two dimensions is;

(4.3)

(

)

(4.4)

[ (

)]

(4.5)



Where i, j = 1, 2; and x1 and x2 represent the horizontal and vertical directions; u1 and u2 are

the corresponding mean velocity components; P is the dynamic pressure; ρ is the density of

the fluid; τij is the Reynolds stress component which contains the fluctuating part of the

velocity, and is expressed in terms of a turbulent viscosity VT and the mean flow gradients

using the Boussinesq approximation;

[ (

)]

(4.6)

μ the turbulent kinetic energy and δij is the kronecker delta function.

The equations that describe the shear stress transport k-ω (SST) model are described by the

expressions below:

Kinetic eddy viscosity

)max( 211

1

SFa

kavT

(4.7)

Turbulence kinetic energy

j

Tk

j

k

j

jx

kvv

xkP

x

kU

t

k)(

(4.8)

Specific dissipation rate

iij

Tw

jj

jxx

kF

xvv

xS

xU

t

1)1(2)( 21

22

(4.9)

Closure coefficients and auxiliary relations

22

500,

2[maxtanh

y

v

y

kF

(4.10)

k

x

UP

j

iijk 10,min

(4.11)



4

221

24,

500,maxmintanh

yCD

k

y

v

y

kF

k

w

(4.12)

10

2 10,1

2maxii

wkwxx

kCD

(4.13)

The below represents the constants of the SST k-omega model

)1( 1211 FF ; 44.0,9

521

; 0828.0,

40

321

; 100

9

;

1,85.0 21 kk ;

856.0,5.0 21 ww

4.5 BOUNDARY CONDITIONS

The boundary conditions employed in this simulation case are as follow:

I. At the inlet, uniform velocity is specified with U1 = Uo, U2 = 0; where 1 and 2

represents the x and y Cartesian directions. The turbulent intensity Iu of 0.2% and

turbulent viscosity ratio of 1 is employed so as to satisfy conditions that are presented

in the response model been considered.

II. At the outlet, the backflow turbulent kinetic energy and turbulent dissipation rate are

set to a value of 1. This was employed so as to eliminate any backflow during the

iterations been performed. This implies that an extrapolation of the conditions of the

outflow plane was performed from within the domain; and thus has no impact on the

upstream flow. It was considered that while there might be no influence of the

backflow on the obtained solutions, it might pose a difficulty in the convergence of

the iterative solutions of the governing equations.

III. No slip condition is applied on the cylinder surface with U1 = U2 = 0.

4.6 COMPUTATIONAL DOMAIN

4.6.1 DESCRIPTION OF THE FLOW CASE

The flow around a circular cross section of diameter D was placed in a computational space

where the uniform cross-flow velocity (U= 1.0 m/s). This is as shown in the figure 17:



Figure 17: Computational Domain

The dimensions of the inlet and exit boundaries are selected so as to minimize the effects of

the boundary condition on the flow in the vicinity of the cylinder. Behr et al (1995) studied

the effects of the lateral distance from the cylinder in the computational domain on the effects

such as mean drag coefficient; the non-dimensional frequency of the vortex shedding

(Strouhal number); and the amplitude of the lift coefficient and prescribed that in order to

achieve computed values for the Strouhal number without high artificial value, the lateral

boundaries should be placed a distance of 8 cylinder diameters and the outflow boundaries

should also be at about 30 cylinder diameters from the cylinder inconsideration. The

computational domain in this case is extended 64 cylinder diameters around the cylinder

centre which represents 32 cylinder diameters from the cylinder wall.

4.6.2 FLUID PROPERTIES

Fluid properties are assumed to be constant as shown in Table 2:

Fluid Properties (at 4oC)

Density 917 kg/m3

Dynamic Viscosity 1.5726 E-03 m2/s

Table 2: Fluid Properties at 4oC

4.6.3 FLOW PHYSICS

The Reynolds number is based on the free stream velocity and the cross sectional diameter of

the rigid pipe as described in equation (2.1). The corresponding Reynolds numbers as

obtained from the current speed past the rigid jumper of probabilities of current speed in the

location of the rigid jumper.



Current Speed

(m/s)

Percentage

Occurrence of

Currents

Reynolds Number

0.05 0.05 11796

0.1 2.23 23593

0.15 19.31 35389

0.2 32.27 47185

0.25 22.58 58982

0.3 11.10 70778

0.35 5.79 82574

0.4 3.02 94371

0.45 2.36 106167

0.5 1.00 117963

0.55 0.34 129760

Table 3: Describing the percentage occurrence of currents and current velocities in a

selected region offshore West Africa (Hariharan, Cerkovnik and Thompson, 2004).

4.7 CASE SETUP

The fluent case was set up using constant fluid properties and boundary conditions as

described in sections 4.3 – 4.5. Constant values of velocity and viscosity were employed

while making the density equal to the Reynolds numbers as obtained in Table 4.6.3. The

reference value for the area which is presented as the projected area of a cylinder in cross

flow is also specified. The value is described as the diameter of the cylinder, which is unity.

4.8 SOLUTION METHODS

A second order upwind scheme is used in the discretization of all of the governing equations

(turbulent transport, momentum transport and pressure-correction) and time into a system of

algebraic equations; this is employed so as to be able to attain a higher order of accuracy of

the solutions.

FLUENT solved the system of algebraic equations using the Least Squares cell based

iterative method in conjunction with the algebraic multi-grid (AMG) method solver at the cell

faces through a Taylor series expansion of the cell centred solution about the cell centroid.

