SIMULATION OF VORTEX INDUCED VIBRATION OF A BLUFF

BODY STRUCTURE

SITI NATASHA BINTI MALIK FESAL

A thesis submitted in fulfillment of the requirement for the award of the

Master of Mechanical Engineering

Faculty of Mechanical and Manufacturing Engineering

Universiti Tun Hussein Onn Malaysia

JULY 2015

v

ABSTRACT

Understanding Vortex induced vibration (VIV) phenomenon is essential as it plays

important role in designing marine risers which used in oil extraction from the

seabed to the offshore platforms were exposed to external flows that may trigger

dangerous VIV oscillations. The present two-dimensional numerical simulations of

circular bluff body is a continuation of previous efforts trying to study the effect of

frequency and amplitudes of the cylinder oscillation that is confined in the cross-flow

and inline flow separately. The k-ε turbulence model is used to simulate the turbulent

flow to evaluate the drag and lift coefficient of circular towards the flow

characteristics which used time independent test (transient) and tested at different

Reynolds number between 10000 and 100000 with uniform velocities of 1.35m/s and

13.5m/s. Results from dynamic response of a cylinder bluff body vibrating at

frequencies variation of 1.48 Hz, 2.77 Hz and 3 Hz within 0.3m, 0.5m and 0.7m

amplitudes variation were observed in this study. It is shown that for inline flow, the

vibration at 0.3m amplitude is significantly low for drag and lift coefficient value for

Re at 100000 compared to Re at10000. Meanwhile for the cross flow value it is

observed that gives high percentages with 39% of drag coefficient and with 59% of

lift coefficient compare to inline flow at high amplitude. However at low mode

amplitude the cross flow contributes more with 19% of drag coefficient and 11% for

lift coefficient compare to inline flow. The result also show that the cylinder oscillate

higher at frequency shedding value with higher magnitude for the cross flow

compare to the inline flow. Consequently, in order to get better performance, the

vortex modes in the wake of oscillating cylinder have been found to be dependent on

the amplitude distribution along the length of the model. The results concludes that

in order to avoid inevitable vibration it is advisable by increasing damping or splitter

when designing marine riser to generate more stable vortex shedding frequency.

vii

ABSTRAK

Memahami getaran Vortex disebabkan (VIV) fenomena adalah penting kerana ia

memainkan peranan penting dalam penaik laut yang digunakan dalam pengekstrakan

minyak dari dasar laut untuk platform luar pesisir telah terdedah kepada aliran laut

yang boleh mencetuskan ayunan VIV yang merbahaya. Simulasi berangka dua

dimensi masa ini badan pembohongan bulat adalah kesinambungan daripada usaha

sebelum ini cuba untuk mengkaji kesan frekuensi dan amplitud ayunan VIV.Kajian

ke atas ayunan silinder pada keadaan menegak dan melintang diuji berasingan.

Model gelora k-ε digunakan untuk menilai pekali seretan dan angkat serta ujian

terhadap masa dan pada nombor Reynolds yang berbeza diantara 10000 dan 100,000

dengan halaju seragam 1.35m/s dan 13.5m/s. Hasil tindak balas dinamik badan

pembohongan menunjukkan silinder bergetar pada perubahan frekuensi dan

amplitude pada 1.48 Hz, 2.77 Hz dan 3 Hz dan nilai amplitud adalah 0.3m, 0.5m dan

0.7m. Ia menunjukkan bahawa bagi aliran dalam baris, getaran di 0.3m amplitud

adalah lebih rendah untuk pekali seretan dan angkat pada nombor Reynolds 100000

berbanding dengan nombor Reynolds 10000. Bagi nilai aliran menegak memberikan

nilai peratusaan sebanyak 39% untuk pekali seretan dan 59% untuk pekali angkat

jika dibandingkan pada aliran melintang pada amplitud tinggi. Walaubagaimanapun

pada amplitude yang rendah,aliran menegak menyumbang lebih pada pekali seretan

sebanyak 19% dan 11% untuk pekali angkat jika dibandingkan pada aliran

melintang.Keputusan ini menunjukkan silinder berayun lebih pada frekuensi

shedding dengan nilai magnitud yang rendah pada aliran melintang. Kesimpulan

mendapati bahawa mod vorteks di tengah-silinder berayun untuk bergantung kepada

nilai amplitud di sepanjang model.Untuk mengelakkan getaran musnah,dinasihatkan

menambah badan peredam dan badan pembohongan tajam untuk menjana frekuensi

yang lebih stabil.

vii

CONTENTS

TITLE i

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENT iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENT vii

