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Planar Dynamics of Inclined Curved Flexible Riser Carrying Slug Liquid-Gas Flows

Bowen Ma & Narakorn Srinil*

Marine, Offshore & Subsea Technology Group, School of Engineering, Newcastle University, Newcastle Upon Tyne, NE1 7RU, United Kingdom

Abstract

Flexible risers transporting hydrocarbon liquid-gas flows may be subject to internal dynamic fluctuations of

multiphase densities, velocities and pressure changes. Previous studies have mostly focused on single-phase

flows in oscillating pipes or multiphase flows in static pipes whereas understanding of multiphase flow effects

on oscillating pipes with variable curvatures is still lacking. The present study aims to numerically investigate

fundamental planar dynamics of a long flexible catenary riser carrying slug liquid-gas flows and to analyse the

mechanical effects of slug flow characteristics including slug unit length, translational velocity and fluctuation

frequencies leading to resonances. A two-dimensional continuum model, describing the coupled horizontal and

vertical motions of an inclined flexible/extensible curved riser subject to the space-time varying fluid weights,

flow centrifugal momenta and Coriolis effects, is presented. Steady slug flows are considered and modelled

by accounting for the mass-momentum balances of liquid-gas phases within an idealized slug unit cell

comprising the slug liquid (containing small gas bubbles) and elongated gas bubble (interfacing with the liquid

film) parts. A nonlinear hydrodynamic film profile is described, depending on the pipe diameter, inclination,

liquid-gas phase properties, superficial velocities and empirical correlations. These enable the approximation

of phase fractions, local velocities and pressure variations which are employed as the time-varying, distributed

parameters leading to the slug flow-induced vibration (SIV) of catenary riser. Several key SIV features are

numerically investigated, highlighting the slug flow-induced transient drifts due to the travelling masses,

amplified mean displacements due to the combined slug weights and flow momenta, extensibility or tension

changes due to a reconfiguration of pipe equilibrium, oscillation amplitudes and resonant frequencies. Single-

and multi-modal patterns of riser dynamic profiles are determined, enabling the evaluation of associated

bending/axial stresses. Parametric studies reveal the individual effect of the slug unit length and the

translational velocity on SIV response regardless of the slug characteristic frequency being a function of these

two parameters. This key observation is practically useful for the identification of critical maximum response.

Keywords: multiphase flow, slug liquid-gas flow, catenary riser, flow-induced vibration, inclined pipe.

* Corresponding author: narakorn.srinil@newcastle.ac.uk; Tel. +44 191 208 6499; Fax +44 191 208 5491

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Nomenclature

I. Riser/pipe and material properties

A (Ao) Inner (outer) area Ar Cross-sectional area c Viscous damping coefficient d (D) Inner (outer) diameter E Young’s modulus fn Natural frequency of submerged and

empty pipe fo Oscillation frequency fn1 Fundamental natural frequency L Total length I Moment of inertia mr Structural mass per unit length mt Total mass per unit length Tdp Tension reduction from top to bottom T (To) Static tension (pretension) Tm Mean tension due to slug flow Tb Bottom mean tension due to slug flow u (v) Horizontal (vertical) dynamic

displacement from its static equilibrium um (vm) Horizontal (vertical) mean drift

( )u v Horizontal (vertical) velocity urms (vrms) Horizontal (vertical) root-mean-squared displacements

( )u v Horizontal (vertical) displacement gradients

( )u v Horizontal (vertical) acceleration

( )u v Horizontal (vertical) curvature ( )u v Horizontal (vertical) angular velocity

( )x y Horizontal (vertical) static slope x (y) Horizontal (vertical) static coordinate

β Inclination angle ( )x y Horizontal (vertical) static curvature

σa Axial stress due to longitudinal strain σt Combined axial-bending stresses σu (σv) Horizontal (vertical) bending stress Poisson’s ratio Structural damping ratio Local curvature in normal direction

II. Fluid properties and correlation variables

Af (Ag) Cross-sectional area of liquid film (gas) B Distribution parameter for average velocity of dispersed bubbles