The use of AMG scheme does reduce the number of iterations (and thus, computational time)

required to obtain a converged solution, particularly when the model contains a large number

of control volumes such as the present study.

The pressure based solver (pressure-velocity coupling) was employed in this study. It

functions based on an algorithm which belongs to the class of methods as detailed in (Chorin,



1968) and referred to as the projection method. In this technique an equation of pressure

(pressure correction) is used in obtaining a solution for continuity of the velocity field (mass

conservation). The solutions for continuity are solved until the convergence of all the non-

linear coupled governing equations is reached.

The PISO pressure-velocity coupling scheme (Pressure-Implicit with Splitting of Operators)

as described in (Issa, 1986) is employed in this thesis. It constitutes a part of the SIMPLE

family of algorithms and is based on a higher degree of approximation for the relationship

between the corrections for pressure and velocity (momentum or neighbour correction).

The PISO algorithm is coupled with a skewness correction also allows FLUENT to achieve

solutions on highly skewed meshes in approximately the same number of iterations as

required for more orthogonal meshes, this helps most importantly around the cylinder wall

where it is most evidenced that the most skewed cells are located due to the fine mesh which

is used in resolving the boundary layer. The PISO algorithm takes more CPU time per solver

iteration because of the additional corrections it performs, but it decreases appreciably the

number of iterations required for convergence, especially for transient problems. The PISO is

also employed in this case due to its ability to sustain its stability even when larger time steps

are employed.

Convergence is achieved when the residuals of the turbulent transport, momentum transport

and pressure-correction equations reach a pre-set value. In each case the iterations were run

until the scaled residuals dropped to 10-4

. The solution convergence is observed dynamically

by observing the residuals, forces, and surface integrals against the elapsed time.

4.9 TIME STEP SIZE CONSIDERATION

An implicit time stepping approach is used i.e. the discretization involves the integration of

every term in the differential equations over a time step ∆t.

There are different ways that can be employed in determining an adequate time step for any

individual simulation and this majorly depends on how fine or coarse the grid been used is,

however one of the proposed methods in determining the most adequate time step size is

dependent on fulfilling the condition presented by Courant et al (1967) for obtaining the

convergence of the solutions of partial differential equations (of which the Navier Stokes

equations is one) and is popularly regarded as the CFL condition. While it is generally not

accepted as a sufficient condition of convergence, it is popularly used in the implicit time

stepping computer simulations of the numerical solutions of the partial differential equations.



The condition in summary states that the time step size in implicit time marching systems of

computer simulations must be less than a certain time otherwise the solution would produce

unstable conditions and roughly incorrect results. In this case where the subject of interest is a

sinusoidal function (vortex shedding frequency) it implies that the time step size must be of a

magnitude which is equivalent or less than the time required for the wave to travel between

two adjacent grid points in the computational grid point.

The CFL Condition given for implicit time stepping solutions is given by the equation below.

( )

⁄

(4.13)

Uo represents the inlet velocity, which is equal to unity; ∆t is the time step size, is the

minimum face area on the mesh as is obtained from the FLUENT mesh check option as

1.0544 × 10-4

. Therefore, from equation (4.13), the time step size is given as 0.01027

seconds

This was considered too small and puts so much constraint on the available computational

resource available and moreover the attainable accuracies of using such small time steps are

regarded as generally negligible. The selection of a time step size which conforms to the CFL

condition and is reasonably accurate is then pertinent.

The time step hence implemented in the simulations is as prescribed in (Atlar, Unal and

Goren, 2010). They suggested and implemented a time step size which is 1/200th

the period

of a single vortex shed. In this instance this amounted to a time step size of 0.026; this

represented a compromise between using the time step size as prescribed by the CFL

condition and been able to obey the cell courant number criterion.

In order to obtain fully converged quantities of the key flow variables, the simulations were

performed over 10,000 time steps.

4.10 POWER SPECTRAL DENSITY

The power spectral density (PSD) is used in the statistical signal processing. It describes how

the power of a signal as is distributed over a known frequency. It is used in the later parts of

this report for analyzing the measured lift coefficients in order to obtain the dominant

frequency. This is obtained using the Fast Fourier Transforms (FFT) available in MATLAB.



4.11 MODAL ANALYSIS

The natural frequencies and mode shapes are identified through a modal analysis of the

structure.

The natural frequencies (mode shapes likely to be excited by the shedding frequencies

induced by current are then identified by a velocity screening analysis. The mode shapes are

classified with respect to the structural planes.

The modal analysis of the pipe structure was performed in three dimensions using ANSYS

Workbench whilst employing a BLOCK LANCZOS SOLVER. In the ANSYS workbench

interface, the rigid structure was created using a swept volume method in the YX plane which

implies that the inline direction of the fluid structure interaction occurs in the Z - direction;

whilst a cross flow is induced in the Y-direction.

The dimensions of the structure are as shown in table (4.10) and the material properties for

the structure was specified as structural steel since it bears semblance to popularly known

steel materials used in the offshore oil and gas industry such as API X65. The resulting

modes and modes shapes are presented in the subsequent sections.

Figure 18: Cross-section of the rigid jumper



Length Dimensions (m)

AB 4

BC 4

CD 10

DE 32

EF 22

FG 4

GH 4

Table 4: Dimensions of the single length sections of the rigid jumper

The single length pipes are connected at the ends via the dimensions of a 90 degree long

radius elbow which is as described in ANSI B16.9. This helps in attaining a correct bend

radius. The ends of the jumper are fixed as would be in service so as to obtain its natural

frequencies.