LIST OF FIGURES x

LIST OF TABLES xiii

LIST OF SYMBOL AND ABBREVIATION xiv

LIST OF APPENDICES xv

CHAPTER 1 INTRODUCTION 1

1.1 Research Background 7

1.2 Problem Statement 9

1.3 Objective of Study 10

1.4 Research Scope 4

1.5 Significant of Study 11

CHAPTER 2 LITERATURE REVIEW 7

2.1 VIV phenomenon. 8

2.2 VIV Force Direction 9

2.3 Flow around Cylinder Structure 10

2.4 Vortex Shedding 12

2.4.1 Von Kármán Vortex Street 13

2.5 Drag Force and Lift Force 16

2.6 Strouhal Number 18

2.7 VIV Equation 19

2.8 Cross Flow and In Line Vibrations of 22

viii

Cylindrical Structure

2.9 Reynolds Number, Re 24

2.10 Navier Stokes Equation 27

2.11 Computational Fluid Dynamics (CFD) 28

2.12 Turbulence model 30

2:13 Turbulence modeling 32

CHAPTER 3 METHODOLOGY 32

3.1 Research Flow Chart 33

3.2 Problem Set-Up 35

3.3 Numerical Modelling 36

3.3.1 Domain Modelling 36

3.3.2 Meshing 38

3.3.3 Solver Setting 39

3.3.4 Material Properties 40

3.3.5 Boundary Condition 41

3.3.4 Dynamic Mesh parameters and 43

Zones

3.3.5 Solution Control 44

3.3.6 Monitoring Output 45

3.3.7 Determining the Cylinder Area 45

3.3.8 Iterate 46

CHAPTER 4 RESULTS AND DISCUSSION 45

4.1 Analysis on Flow around Circular Cylinder 46

4.1.1 Comparison with Others Work 48

Towards The Fluid Flow

4.2 VIV of Cylinder With One Degree of Freedom 49

4.2.1 The Effect of Frequency towards 50

The Fluid Flow

4.2.2 Effect of Amplitude towards 51

The Fluid Flow

4.23 Effect of Pressure Distribution 54

Towards The Fluid Flow

4.4 Vortex Shedding In The Wake 58

ix

CHAPTER 5 CONCLUSION AND RECOMMENDATIONS 611

5.1 Conclusion 610

5.2 Recommendations 63

REFERENCES 65

APPENDICES 70

x

LIST OF FIGURES

2.1 Velocity Vector Plot Depicting VIV 8

2.2 Force direction on the model 9

2.3 Flow around circular cylinder 11

2.4 Sketch of the physical uniform flow and 13

2.5 Von Kármán Vortex Street at increasing 14

2.6 Von Kármán Vortex streets forming in the wake 15

2.7 Kármán “vortex street” behind the circular cylinder 16

2.8 Vibrations of forces acting around a circular cylinder 17

submitted to vortex shedding.

2.9 Strouhal number as a function of Reynolds number for 18

2.10 Effect of Reynolds number on Strouhal number for 19

2.11 One Dimension Spring-Pot System 19

2.12 Definition sketch of vortex-induced vibrations 22

2.13 Frequency response against reduced velocity 23

2.14 Fingers of Dye, filament-line sketch of the vortex 25

2.15 Schematic view of the CFD 29

3.1 Project flow chart 33

3.2 CFD workflow 34

3.3 General Problem Set Up 35

3.4 Geometry Process created in ANSYS Fluent Design Modeller 37

3.5 Computational geometry and boundary condition 37

3.6 Meshing of circular bluff body 38

3.7 An unstructured triangular mesh around a circular cylinder 39

3.8 Display Panel for Solver Setting. 40

3.9 Label of boundary types 41

3.10 Velocity Inlet Control Panel 42

xi

3.11 Solution methods and controls panel 44

4.1 Time histories of the drag and lift coefficients for Re =10000 46

4.2 Contour of velocity magnitude at Re = 10 000 48

4.3 Contour of vorticity velocity at Re = 10 000 48

4.4 Mean drag coefficient as function of reduced velocity. 50

4.5 Lift Coefficient as a function of reduced velocity Error! Bookmark not

defined.