C Ratio of maximum to mean velocities of the liquid slug

Cfg, n Two empirical coefficients of Blasius correlation for friction factor

Dhf (Dhg) Hydraulic diameter of liquid film (gas) ff (fg) Liquid- (gas-) wall friction factor fi Interfacial friction factor fs Characteristic slug frequency hf Hydrodynamic film height hs (hc) Initial (critical) liquid film height j Modification factor of bubble rise

velocity kp Pressure fluctuation parameter

Lf (Ls) Film (liquid slug) length Lu Slug unit cell length ma Still-water added mass per unit length mi Phase mass per unit length (i=1 for liquid,

2 for gas) Ns Number of full slug units along riser N Order of slug frequency components P Pressure induced by internal flows Pdp Pressure drop from the bottom inlet Pm Flow-induced mean pressure component Pt (Pb) Outlet (inlet) mean pressure Po External hydrostatic pressure ∆Pf (∆Ps) Pressure drop in film (slug liquid) zone ∆Pu Total pressure drop over slug unit ∆Pg Pressure drop due to gravity effect Re Reynolds number Rf (Rs) Liquid holdup in film (liquid slug) zone R Liquid holdup over slug unit length R̃ Normalized liquid holdup profile Si Interfacial width Sf (Sg) Liquid film (gas) wetted perimeter Ud Elongated bubble drift velocity in

stagnant liquid U0 Average drift velocity of small

dispersed bubbles U Single-phase flow velocity Ul Liquid velocity in liquid slug zone Ub Average velocity of small dispersed

gas bubbles in liquid slug U∞ Bubble rise velocity

Uf (Ug) Liquid (gas) velocity in film zone Us Mixture velocity Ui Phase velocity (i=1 for liquid, 2 for gas) Ut Slug unit translational velocity uL (uG) Liquid (gas) velocities along slug unit αf (αs) Void fraction in film (liquid slug) zone uls (ugs) Superficial liquid (gas) velocity αu Average void fraction Vg (Vf) Relative gas (liquid film) velocity αvol Gas volumetric fraction of slug unit

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ηl (ηg) Kinematic viscosity of liquid (gas) σ Interfacial surface tension ρl (ρw) Liquid (external water) density τf (τg) Liquid-wall (gas-wall) shear stress ρg Gas density τi Interfacial liquid-gas shear stress ρu Average phase density ϕ Geometric angle with respect to

diameter centre in the film zone Summation of liquid-gas properties

III. Coordinates, differentiations and others

g Gravitational acceleration ∆s Spatial discretization of riser length s Arclength coordinate ∆t Time step t Time ∆z Spatial step for slug flow z Longitudinal slug unit coordinate

attached to a slug unit moving frame y* Normalized vertical coordinate with

respect to vertical span X-Y Global coordinate system ( ) Partial or ordinary differentiation with

respect to arclength coordinate ( ) Partial differentiation with respect to

time f

f

dR

dh

Ordinary differentiation of liquid film holdup with respect to film height

fdh

dz

Ordinary differentiation of film height with respect to slug unit coordinate

du (dv) Infinitesimal dynamic displacement in horizontal (vertical) direction

ds (dsr) Infinitesimal arclength segment in the static (dynamic) state

dx (dy) Infinitesimal static segment with respect to horizontal (vertical) coordinates

IV. Abbreviation

FFT Fast Fourier Transformation SIV Slug flow-induced vibration 1. Introduction

Deepwater risers are key components in the offshore subsea industry for conveying hydrocarbon flows from

seabed wells to ocean surface platforms. Depending on the reservoir fluid properties, such multiphase flows

may consist of two leading gas and liquid (oil and/or water) phases known as the two-fluid, liquid-gas flows.