Figure 19: Cross section of the ANSI B16:9 long radius elbow

As shown in the figure above, A defines the radius of the elbow and is equal to 0.610m.



Figure 20: 3-Dimensional jumper in the ANSYS WORKBENCH environment

The response models used in determining the normalized amplitude of vibration due to the

fluctuating forces acting on the structure require the natural frequency of the structure while it

is immersed in still water. In order to appropriately simulate this condition, the principle of

the submerged weight is employed on the rigid structure.

The submerged weight of the structure is obtained as follows:

The volume and mass of the structure is as calculated from the ANSYS WORKBENCH as

2.4049m3

and 18,878 kg. The weight of the pipe which is obtained from the multiplication of

the mass by the acceleration due to gravity is 185,193.18N.

The displaced weight of the of the structure (buoyancy force) can be obtained using the

following relation

(4.14)

Where v is the volume of the structure; ρsw is the density of sea water

The submerged weight of the structure is the difference between the dry weight of the

structure and the displaced weight of the structure.

(4.15)

WSB; WD and WDW represent the submerged weight; dry weight and displaced weight of the

structure respectively.



From the above; the submerged weight of the structure is given as

From equation (4.15) above the submerged weight is evaluated to be 161,011.31 N

The submerged weight presents an avenue to alter the density of the structure in the ANSYS

WORKBENCH environment; this is of course valid because the acceleration due to gravity

and volume of the structure remains constant irrespective of the structural environment.

This method is also rendered valid as the natural frequency of a structure is dependent on the

associated mass of the structure and its stiffness.

√

(4.16)

Where f is the natural frequency, k is the stiffness and m is the mass of the structure

respectively

The stiffness of the structure is an inherent property which is dependent of the molecular

distributions within the structure. The density of the submerged structure is can then be

obtained from the following equation:

This value is used in describing the density of the structure when immersed in water. This is

henceforth referred to as the immersed steel.

4.12 RESPONSE MODELLING

The prediction of vortex induced vibrations due to the actions of the fluctuating pressure

force acting on a bluff body is very difficult. This is majorly due to the non-linearity that

exists between the interactions of the feedback between the bluff body and the fluid flow.

The model provides the “maximum steady state VIV amplitude response as a function of the

basic hydrodynamic and structural parameters” (DNV-RP-F105, 2006) The response model

has two different approaches to in-line and cross-flow responses hence does not take into

consideration the effect of multiple modes of vibration that may occur at the same location in

the structure



The amplitude response depends on the following hydrodynamic variables according to

(DNV-RP-F105, 2006). These are:

Reduced velocity

Keulegan Carpenter number

Current flow velocity ratio

Turbulence intensity

Flow angle , relative to the pipe

Stability parameter

Seabed gap ratio

Pipe roughness

In this dissertation as stated in earlier sections would only be considering the steady flow of

current past the cylinder, hence the Keulegan Carpenter number, seabed gap ratio and pipe

roughness are eliminated because it describes the ratio of drag forces and inertia forces on

bluff bodies in an oscillatory fluid flow; distance from the seabed whereas the jumper has a

more significant clearance from the mud line (doesn’t lie on the seabed) and the pipe is

considered to be smooth respectively

The variables are defined as follows:

The reduced velocity is defined as

(4.17)

Where Uc is the mean current velocity normal to the pipe; Uw is the significant wave-induced

flow velocity and fn is the natural frequency for a given vibration mode

The current flow velocity ratio (α) is defined by

(4.18)

The stability parameter Ks representing the damping for a given modal shape is given by:

(4.19)

Where me is the effective mass that represents the generalized mass corresponding to each

mode shape; and is the total modal damping ratio which is a sum structural damping, soil

damping and hydrodynamic damping which is within the lock-in is regarded as zero. A value



of 0.005 is recommended for the total modal damping ratio as this analysis does not take into

consideration pipe-soil interaction.

4.12.1 IN-LINE RESPONSE MODEL

The inline response model is represented as shown below

Figure 21: Coordinate representation of the In-Line response model

(DNV-RP-F105, 2006)

This can be can be constructed from the following coordinates

(

) (4.20)

(

) (4.21)

(

) (4.22)

(

)

(4.23)

(

) (4.24)

(4.25)

(4.26)



(

) ( (

) (

)) (4.27)

(

) ( (

) ) (4.28)

Where (AY/D) is the normalized in-line VIV amplitude as a function of VR and Ks. VR

represents the onset inline reduced velocity; end inline reduced velocity; and corresponding

coordinates for reduced velocities in between.

(4.29)

(4.30)

γf = safety factor on natural frequency = 1.15;

γk (safety factor on stability) = 1.3;

γon (factor on onset of Vortex induced vibration) = 1.1.

( ) ( ) are reduction factors which relate the turbulence intensity and

angle of attack.

( ) (

√ ) ( ) (4.31)

( ) ( )

(4.32)

The values of the turbulence intensity and flow angle with respect to the reduction function

are given by the subsequent diagram shown below:



Figure 22: Reduction function with respect to turbulent intensity and flow angle

(DNV-RP-F105)

The angle of attack on the flow relative to the jumper is 90o and hence the value of turbulent

intensity (0.2%) as specified in the computational simulation of the shedding frequency is

justified.