4.6 Mean drag coefficient for different forcing amplitudes 52

4.7 RMS lift fluctuations for different forcing amplitudes 53

4.8 Maximum vertex area velocity magnitude surface on the 54

4.9 Pressure coefficient distribution around circular (Re = 10 000) 55

4.10 Pressure coefficient distribution in Cross flow direction 56

And sequence of cross flow periodic motion

4.11 Pressure coefficient distribution in Inline flow direction 577

And sequence of inline flow periodic motion

4.12 Vortex Pattern For Rigid Cylinder 588

4.13 Vortex Pattern Comparison Cross Flow Direction 59

4.14 Formation of 2S Mode in the initial branch 60

4.15 Formation of 2P mode at the upper and lower branch 60

xiii

LIST OF TABLES

Table 2.1 Flow regime around smooth, circular cylinder in 12

Steady current

Table 2.2 Relationship of Amplitude and Frequency Mode 21

Table 2.3 Classification of disturbance free flow regime 26

Table 2.4 Characteristics of laminar and turbulent flow in pipes 31

Table 2.5 Features of Turbulent Flows 31

Table 3.1 Number of element and node for circular cylinder 39

Table 3.2 Boundary condition 41

Table 3.3 Solutions method details for Fluent 47

Table 4.1 Physical quantities in the flow past a rigid cylinder, 49

At Re = 10000

xiv

LIST OF SYMBOL AND ABBREVIATION

VIV Vortex Induce Vibration

FSI Fluid Structure Interaction

CFD Computational Fluid Dynamic

PISO Pressure Implicit with splitting operators

r.m.s root mean square

CF Cross Flow

IL Inline Flow

1dof One degree of freedom

St Strouhal Number

f Frequency

fex Frequency Oscillation

fst Frequency Shedding

d Width of shedding body

𝑈 Fluid velocity

Re Reynolds Number

𝑈𝑟 Velocity Reduction

𝐴𝑦 Vibration Amplitude

𝜇 Free stream velocity

𝐷 Diameter of the bluff-body

𝑣 Kinematic viscosity of the fluid

Ρ Density of the fluid

𝐶𝑙 Lift coefficient

𝐶𝑑 Drag coefficient

F Amount of pressure

A Area

xv

LIST OF APPENDICES

A User Define Function 70

B Parameter Analysis and Sample Calculation 72

C Mesh Updating For Cross Flow and Inline Flow 77

D Analysis Data on Cylinder 80

E Project Gantt Chart 105

CHAPTER 1

INTRODUCTION

1.1 Research Background

Vortex-induced vibrations of bluff bodies have been studied for a long time due to

their importance in both academic researches and engineering applications. The

prediction of offshore structures subjected to VIV is one of the most challenging

tasks since VIV is a subset of the broader field, the complicated fluid-structure

interaction (FSI) phenomenon is involved. For the purpose of understanding this

issue, numerous experimental and numerical simulation studies have been carried out

on this fluid-structure problem, often dealing with only one degree of freedom (1dof)

due to the difficulties and complexities associated mainly with VIV covering from

rigid cylinders studies to elastic structured cylinder. For instance, Zhang et.al. (2009)

confirms that VIV caused by vortex shedding from the cylinder is a typical fluid-

interaction problems.

Generally there are two types of motion involved in order to perform the

simulation either free induced motion or force induced motion. The phenomenon of

vortex shedding lock-in due to either forced or self-induced cylinder oscillations in

crossflow has been extensively studied over the past decades (Konstantinidis et.al.,

2003). According to Zhou et.al (1998), different experimental approach to investigate

FSI depending on the studies set out to be obtain. For example if the effects of

structural vibrations on the wake behavior are of interest only, the approached

8

adopted is to force the structure to vibrate usually in one direction at prescribe

amplitude and frequency.

According to Tang et.al (2013), several researcher pointing out experimental

works focused on high Reynolds numbers such as Feng (1968), Brika and Laneville

(1993) and Khalak and Williamson (1999) was among the first researchers to

investigate the VIV problem. Mittal and Kumar (1999), Shiels et al (2001) and

Plazeck et al (2009) among the namely few researchers carried out numerical

simulations of an elastically mounted circular cylinder subjected to VIV.

Observation by Pattel (2010) shows that flow is very sensitive to the changes

of Reynolds number, a dimensionless parameter representing the ratio of inertia force

to viscous force in a flow. The periodic flow force, generated by the periodic vortex

shedding, affect the cylinder vibration as well as the fluid flow around cylinder, the

fluid force and the vortex patterns ,thus forming the complex fluid-structure

interaction. Zhang et.al (2009) confirmed that flow induced vibration of an elastic

circular cylinder is a nonlinear case.

The VIV of cylinders bluff body influences the dynamics vibrations of

offshore riser tubes bringing oil from the seabed to the surface.as well as others,

submarine towed array cables and ship moorings in water as well as atmospheric

problems including smokestacks, cellular towers and buildings.

According to Michael (1994) the most essential VIV parameters are the lift

coefficient, the shedding frequency (Strouhal number), the correlation length, and the

shedding frequency bandwidth and are to be obtained from experimental model test.

Similarly, lock-on can also be observed when the cylinder is stationary and

the incident mean flow has a periodic component superimposed upon it (Barbi et al.,

1986; Armstrong et al., 1986; Hall and Griffin, 1993). This type of flow, termed

perturbed flow here, is equivalent to in-line oscillations of the cylinder in a steady

incident flow when the perturbation wavelength is large compared to the cylinder

diameter (Griffin and Hall, 1991)

There are several others of important parameters in determining VIV

response due to vortices. They are reduced velocity, the dimensionless amplitude is

the main parameter in performing experiment regarding vortex induce vibration.

Reduce velocity is a length of the path per cycle / diameter of the cylinder or can be

written as U/fex D where U is a flow velocity, fex is the frequency of oscillation of a

vibrating body and D is cylinder diameter (characteristic width). The Norwegian

9

Marine Research Institute (MARINTEK) concluded that among the drag effects of

VIV on marine risers are reducing fatigue life time, increase axial tension, increase

extreme loads and increase drag resistance.

Previous research had been conducted extensively to study the interaction

between vortices and structures. According to Song et al. (2011), majority of the

research is focused on rigid cylinders interaction instead of flexible structures. There

are few numerical simulation performed in finding VIV response with flexible

structures such as Bai et al. (2005).

