Dissertation Body
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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.”
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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
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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.
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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.
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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
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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
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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
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APPENDIX B – PLOT OF COEFFICIENT OF LIFT AND POWER SPECTRAL
DENSITIES ............................................................................................................................. 84
B-1- COEFFICIENT OF LIFT ............................................................................................ 84
B-2: POWER SPECTRAL DENSITIES ............................................................................. 86
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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
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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
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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
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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
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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
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(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
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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.
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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:
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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”
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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)
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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.
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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).
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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.
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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.
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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)
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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
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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
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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
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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
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( )
( ) (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.
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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.
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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.
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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;
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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,
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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
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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
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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.
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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
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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.
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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.
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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).
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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.
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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:
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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:
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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.
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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.
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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).
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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)
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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)
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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:
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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.
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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,
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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.
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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.
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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
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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.
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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.
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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
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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
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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)
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(
) ( (
) (
)) (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:
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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:
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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)
( )
( )
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( )
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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.
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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.
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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
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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
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( )
( )
( )( )
( )
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%.
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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:
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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
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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.
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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
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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
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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
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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)
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Figure 27: Coefficient of Lift against flow time (seconds) (Re = 11796)
Figure 28: Power Spectral Density of the Coefficient of Lift (Re = 58982)
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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:
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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.
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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).
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
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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
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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.
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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
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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
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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
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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
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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
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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)
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APPENDIX B – PLOT OF COEFFICIENT OF LIFT AND POWER SPECTRAL
DENSITIES
B-1- COEFFICIENT OF LIFT
Plot of Lift Coefficient for Reynolds
Number = 11796
Plot of Lift Coefficient for Reynolds
Number = 23593
Plot of Lift Coefficient for Reynolds
Number = 47185
Plot of Lift Coefficient for Reynolds
Number = 58928
Plot of Lift Coefficient for Reynolds
Number = 70778
Plot of Lift Coefficient for Reynolds
Number = 82574
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Plot of Lift Coefficient for Reynolds
Numbers = 94371
Plot of Lift Coefficient for Reynolds
Numbers = 106167
Plot of Lift Coefficient for Reynolds
Numbers = 117963
Plot of Lift Coefficient for Reynolds
Numbers = 120760
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B-2: POWER SPECTRAL DENSITIES
Power spectral density for Reynolds
number = 11,796
Power spectral density for Reynolds
number = 23,593
Power spectral density for Reynolds
number = 35,963
Power spectral density for Reynolds
number = 58,928
Power spectral density for Reynolds
number = 70,778
Power spectral density for Reynolds
number = 82,574
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Power spectral density for Reynolds
number = 94,371
Power spectral density for Reynolds
number = 106,167
Power spectral density for Reynolds
number = 117,963.