4.12.2 CROSS FLOW RESPONSE MODEL

The cross flow response model is as generated below:



Figure 23: Coordinate Representation of the Cross-Flow Model (DNV-RP-F105, 2006)

The following plot can be generated using the following coordinates

(4.33)

( )

(

) (4.34)

(

) (

) (4.35)

(4.36)

(

) (

) (4.37)

(

) (

) (

) (4.38)

(

) (

) (4.39)

(

) (

) (4.40)

( )

( )



( )



CHAPTER FIVE

5.0 VERIFICATION

Verification in its broadest sense is defined as a process for assessing simulation numerical

uncertainty and when conditions permit, estimating the sign and magnitude of the numerical

error itself and the uncertainty in that error estimate (Stern et al, 2001). The errors in a CFD

code can be broadly classified into two types. They are namely:

Modelling Errors and uncertainty: due to the assumptions and approximations in the

mathematical representation of the physical problem (Stern et al, 2001). These include

specification of boundary conditions, choice of turbulence models, convergence

conditions and geometry.

Numerical errors and uncertainties: are largely due to the use of iterative solution

methods and specification

A verification process was performed on the solutions of the computational processes

performed in the course of this thesis. In relation to the structural components, a grid

dependence study was carried out, while in the study of the flow past a bluff body, an

uncertainty assessment was carried in two parts. They include the errors due to grid size and

time steps; the uncertainty process as described by Freitas (1993) was carried out.

A total combination of 5 grid densities ranging from fine through to coarse mesh densities

were used including 5 time steps. The refinement factors used in the variance of the mesh

density and time steps are equal and are as outlined below:

Grid Refinement factor (discretization) Refinement factor (Temporal)

G0 do to

G1

G2

G3

G4

Table 5: Refinement factors; do is the distance of the first node to the cylinder wall and

to is the initial time step size as obtained in section 4.9.



The refinement factors were as recommended by (Viola, 2011), but is also vetoed by (Freitas,

1993) whom recommended a uniform refinement factor of approximately 1.3 which is also

acceptable by the Journal of Fluid Engineering. The choice of refinement is of utmost

importance as values less than or equal to one produces small changes to the solutions

obtained and sensitivity of the control parameters been evaluated would be difficult to obtain

and/or quantify.

The details of the mesh densities and corresponding number of elements and nodes are

presented in the table below:

Grid Number of

Nodes Number of elements

G0 59400 59100

G1 46560 46320

G2 35532 35344

G3 27600 27450

G4 21480 21360

Table 6: Mesh densities

5.1 DISCRETIZATION UNCERTAINTY

In order to ascertain the uncertainty of the solutions obtained from the selected grid G0, a

discretization uncertainty methodology acceptable for publications in the Journal of Fluid

Engineering was implemented. It prescribes for the solution of the uncertainty due to only

discretization a set of (3) significantly completely different grids be employed. The outline

presented herein is called the fine grid convergence index GCIfine and is as enumerated as

follows:

The control parameter employed in this study are averaged values of the coefficient of drag

as obtained from each simulation spanning from fine to coarse as given by the refinement

factors given in Table (5.0a) whilst taking the precaution of neglecting the transient values.



Control Parameter

(Drag coefficient) Averaged Values

G0TS0 ( 0) 1.5199

G1TS0 ( 1) 1.5359

G2TS0 ( 2) 1.5430

G3TS0 ( 3) 1.5105

G4TS0 ( 4) 1.4995

Table 7: Averaged control values as obtained from the simulation of flow past a

cylinder (Re = 11769)

The chosen three significant grids are G0TS0; G2TS0 and G4TS0.

The first step requires the definition of a representative mesh or grid size and where integral

values such as the coefficient of drag is used as the control parameter in two dimensional

simulations the equation below defines such:

∑ (

) (5.0)

Where h is the representative mesh size; is the area of the ith

cell (and is obtained from

the total surface area of the grid employed) and n is the total number of cells used for

computations.

(

)

The value of h for G0 would be equal to

Similarly, the values of h for G2 and G4 are evaluated to be 0.3012 and 0.388.

The condition of continuity which is is satisfied. The refinement factor “r” is

defined as:

The values obtained for these ratios are



The apparent order of accuracy “p” is obtained from the fixed point iteration solution of the

equation below; with the initial guess been equal to the first term.

( )| | | | ( )| (5.1)

(5.2)

(5.3)

Therefore

(

) (5.4)

This is indicative of an oscillatory convergence as described in Freitas (1993).

( ) (

) (5.5)

( ) (

)

The equation (5.1) is solved iteratively after substituting the necessary values of equations

(5.2 -5.5) in MATLAB setting the initial value of iteration to 0; the M-script and solution

script are attached in appendix A.

The order of accuracy “p” is obtained as 2.5

The extrapolated values of the control parameter between the range been considered are

obtained using the equations that follows:

( )

( )

( )( )

( )

And



( )

( )

( )( )

( )

The approximate relative error is calculated using the equation (5.6).

|

| (5.6)

|

|

The extrapolated relative error is calculated using the equation (5.7)

|

| (5.7)

The fine grid convergence index which determines the uncertainty of the reported solution

due to the discretization errors is calculated using the equation (5.8)

(5.8)

The discretization uncertainty is defined in terms of percentage and in this case is equal to

2.098%. It was earlier stated that this solution due to the discretization was oscillatory; this

implies that the values of uncertainty in the solution cannot but be determined ambiguously.

In this case, the uncertainty of the solutions determined from the grid lies in amplitude

defined by Roache (1998) as:

( ) (5.9)

Where U represents the uncertainty; and SU and SL represents the upper and lower bounds of

the solution oscillations. The solution to the above equation requires more than the three

solutions that would have been hitherto considered for the verification procedure.