The computational method is used to analyse a bluff body responses when

exposed into specific fluid flow condition. A model is developed using Fluent 15, an

application tools for fluid dynamics analysis from ANSYS Inc, whilst the principle

data and flow condition is taken from numerical simulation. Both results will be

compared to find dissimilarities as many discrepancies occur between measurements

and predictions from empirically based codes and CFD is always reported (Song et

al., 2011).

1.2 Problem Statement

Unsteady oscillatory phenomenon, which causes the pressure distribution around the

cylinders to fluctuate, resulting in forces perpendicular to the flow and bluff body

structure. These forces excite forced oscillations of the cylinder known as vortex-

induced vibrations. When the frequency of VIV approaches one of the natural

frequencies of the structure, the amplitude of vibration is enhanced through a

resonant phenomenon known as lock-in or synchronization of the wake. Lock-in and

oscillation frequencies can also take place when a cylinder oscillates in line with the

incident flow (Griffin and Ramberg, 1976; Ongoren and Rockwell, 1988)

Although the phenomenon of VIV of bluff bodies has been studied

extensively, the vast majority of previous studies have concentrated solely on

transverse vibrations since the fluctuating lift is the dominant force (Konstantinidis

et.al. 2003).Various response curves have been measured showing the amplitude,

frequency, and phase of cylinders undergoing VIV. However, since the fluctuating

forces responsible for these oscillations have unsteadiness in both lift and drag, the

10

role of stream wise vibrations cannot be ignored. To understand the relationship

between frequencies, amplitude, the motion of the cylinder as well as vortex

shedding flow, studies on the flow past a circular cylinder undergoing forced

vibration should be carried out.

1.3 Objective of Study

The objectives of this research are:

To investigate the effect of the frequency, amplitude, subjected to cross-line and in-

line motion of the bluff body on the flow velocity and pressure in the time response

(unsteady state).

1.4 Research Scope

The scopes can be listed as:

i. Develop the Computational Fluid Dynamics (CFD) model for the flow

simulation of a bluff body.

ii. Cross-sectional area of the bluff bodies 0.0079m2 for circular

iii. Shape is allow to move in in-line flow (X-direction) and cross-flow

(Y-direction)

iv. Test at frequency variation motion of the shape ranging from 1.39 Hz,

2.77 Hz and 3.0 Hz.

v. Test at amplitude variation motion of the shape ranging from 0.3m,

0.5m and 0.7m.

vi. Test at Reynolds number variation ranging between 10000 and

100000.

11

1.5 Significant of Study

The effect of circular cylinder oscillation in lateral (cross flow) and transverse

motion (inline flow) towards the flow profiles will be investigated using a

simulation. Addressing from low to high Reynolds numbers within the forcing

frequency of the structure will be varied accordingly at various values of amplitude

in order to analyze the difference. It is believed that such simulations are important

for understanding vortex induced vibrations, characterized by controlled oscillations.

The knowledge gained from this study later may contribute significantly to the

understanding of the VIV and for testing and validation of numerical simulations of

the flow around stationary and vibrating bluff bodies which are of great importance

in engineering structures especially cylindrical structures, such as subsea pipelines

and marine risers.

CHAPTER 2

LITERATURE REVIEW

This chapter introduces the background theories of CFD and the techniques for

solving fluid flow problems. Understanding CFD is important in investigating the

influences and impacts of fluid flow. The chapter consists of two main sections, by

outlines the governing mathematical equations for solving the turbulence models of

the flow around bluff bodies and roles of CFD in the study of fluid flow. In first

section, which is the major part of this chapter describes the basic concept of vortex

shedding and vortex induced vibrations are explained and previous research is

examined to illustrate the relevant parameters associated with vortex-induced

vibrations. Past simplifications in analysing vortex-induced vibrations are shown and

the relevant non-dimensional parameters for properly modelling and scaling the

forces associated with vortex induced vibrations are defined. The emphasis is given

to the last section of this chapter where the problem of mode ratio variation is

introduced with an explanation of the work performed by Dahl (2008).

Characteristics of bluff-body flows and important parameter that control such flows,

the Reynolds number, the Strouhal number, drag coefficient and lift coefficient, also

were discussed.

8

2.1 VIV phenomenon

Most of the interest in the flow around a cylindrical body oscillating

transversely to a free stream is due to its relevance to vortex-induced vibration

(VIV). The literature review shows that there is a collective opinion among experts

regarding the VIV phenomenon that arises when a body is placed in a flow and the

fluctuating lift force due the asymmetric formation of vortices in the wake causes the

body to vibrate. The primary reason for the formation of these vortices is the

frictional shear stress arising within the boundary layer, which denotes a very thin

layer in the neighborhood of the body (Gamino., 2013; Schlichting 1968). This

phenomenon of alternating vortex shedding is depicted in a two-dimensional plane as

shown in Figure 2.1.

Figure 2.1: Velocity Vector Plot Depicting VIV (Gamino, 2013)

VIV is “a direct consequences of lift and drag oscillations due to the vortex

shedding” Wanderley et.al. (2007); Bearman (2000) define for a moving cylinder

“the fluid interacts strongly with the cylinder motion and the vortex shedding

frequency is captured by the body natural frequency over a wide range of flow

speed”. For a fixed or rigid body, the vortex shedding frequency is a function of

Reynolds number only. Zhau (2013) investigated that a waves are modelled by

oscillatory flow when the hydrodynamics around slender cylindrical structures. He

also states that “a highly level of fatigue damage in a relatively short period of time

for risers that exposed to harsh ocean environments”. Bai (2005) define “VIV occurs

anytime when a sufficiently bluff body is exposed to a fluid flow that produces

vortex shedding at, or near, a structural natural frequency of the body”. VIV can

9

occur with high dangerous amplitudes as the continuous periodic vibration of the

structure could make it susceptible to fatigue failure.

Computational fluid dynamics (CFD) simulations is presented as another

option as an alternative to response models can be applied for VIV assessment to

overcome the inherent limitations of the state-of-practice engineering approach

(Gamino, 2013).

Generally modelling VIV riser behavior is dependent on a number of

empirical parameter such as Strouhal number, correlation length and lift coefficient.

Other related flow parameter such as Reynolds number, surface roughness,

Keulegan- Carpenter number, and turbulent intensity. Commonly lift coefficient and

Strouhal number is the most of the tests provide data for all parameter

2.2 VIV Force Direction

The vortex shedding induces fluctuating forces on the body and, if it is non-rigid,

causes it to oscillate in the transverse and in-line direction, thus generating a periodic

variation in the force components on the cylinder. The force components can be

divided into cross-flow (CF) and in-line (IL) directions, which is important when

doing further fatigue analysis. According to Hill (2013), each time a vortex sheds, a

force is generated both in the in-line and cross-flow direction, causing an oscillatory

multi-mode vibration as shown in Figure 2.2.

Figure 2.2: Force direction on the model

10

Consequently, as vortices are being shed on the cylinder surface, the cylinder

experiences forces which are periodic in nature. These forces cause the cylinder to

continuously vibrate as long as vortices are shed. Chakrabarti, (2005) listed that

important hydrodynamic quantities that influence VIV are: Reynolds number, lift

coefficient, correlation of force components, shedding frequencies and their

interactions, added mass or mass ratio and damping

Dynamic force in CF (transverse) direction to the flow due to vortices

reactions is referred to lift force. Transverse cylinder vibration has a large effect on

vortex shedding. The correlation of vortex shedding along the cylinder axis increases

by transverse cylinder vibration. Increased transverse vibration amplitude (Ay) also

increases the ability of the vibration to lock in or synchronize the shedding frequency

(Blevins, 2001).

According to Michael (1994), when motion of the cylinder organizes the

wake and cause the shedding frequency to abruptly jump from its nominal value fst to

a value equal to the oscillation frequency, fex, the shedding frequency said to be in

locked-in, locked on or synchronized to the cylinder frequency. The first

phenomenon reported by Bishop in 1964.

Vortices shed far from the cylinder when cylinder vibration near to its

maximum displacement. Zdravkovic (1982) agrees that the vortex is shed from the

side opposite the side experiencing maximum displacement when vibration

frequency is below the natural shedding frequency and vortex is shed from the same

side as the maximum displacement when vibration frequency above the shedding

frequency (Blevins, 2001).

Stansby (1979) found that in several key point regarding lock-in

phenomenon including shedding behavior at the edges of the lock-in frequency range

is similar for uniform and shear flow while the minimum amplitude to ignite lock-in

is increase with the increase of Re.

2.3 Flow around Cylinder Structure

When a fluid flow past a bluff body, in this case, a cylindrical structure whose

direction is perpendicular to the axis of the bluff- body such as cross-flow ,Cl the

11

bluff-body structure will try to vibrate in a direction normal to the flow direction

(Akaydin et.al 2010). This flow induced vibration configuration is controlled by the

Reynolds number (Re), could be caused by the turbulence generated by the flow

around and in the wake of the cylinder.

When sufficiently large Reynolds number (Re) exceeds a critical value,

somehow the cylinder might experience excitations or vibrations, as the unsteady

flow around the cylinder separates from a wider section of the body giving rise to

periodic vortex shedding from either side of the body, forming pressure differences

causing a net force exerted on the bluff-body in the direction perpendicular to the

flow and creating vortices in a repeating of swirling vortices known as the von

Karman vortex street.

Flow patterns around of bluff bodies under either steady or turbulent ongoing

flows have been formed by either 1 shear layer or 2 shear layers on one side or both

sides of bodies due to the flow-body interactions depends on characteristics of bluff

bodies by means sectional shape and dimension and characteristics of ongoing flow

such as steady or unsteady, wind velocity, attack angle and even in-flow movement

of body, the around-body flow shear layer can be either stability or instability (Hoa,

2005).

Figure 2.3: Flow around circular cylinder (Gamino, 2013)

12

2.4 Vortex Shedding

Reynolds (1883) was the first to propose a criterion for differentiation between

laminar and turbulent flows in his classic dye visualization and suggested a critical

value of for the upper limit of laminar flow. In a second paper in 1895, he showed by

time-averaging the Navier-Stokes equations that new extra convection terms

appeared in turbulence which have the units of stress and are therefore called

Reynolds stresses.