The uncertainty due to discretization is thus given as

(( ) ( )) (5.10)

( ) = 0.0102 = 1.02%.



The uncertainty of the solutions herein obtained lies within the bounds of ±1.02%.

The temporal uncertainty calculations follow the same sequence of calculations except that

the values of the control parameters do change. In performing this bit of calculating the

simulation numerical uncertainty; the selected grid (fine) was used in performing simulations

where in the time step size was increased by the refinement factors as shown in Table 5.0a;

the obtained control parameters are as shown in the table below:

Control Parameter

(Drag coefficient) Averaged Values

G0TS0 ( 0) 1.5199

G0TS1 ( 1) 1.5135

G0TS2 ( 2) 1.4975

G0TS3 ( 3) 1.4994

G0TS4 ( 4) 1.4837

Table 8: Averaged Drag Coefficients as obtained from the simulation of flow past a

cylinder with varying time steps (Re = 11769)

The value of S as obtained during the calculation of the temporal (time step) uncertainty

implies a condition of monotonic convergence and all of the solutions obtained lie

asymptotically to the condition of convergence.

The variation of convergence conditions obtained during this verification exercise could

imply that as the time step with which the solution of the Unsteady Reynolds Averaged

Navier Stokes Equation was been carries out was increased; the solutions obtained couldn’t

capture the magnitude of unsteadiness in the solution.

The report of this verification exercise yields the discretization uncertainty and temporal

uncertainty and as are as shown in the table:



5.2 GRID DEPENDENCE STUDY

A grid dependency study was carried whilst carrying out the modal analysis of the rigid

jumper structure. Utmost care was taken to preserve the element shapes; this was

accomplished by testing the model through a Finite Element Modeller component of the

ANSYS WORKBENCH before the modal analysis was performed.

Uncertainty due to Grid Uncertainty due to Time steps

= Coefficient of Drag

(with Oscillatory

convergence)

= Coefficient of Drag (with

Monotonic convergence)

r21 1.291 1.291

r32 1.288 1.288

0 1.5199 1.5199

2 1.5430 1.4975

4 1.4995 1.4837

p 2.48 1.87

ext20

2.3 1.48

ext42

1.59 1.61

ea20

1.519% 1.519%

GCIfine20

2.098% 1.578%

Table 9: Verification Results



IMMERSED STEEL STEEL IN AIR

Number of

Elements 440000 330000 132000 440000 330000 132000

Mode Frequency (Hz) Frequency (Hz)

1 0.2632 0.2646 0.2646 0.2454 0.2468 0.2467

2 0.7024 0.7056 0.7055 0.6550 0.6580 0.6579

3 0.8811 0.8864 0.8863 0.8216 0.8265 0.8264

4 1.3690 1.3753 1.3751 1.2765 1.2824 1.2822

5 1.7094 1.7182 1.7180 1.5939 1.6021 1.6019

6 1.8047 1.8158 1.8158 1.6828 1.6932 1.6931

7 2.2675 2.2810 2.2810 2.1143 2.1269 2.1268

8 2.3891 2.4007 2.4004 2.2277 2.2385 2.2382

9 3.0650 3.0795 3.0791 2.8579 2.8714 2.8710

10 3.7198 3.7408 3.7406 3.4684 3.4880 3.4878

11 3.8737 3.8953 3.8951 3.6119 3.6321 3.6319

12 5.3453 5.3764 5.3762 4.9841 5.0132 5.0129

13 5.7226 5.7578 5.7577 5.3359 5.3687 5.3687

14 6.5198 6.5553 6.5548 6.0792 6.1124 6.1119

15 7.9261 7.9749 7.9748 7.3906 7.4360 7.4360

16 8.7031 8.7454 8.7444 8.1151 8.1545 8.1536

17 10.0454 10.0984 10.0976 9.3667 9.4161 9.4153

18 11.0001 11.0661 11.0654 10.2568 10.3183 10.3177

19 11.0047 11.0688 11.0687 10.2611 10.3209 10.3208

20 13.5673 13.6490 13.6487 12.6505 12.7268 12.7265

Table 10: Grid dependence

Convergence of the solutions is observed between as the mesh is refined further. The

solutions obtained from the second refinement are used for all calculations hence forth.



CHAPTER SIX

6.0 RESULTS AND DISCUSSION

6.1 INTRODUCTION

The results of all the simulations and other findings not hitherto discussed are outlaid in this

chapter. It presents an explanation of trends for all of the varied properties and their attendant

consequences.

6.2 MODAL ANALYSIS

The structural response of the jumper included nonlinear geometric effects such as large

displacements and rotations. These displacements have their characteristic mode shapes and

frequencies; particularly of interest were single mode shapes that were observed in the Z and

Y directions. These two axes represents the direction of the inline and cross-flow response

due to vortex induced vibrations should the corresponding frequencies be low enough to be

excited according to the DNV standard as highlighted in equation (3.7). A total of 20 modes

were extracted with the boundary conditions idealizing a typical subsea jumper as installed

empty between two subsea facilities and the mode shape were observed to be principally in-

plane or out-of –plane.

The table below shows the results of the natural frequency of the rigid jumper under

observation both in air and in still water.