Theodore von Kármán is the person who first stimulated widespread interest

and published the first theoretical study of vortex streets in 1911. In 1981 Schatzman,

who studied the analysis of a model for the Von Kármán Vortex Street found that the

linear stability of the point vortex has been generalized to vortices of finite size and

can stabilize the array (Azman, 2008).The vortex shedding around bluff bodies can

be classified by some following kinds (Hoa, 2005; Matsumoto 1999):

Table 2.1: Flow regime around smooth, circular cylinder in steady current

(Hall-Stinson, 2011; Asyikin, 2012)

No separation

Creeping flow

Re < 5

A fixed pair of symmetric vortices

5 < Re < 40

Laminar vortex street

40 < Re <200

Transition to turbulence in the

wake

.

200 < Re < 300

Wake completely turbulent

A. Laminar boundary layer

separation

300 < Re < 3x105

Subcritical

13

(Continued)

A Laminar boundary

layer

separation

B. Turbulent boundary

layer separation: the

boundary layer partly

laminar partly

turbulent

3x105 < Re < 3.5x105

Critical (Lower transition)

B. Turbulent boundary layer

separation: the boundary

layer partly laminar

partly turbulent

3.5x105 < Re < 1.5x106

Supercritical

C. Boundary layer

completely turbulent at

one side

1.5 x 106 < Re < 4 x106

Upper transition

C. Boundary layer

completely turbulent at

two sides

4 x l06 < Re

Trans critical

2.4.1 Von Kármán Vortex Street

Numerous numerical and empirical methods have been studied in the analysis of

uniform flow past a rigid 2D cylinder. One of the parameters engineers study is the

formation of vortices behind the cylinders wake, which is illustrated in Figure 2.4:

Flow

Velocity Cylinder’s

wake

Figure 2.4: Sketch of the physical uniform flow and

the location of the cylinder’s wake

14

A von Kármán vortex street is the description given to an alternating pattern

of vortices. When a fluid flows over a blunt, 2 dimensional body, vortices are created

and shed in an alternating fashion on the top and bottom of the body (Bjswe et.al,

2011). Given that the body is symmetrical, this phenomenon will initially be

symmetrical but will eventually turn into the classical alternating pattern. Figure 2.4

and Figure 2.5 is a good depiction of a common von Kármán vortex street. This

behavior was named after Theodore Von Kármán for his studies in the field.

The frequency of Karman vortex shedding can be estimated by the Strouhal

number. A vortex shed from the other side causes a force in the opposite direction.

This alternate vortex shedding characterized by definite periodicity would cause

fluctuating pressure forces on the surfaces of the body due to which the body

undergoes vibrations. The turbulence represents the random fluctuations of the flow

and is not periodic. The turbulence excited vibration can occur at any flow velocity

with increasing amplitude, flow velocity and a frequency around the natural

frequency of the bluff-body structure. Thus the bluff-body is set to vibrate due to the

alternate shedding of vortices named as Vortex-Induced Vibration (VIV).

Figure 2.5: Von Kármán Vortex Street at increasing

Reynolds numbers (Bjswe et al.,2011)

15

Figure 2.6: Von Kármán Vortex streets forming in the wake

Of a Bluff-body on the onset of turbulence (Hill, 2013)

A research shown that when Reynolds number exceeded 100, the von

Karman vortex street occurred. When the Reynolds number was higher, the von

Karman vortices lasted for a shorter a period. As a result, the distance between the

vortices decreased. As shown in Figure 2.6, the results of the analysis are in good

agreement with the visualized outcome derived from the theoretical (Sato et al.,

2012)

Kármán vortex street phenomenon was known constant K = h/a, where h is

the transverse distance between the two rows of vortices and a is the longitudinal

space period, as indicated in Figure 2.7 .There are two vortex sheets at each side of

the body stemming from the boundary layer separation, which roll up into the

“vortex streets” in the wake. The Karman “vortex streets” were deduced in ideal

situations such the nonviscous fluid and the concentration of vorticity. Actually, in a

viscous flow, in order to keep the moment of vorticity, which is equal to the mean

vorticity per unit length multiplied by h, as a constant, the transverse distance h has

to increase with time since the mean vorticity decreases with time due to the

interaction of the two vortex rows (two separating shear layers) with vorticities of

opposite sign (Wu & Kareem, 2012)

16

Figure 2.7: Kármán “Vortex Street” behind the circular cylinder

(Wu & Kareem, 2012)

2.5 Drag Force and Lift Force

The drag force acts in a horizontal direction or in line flows on the cylinder surface,

depends on shape and orientation of a body. The average drag on a cylinder vibrating

at or near the vortex shedding frequency is also a function of vibration amplitude.

The high in-line vibration amplitude will increase the drag force. Drag coefficient,

Cd is calculated as follow number is used to describe the relationship between vortex

shedding frequency and fluid velocity and is given by;

𝐶𝑑 = 2𝐹𝐷

𝜌 𝑈2𝐷 (2.1)

Where ⍴ is the fluid density, U is the flow velocity, D is the cylinder diameter and Fd

is the sum of the pressure force

Lift coefficient, Cl is calculated similarly but vertical force is considered

rather than horizontal force. The force of the cross-flow direction is commonly

named as the lift force (FL) which is shown in Figure 2.8.

𝐶𝑙 = 2𝐹𝑙

𝜌𝑈2𝐷 (2.2)

Where F is the amount of pressure and viscous force components on the

surface of the cylinder acting in the along-wind direction In experiments, dynamics

lift force must be make sure not be well correlated along the test section otherwise,

17

lower lift coefficient value will be reported. In experiment, pressure transducer is

replaced force transducer to obtain more accurate results because they measured the

pressure distribution of the fluid around the cylinder’s circumference. Lift force

frequency is characterized by the Strouhal number

Figure 2.8: Vibrations of forces acting around a circular cylinder submitted to vortex

shedding. (Cunff et.al, 2002)

The Strouhal number is related to the Cd of the flow. In sub-critical Reynolds

number region (100 < Re < 10,000), increase in the Strouhal number is generally

accompanied by a decrease of the Cd (Pattel, 2010).

Theoretically, the drag force is oscillates at twice the frequency of the lift

force for the flow past a circular cylinder or generally flow involving separation.

When a vortex is shed from the top of the cylinder, a suction area is created and the

cylinder experiences lift. Half a cycle later, an alternate vortex is created at the

bottom part of the cylinder. Throughout the process, the lift force changes alternately

in a complete cycle of vortex shedding but the cylinder experiences drag constantly.

It is important that any turbulence models can simulate all the forces coefficient

correctly for the analysis of the flow past circular cylinder (Pattel, 2010).

18

2.6 Strouhal Number

The Strouhal number depends on Reynolds number, shape and dimension of body,

and wind or water velocities. The increasing of Reynolds number will affect Strouhal

number increases or decreases in certain range of Reynolds number. The increasing

size or dimension (D) will increase the Strouhal number, the increasing of velocity

(U) and change of attack angle will decreasing the Strouhal number (Hoa, 2005). It

should be noted that the Strouhal number can be influenced by the Reynolds number

in some cases, for circular cylinders, the Strouhal number seems to stay constant on

the certain range of Reynolds number between 1.5x102 and 1.5x105 (Hoa, 2005,

Blevin 1977, Simui&Scanlan 1978).

Figure 2.9: Strouhal number as a function of Reynolds number for

Smooth circular cylinder (Hall-Stinson, 2011)

19

Figure 2.10: Effect of Reynolds number on Strouhal number for

2D circular cylinder (Simui and Scanlan, 1978)

The technique for compensating of the influence of the Reynolds number

difference between full scale structure and experimental model should be required

due to difficulty and possibility to simulate the Reynolds number using tool of wind

tunnel experiments, because it can effecting on evaluation of vortex-induced

vibration (Hoa,2005;Shimada & Meng,1998).

2.7 VIV Equation

In the one degree of freedom vortex induced vibration problem, an elastically

mounted circular cylinder is situated in a free stream of fluid shown in Figure 2.6

Figure 2.11: One Dimension Spring-Pot System (Asyikin, 2012)

20

The structure is conditioned as a vibration-free elastically cylinder by means the

cylinder is free to respond the vibrations which is useful to find amplitude, frequency

and phase angle of a vibrating cylinder and helpful in studying flow-visualization of

the wake of the cylinder.

The differential motion equation of the system shown in Figure 2.11 is

formulated as:

my(t) + cy(t) + ky(t) = F(t) (2.3)

Where m is the total mass of the system and dot over the symbols indicates

differentiation with respect to time, c is the coefficient of viscous damping and k

denotes the stiffness of the spring.

𝑚∗ = 4𝑚

𝜋𝜌𝐷2 (2.4)

𝜁 =𝑐

𝑐0 (2.5)

𝑉𝑟 =𝑈∞

𝑓𝑛𝐷 (2.6)

Where the critical damping, C0

c 0 = 2√𝑘𝑚 (2.7)

And the natural frequency of the body in vacuum

𝑓𝑛 =√𝑘/𝑚

2𝜋 (2.8)

The mass ratio represents the ratio between the inertial forces of the body and

the hydrodynamic forces of the fluid. For VIV problems in water m∗ is very low,

21

while in air it is usually one or two orders of magnitude larger. The excitation

frequency in air has generally a value similar to the frequency of the cylinder’s

response close to the body natural frequency fn.

The synchronization of vortex shedding and the body frequency takes place

over a range of reduced velocities, whose width is proportional to the oscillation

amplitude.

Table 2.2: Relationship of Amplitude and Frequency Mode (Asyikin, 2012)

For a free vibration system, such as no external forces working on the system

(F=0), the solutions can, therefore, be differentiated into two conditions: without and

with viscous damping.

According to King et al (1974), Vandiver (1987) have found that if a cylinder

is mounted in the in-line direction, the oscillating drag produced by vortex shedding

may induce in-line vibrations. In general these vibrations will occur at roughly twice

the frequency for transverse vibration. These vibrations pose numerous problems

with underwater pipelines because larger vibrations will typically occur at very low

reduced velocities.