Natural Frequencies (Hz)

Mode In still water In air Mode Shape

Effective Mass (Kg) in

still water per unit length

of the free span

1 0.2646 0.2467 Out of Plane 278.8

2 0.7056 0.6579 In-plane 17.39

3 0.8864 0.8264 Out of Plane 18

4 1.3753 1.2822 In-plane 316.01

5 1.7182 1.6019 In-plane 129.183

6 1.8158 1.6931 Out of Plane 75.86

7 2.281 2.1268 Out of Plane 1.33

8 2.4007 2.2382 In-Plane 26.1519

9 3.0795 2.871 In-Plane 72.22

10 3.7408 3.4878 Out of Plane 0.822

11 3.8953 3.6319 Out of Plane 1824.88

12 5.3764 5.0129 In-Plane 11.1088

13 5.7578 5.3686 Out of Plane 55.6498



Natural Frequencies (Hz)

Mode In still water In air Mode Shape

Effective Mass (Kg) in

still water per unit length

of the free span

14 6.5553 6.1119 In-Plane 29.048

15 7.9749 7.436 Out of Plane 71.4604

16 8.7454 8.1536 In-Plane 128.073

17 10.0984 9.4153 In-Plane 48.2193

18 11.0661 10.3177 In-Plane 21.9785

19 11.0688 10.3208 Out of Plane 35.3162

20 13.649 12.7265 Out of Plane 3.25802

Table 11: Mode Shapes, Natural Frequencies and Effective masses of the Rigid Jumper

Spool (immersed in still water and dry).

The lowest natural frequency is observed to be out-of plane and is approximately a third of

the consecutive mode frequency which is observed to be in-plane. The first two modes are as

shown in the figures below:

Figure 24: Mode shape (1) out-of- plane translating to In-Line VIV



Figure 25: Mode shape (2) in-plane- translating to Cross- flow VIV; shaded portion

represents the undeformed model

There is an observed increase in the natural frequency in comparison with the natural

frequencies of the structure in air. This is presumed to be as a result of the increased stiffness

as compared to the reduced weight of the structure.

The numbers of modes considered for analysis were ten (10). This is because using the

constant Strouhal number in the subcritical regime in which this analysis takes place, the

primary vortex shedding frequency is approximately 0.27Hz and that the most significant

excited frequencies of the structures are more than 9; which in this case is 2.43Hz. It was also

observed that the rotational component of the modes were significant increasing after the

mode 10.

6.3 VORTEX SHEDDING FREQUENCY

In figure 26, the values of the dimensionless lift force acting on the cylinder plotted against

the flow time is shown. It is a direct consequence of the vortices been shed from the lower

and upper surfaces of the cylinder and hence can be used in determining the vortex shedding

frequency of the flow past the cylinder.

The vortex shedding frequency was determined from the spectral density distribution

function of the measured lift fluctuations (FFT Technique). In the range of Reynolds numbers



observed, a single vortex shedding frequency was clearly observed as shown in the figure 28;

this figure only stands as an example as the other figures and the related M-script is attached

in appendix (A).The maximum lift on the cylinder occurred at the same value of flow speed

and continued to oscillate at this frequency at higher Reynolds numbers where the correlation

of the shedding of vortices along the cylinder span have reduced and the effects of

turbulence due to the instability of the boundary layer had been obliterated. However in the

lower bounds of the Reynolds numbers considered, the maximum lift exceeded the value of

the flow speed.

Figure 26: Coefficient of Lift against flow time (seconds) (Re = 58982)



Figure 27: Coefficient of Lift against flow time (seconds) (Re = 11796)

Figure 28: Power Spectral Density of the Coefficient of Lift (Re = 58982)



The table below shows the obtained values of the vortex shedding frequency.

Reynolds

Numbers

Vortex Shedding Frequency

(Strouhal Number)

11796 0.1936

23593 0.1936

35389 0.1936

47185 0.2041

58982 0.2153

70778 0.227

82547 0.227

94371 0.2331

106167 0.1836

117963 0.2233

120760 0.2233

Figure 29: Vortex shedding frequencies for observed Reynolds Numbers

The above vortex shedding frequencies refer to the corresponding Strouhal numbers in this

case. This is because as the equation (2.3) infers if the flow velocity and diameter are equal to

unity as employed throughout all the simulations of the flow past the cylinder in this case,

then the Strouhal number is equal to the vortex shedding frequency.

It was observed that during each cycle of vortex shedding a vortex is alternately shed from

the upper and lower surfaces of the cylinder. The travel downstream to form the Von-

Karman vortex street as shown in the figure below:




clearly depicting the 2S mode of vortex shedding- two single vortices are shed at each

cylinder oscillation cycle.




depicting the obliteration of the shed vortices due to the turbulent diffusion as described

in Roshko (1952).

6.4 EFFECTS OF REYNOLDS NUMBERS ON FLOW PROPERTIES

The Reynolds Numbers covered in this study spanned within the subcritical region and was

fully spread over three regimes as described in the Zdravkovich (1997) classification.

6.4.1 STROUHAL NUMBER

It was observed that the Strouhal number increased with increasing Reynolds numbers. In the

lower regimes of the Reynolds numbers considered, the Strouhal number increased

asymptotically with the constantly reported uniform Strouhal number for the subcritical range

until a divergence was observed at Re = 58982 additional simulations carried out for

Reynolds numbers in between ranges (58982 – 94371) yielded no significant difference in

trends. The observed dip noticed at Re = 106167 corresponds to the empirical value of 0.186



which Norberg (2003) predicts to be a critical value of Strouhal Number and states occurs at

Re = 1.6 × 105. This point marks a transition phase between the TrSL3 and TrBL. In this

phase the relative shedding bandwidth passes through a maximum. However, the continued

increase of the Strouhal number after this dip is due to the reattachment of the turbulent shear

layers.

The obtained values are compared with the empirical straight line fit of many experiments as

obtained in Norberg (2003) and Blevins (2009).

Figure 32: Flow past a Cylinder: Variation of Reynolds Number against shedding

frequency

6.4.2 DRAG

The unsteady component of the drag was found in all variations of the Reynolds numbers to

be approximately 0.22 as shown in the diagram below:

0

0.05

0.1

0.15

0.2

0.25

1.E+04 1.E+05

Vo

rte

x Sh

ed

ing

Fre

qu

en

cy (

fs)

Reynold Number (Re)

Present Simulation 0.2%Tu

Norberg (2002)

Blevins (2009)

Achenbach,1982



Figure 33: Unsteady component of drag (Re = 23,593)

The reference area used in the determination of the coefficient of drag during all of these

simulations is the unit area specified by the diameter of the cylinder and span-wise length of

the cylinder, which in a 2-Dimensional Case is equal to unity (1).

In the subcritical regime, there is a rapid decrease in the mean drag coefficient. A decreasing

trend in the coefficient of drag was observed as Reynolds numbers was increased, this is

primarily due to the delayed separation of the turbulent boundary layer which is formed as

the flow transits from the laminar separation that occurs in the front of the cylinder at the

lower Reynolds numbers. The reduction which in effect continued past the known boundaries

for the sub-critical range i.e. (1 – 1.2) is due to the effects of the free stream onset turbulence

which as Blackburn and Melbourne (1996) found out hasten the transition into the

supercritical Reynolds regime.

The above variation between the coefficient of drag and the Reynolds Number follows

Roshko (1954) conclusion of the inverse relationship that exists between these two variables.



Figure 34: Present Simulation: Averaged Values of the Coefficient of Drag (Blue);

Scatter Diagram of the Coefficient of Drag of a Cylinder in atmospheric turbulence.

(Yuji, 2004) (Brown)

Figure 35: Pressure distribution across the transverse section of the cylinder in cross

flow

Roshko (1954) asserts from experiments that the coefficient of pressure between (30o - 35

o)

should be zero and this is assumed to be the position of the cross-flow (stagnation point).

The coefficient of pressure is directly related to the point of separation of the turbulent layer

from the cylinder surface. The point of separation is the point at which the boundary layer

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0.00E+00 2.00E+04 4.00E+04 6.00E+04 8.00E+04 1.00E+05 1.20E+05 1.40E+05

Co

eff

icie

nt

of

Dra

g

Reynolds Numbers

Present Simulation

Yuji (2004)

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

Co

eff

icie

nt

of

Pre

ssu

re (

Cp

)

Distance along the Cylinder axis(Degrees)

Re= 35389

Re= 82547

Re = 117963



becomes unstable due to the adverse pressure gradients. It can be implied as shown above

that the angle of separation of the boundary layer from the surface area of the cylinder

increases from about 70o in lower sub critical Reynolds numbers to about 81

o with increasing

Reynolds numbers.

6.4.3 RESPONSE MODELLING

6.4.3.1 DNV SCREENING

The response modeling commences with the screening criterion as shown in the equation

(3.7). The reduced velocity is directly related to the shedding frequency by the equation

(2.2)

Simplifying the equation (3.7) above by substituting (2.2) into (3.7);

(6.4.3.1)

Mode Natural

Frequencies

Safety factor applied

on Natural

frequencies

Vortex Shedding

Frequencies

Safety factor applied

on vortex shedding

frequencies

1 0.2646 0.2301 0.1936 0.2130

2 0.7056 0.6136 0.1936 0.2130

3 0.8864 0.7707 0.1936 0.2130

4 1.3753 1.1959 0.2041 0.2245

5 1.7182 1.4941 0.2153 0.2368

6 1.8158 1.5790 0.2270 0.2497

7 2.2810 1.9835 0.2270 0.2497

8 2.4007 2.0875 0.2331 0.2564

9 3.0795 2.6778 0.1836 0.2020

10 3.7408 3.2528 0.2233 0.2632

0.2233 0.2632

Table 12: DNV Screening Criterion

It was observed from the above that Mode 1 (i.e. the fundamental frequency) would only be

excited by the first four flow velocities been considered. It also shows according to the DNV

criterion that all the vibration modes of the structure would be excited even with the lowest

flow velocities. This indicates that the jumper is excited into the resonant mode at all



frequencies; however it must be noted that the response model of the DNV RP-F105 is based

on free span phenomena which is dominated by resonance.

6.4.3.2 RESPONSE MODELLING

The coordinates of the response model as calculated using the criteria as listed in section 412.

It must however be stated that for the excited modes only the highest flow velocities at which

they are excited at are considered in the plot. This is following the reasoning that the highest

amplitude of oscillation would be located at these velocities. The coordinates are as obtained

and shown in the table below:

Mode

Effective

Weight (Per

Unit Length

of the Pipe

Span)

Velocitie

s

VIL

R,onse

t V

ILR,1 AY,1/D V

ILR,2 AY,2/D V

ILR,end

1 279

0.05 1.2943 1.8587 0.0564 3.7581 0.0415 3.841

0.1 1.2943 1.8587 0.0564 3.7581 0.0415 3.841

0.15 1.2943 1.8587 0.0564 3.7581 0.0415 3.841

0.2 1.2943 1.8587 0.0564 3.7581 0.0415 3.841

3 18.12

0.05 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.1 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.15 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.2 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.25 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.3 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.35 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.4 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.45 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.5 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

0.55 0.9091 2.6288 0.172 4.3088 0.0742 4.4572

6 75.821

0.05 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.1 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.15 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.2 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.25 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.3 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.35 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.4 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.45 0.9091 2.3733 0.1464 4.187 0.067 4.3209