Studies by Naudascher (1987), Vandiver (1987), Torum et al, (1996) show

that the in-line vibrations are an order of magnitude smaller than the transverse

vibrations. Inline vibrations can also occur by themselves and constitute a fatigue

problem for the free-span length of pipes on the seafloor. They are usually not a

concern for risers since they appear at lower reduced velocity. Axial vibrations (Huse

et al., 1998) are a by-product of cross-flow vibrations and are observed when the

axial deformation due to the cross-flow vibrations triggers a resonant axial mode

(Cunff et al., 2002)

Vibration motion Max Amplitude Oscillation Frequency mode

Cross Flow 𝑽𝒓 ≈ (𝟏/𝑺𝒕 ) f n = f s

Inline Flow 𝑽𝒓 ≈ ( 𝟏/𝟐𝑺𝒕 ) f n = 2 f s

22

2.8 Cross Flow and In Line Vibrations of Cylindrical Structure

Vibrations of structure emerge as the result of periodic variations in the force

components due to the vortex shedding. The vibrations can be differentiated into

cross flow and in-line vibrations. The cross-flow vibration is caused by the lift force

whilst the in-line one is caused by the drag force. Both vibrations are commonly

called the vortex induced vibrations.

Figure 2.12: Definition sketch of vortex-induced vibrations (Asyikin, 2012)

When the frequency of oscillation is close to the Strouhal frequency, the cross

flow force contains a dominant fluctuating component with the vortex-shedding

period, while the Inline Flow forces contain a steady drag component and a

fluctuating component with half the vortex-shedding period. Vortex formation locks

on to the frequency of cylinder vibration within a relatively narrow range around the

Strouhal frequency called the wake capture region. Under free vibration conditions,

the cross flow response is typically larger than the inline response by a factor of at

least two.

23

A bluff body undergoes perfect vortex-shedding resonance if its inline natural

frequency (fx) is twice the transverse natural frequency (fy), and the Strouhal

frequency is near or at the transverse natural frequency, resulting in fluid excited

motions in both inline and transverse directions ( Dahl et al, 2007 ).

Numerical simulations was conducted by Tang in 2013 for VIV of an

elastically mounted circular cylinder freely oscillating in both cross-flow and in-line

directions. All of the numerical results herein were obtained at Re = 100, mass ratio

𝑚∗ = 10.0, and structural damping ratio 𝜉 = 0.01, low combined mass damping

parameter 𝑚∗𝜉 = 0.1.The reduced velocity V𝑟 was varied from 3.0 to 10.2, by

changing the natural frequency f𝑛 of the cylinder. One of the important phenomena

associated with VIV is the synchronization defined as the vortex shedding frequency

matches the natural frequency of the cylinder.

This is well demonstrated in Figure 2.8 where the two vertical dashed lines

illustrate the boundaries of the synchronization region. Over the synchronization

range, it is observed that the vortex shedding frequency is modulated to the natural

frequency of the elastically mounted circular cylinder and the oscillations are

dominated by the oscillating frequency. In this reduced velocity range, multiple

frequencies are presented in the spectra of the response making the lift and

displacement fluctuations behave large deviation from the pure-tone oscillations

(Tang et al, 2013).

Figure 2.13: Frequency response against reduced velocity

(Tang.et al, 2013)

24

2.9 Reynolds Number, Re

The flow around a stationary smooth circular cylinder depends on Re, one of the non

dimensionless hydrodynamic numbers that is accounted, denotes the relationship

between inertia forces and the viscous forces in the boundary layer and is categorized

roughly by four regimes which is subcritical, critical, supercritical and trans critical.

In 1833, one of the earliest research by Osborne Reynolds was the first to

systematically investigate the transition from laminar to turbulent flow by injecting a

dye streak into flow through a pipe having smooth transparent walls. His

observations led to identification of a single dimensionless parameter, now called the

Reynolds number, and denoted Re

Re = Inertia Force

Friction Force=

𝜌𝑈𝐷

𝜇 =

𝑈𝐷

𝜈 (2.9)

Where ρ is fluid density, µ is dynamic viscosity, D- is the cylinder diameter, and U is

the fluid velocity. One of the reasons that the Reynolds number is so important is that

its value determines the type of flow around a bluff body.

In 2001 Blevins shows that the composition of the vortex pattern from the

bluff body will change depending on Reynolds number. The experiments examined

here are in the regime where the vortex street is the fully turbulent and include

examples up to a Reynolds number of 3.5 ×105 (Swithenbank et.al, 2008). He

proposed a crude division of the flow regimes is given by:

• 300 < Re < 1.5 ×105 Subcritical regime

• 1.5 ×105 < Re < 3.5 ×105Transitional regime

• Re >3.5 ×105Supercritical regime

One should be aware that the division of flow regimes into Reynolds number

ranges is not definite. Disturbances may have a profound effect on the flow and

change the rearranges for where the various regimes are seen also influenced the

65

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