0.5 0.9091 2.3733 0.1464 4.187 0.067 4.3209



0.55 0.9091 2.3733 0.1464 4.187 0.067 4.3209

7 1.33

0.05 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.1 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.15 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.2 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.25 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.3 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.35 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.4 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.45 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.5 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

0.55 0.9091 2.7032 0.1794 4.3443 0.0763 4.4969

10 0.822

0.05 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.1 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.15 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.2 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.25 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.3 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.35 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.4 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.45 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.5 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

0.55 0.9091 2.7055 0.1796 4.3453 0.0764 4.4981

Table 13: Coordinates of the In-Line Response VIV

Figure 36: Plot of In-Line Response using obtained Coordinates

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 1 2 3 4 5

In-L

ine

VIV

No

rmal

ize

d

Am

plit

ud

e

Reduced Velocity

In-Line Response Model

Mode1

Mode 3

Mode 6

Mode 7

Mode 10



Figure 37: Plot of Cross-Flow Response using obtained Coordinates

The excitation region calculated in relation with respect to the reduced velocity is 0.11 ≤ Vr ≤

6.08 which is indicative that the jumper in consideration would experience both In – Line and

Cross – Flow VIV. The cross-flow response which is due to lift force acting on the cylinder

was more than one diameter which is due to the fact that the safety factor applied to the

structural frequency and vortex shedding frequency into the lock-in region.

The in-line response was observed to be about 10% of the cross flow response and was found

to be very dependent on the mass ratio as against the flow velocity; this was also observed in

Guilmineau and Queutey (2004). Sumer and Fredscoe (1999) defined some boundaries for

the stability of the in-line vibrations which are 1 ≤ Vr ≤ 2.5 and 2.5 ≤ Vr ≤ 4 which represent

the first and second instability region. As observed the in-line response due to VIV for this

jumper exceeds the second instability region, whence the structure vibrates at more than three

times the Strouhal frequency.

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Cro

ss-F

low

VIV

No

rmal

ize

d

Am

plit

ud

e

Reduced Velocity (Vr)

Cross-Flow Response Model

Mode2

Mode4

Mode5

Mode8

Mode9



CHAPTER SEVEN

7.1 CONCLUSION

The results for the fluid structure interactions of a rigid subsea jumper in presented in this

work. It predicts the amplitude of vortex induced vibrations a rigid jumper with a steady

current flowing past using a semi-empirical response model.

It established the effects of Reynolds numbers and free stream turbulence on the flow past a

bluff body i.e. cylinder in two dimensions (Turbulence Intensity = 0.2%). The results

obtained gave general agreement with previously published works. The turbulence model

employed in the simulation also shows good predictions of the transitions in the wall

boundary layers showing an appreciable relationship (trends) at Reynolds number = 106167.

The modal analysis predicted an increase in the natural frequencies of the structure when it is

immersed in still water due to an increase in the stiffness to mass ratio.

Finally, the in-line response of the rigid jumper was found to about 10% of the cross flow

response, but was in a magnitude of 3 times the vortex shedding frequency, establishing the

significance of in-line response of solid structures in the analysis of vortex induced

vibrations.

7.2 RECOMMENDATIONS

This study has sought to put additional information into the public domain about vortex

induced vibrations of a rigid subsea jumper which is a popular piece of kit in the offshore oil

and gas industry.

The rigid jumper in this study has been considered as installed i.e. empty, further areas of

study could include the impact the contained fluid, the application of another response model

(which is not subjected to the empirical methods such as one described in Choudhury, (2011)

in determining the amplitude of vibrations.

With improved computational prowress and technical know-how the full 3- Dimensional

structural analysis coupled with fluid motions may be performed. However a very immediate

step might be to compute the effects of the vibrations on the fatigue life of the structure.



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APPENDIX A – M-SCRIPTS

A-1 – M - Script for fixed point iteration solution of the order of accuracy

clear all

maxIterations = 100;

converged = 0;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Main learning (parameter

estimation) loops

k=2;

P(1)=0;

while k<maxIterations & ~converged

P(k)=(1/0.7605)*abs(-2.207+log(((2.1394^P(k-1))+0.11)/((3.1044^P(k-1))+0.11)));

e=P(k)-P(k-1);

k = k+1;

% Test for convergence

if (e<0.00001)

converged = 1;

end

end

P

A – 2 – M- Script for Power Spectral Density

filename='23593.xls';

a=xlsread(filename);

t=a(:,1);

y=a(:,2);

plot(t,y)

Y=fft(y,5000);

pyy=Y.*conj(Y)/5000;

f=linspace(0,32,length(pyy));

plot(f,pyy)

plot(f(1:2500),pyy(1:2500))

set(gca,'XScale','log')

msgbox('click in the plot')

[X Y]=ginput(1)



APPENDIX B – PLOT OF COEFFICIENT OF LIFT AND POWER SPECTRAL

DENSITIES

B-1- COEFFICIENT OF LIFT

Plot of Lift Coefficient for Reynolds

Number = 11796


Number = 23593


Number = 47185


Number = 58928


Number = 70778


Number = 82574




Numbers = 94371


Numbers = 106167


Numbers = 117963


Numbers = 120760



B-2: POWER SPECTRAL DENSITIES

Power spectral density for Reynolds

number = 11,796


number = 23,593


number = 35,963


number = 58,928


number = 70,778


number = 82,574




number = 94,371


number = 106,167


number = 117,963.