When these flows propagate through a long cylindrical pipe with variable inclinations such as a catenary, S-

shaped or lazy-wave riser, several flow patterns may evolve, depending on the flow-pipe parameters and phase

interfacial characteristics (Barnea et al., 1980; Spedding and Nguyen, 1980; Taitel and Dukler, 1976). It is

recognized that the slug flow pattern, exhibiting an alternate distribution of liquid- and gas-dominant phases,

is problematic for practical operations owing to a sudden change in flow momenta and pressure potentially

causing the pipe resonant oscillations (Bordalo and Morooka, 2018; Hara, 1977; Patel and Seyed, 1989; Wang

et al., 2018). A large-diameter, long and bendable pipe transporting the mixed liquid-gas phases may be further

subject to combined external current and internal flow excitations. Apart from potentially amplifying overall

responses, such complex flow-pipe interactions could lead to a flow assurance issue, operational interruption

and greater engineering solution cost. Although the external flow vortex-induced vibrations and internal single-

phase flow-induced vibrations have been widely studied in the literature (Sarpkaya, 2004; Williamson and

Govardhan, 2004; Wu et al., 2012) and books (Blevins, 1990; Chen, 1985; Paidoussis, 2014), understanding

of fundamental slug flow-induced vibration (SIV) mechanisms is still lacking. There is also a need for a generic

mathematical model and numerical approach to predict and describe SIV features of industrial importance. 

Recently, Miwa et al. (2015) have provided a comprehensive review on research progress in a two-phase

flow-induced vibration in small-diameter pipes such as heat exchangers and power plant components in nuclear

energy applications. Significant vibrations in straight pipes with bends has been experimentally reported under

slug flows causing an intermittent change in momentum flux and resonance between the pressure fluctuations

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versus the pipe’s natural frequencies. Using the Euler’s beam theory, Hara (1973) formulated the equation of

transverse motion of a straight pipe transporting slug flows modelled as a series of piston liquid-gas phases

moving with a unique frequency. Hara showed that, with a single mode approximation, SIV is dominantly

caused by a parametric resonance due to the space-time varying mass distributions. Further, Hara (1977)

compared theoretical and experimental results for a horizontal pipe carrying air-water slug flows, justifying

the use of a simplified phenomenological flow model for predicting SIV. 

For offshore risers with large diameters and long lengths, Patel and Seyed (1989) theoretically described

key features of internal flow-induced static and dynamic forces acting on the pipe, including the fluid weight,

curvature-induced load, centrifugal and Coriolis effects of flow momenta. They also modelled a steady-state

slug flow through a sinusoidal function of fluid densities having a mean component and space-time varying

counterpart with a harmonic slug frequency. They remarked how SIV can amplify dynamic tensions and

modify the riser geometric stiffness. This slug flow idealisation has been applied by Pollio and Mossa (2009)

to a catenary riser, by also accounting for a relationship of the slug wavelength and pipe inclination permitting

a variable slug frequency. An irregular behaviour of riser axial stress was highlighted. More recently, Meléndez

and Julca (2019) applied a sinusoidal density function to model the slug flow in a catenary riser subject to

external current flows. They demonstrated an amplification of riser top tensions depending on the flow rates.

Alternatively, Bordalo and Morooka (2018) modelled steady slug flows as a series of liquid plugs and gas

pockets travelling with a single slug unit velocity. They introduced a mass distribution function depending on

the slug velocity and unit length through liquid-gas densities, and showed that large oscillations may be

generated when the slug frequency nearly resonates with one of the riser natural frequencies. Kim and Srinil

(2018) have implemented such a slug unit cell as an initial input velocity and volumetric fraction at the pipe

inlet for computational fluid dynamics simulations of a subsea jumper transporting slug flows. The effect of

multiple bends on the flow pattern modification was highlighted. Using this slug unit concept, Safrendyo and

Srinil (2018) introduced a slug length randomness into the dynamic simulation of catenary riser carrying slug

flows. They highlighted chaotic riser responses with broad-band oscillation frequencies and greater amplitudes

when accounting for combined SIV and external flow excitations. Nevertheless, these idealised fluid force

models neglect the detailed flow features with variable phase velocities and fractions. To account for a more

realistic slug flow pattern, a three-dimensional flow simulation (Moon et al., 2019) or a hydrodynamic slug

flow theory describing a spatial characteristic of a non-uniformly elongated gas bubble within a liquid film

(Cook and Behnia, 1997; Taitel and Barnea, 1990) should be implemented.  

Modelling of one-dimensional two-phase flows, averaged over a pipe cross section, may be generally

classified into three main categories: homogenous, drift-flux and two-fluid models (Issa and Kempf, 2003;

Monette and Pettigrew, 2004; Montoya-Hernández et al., 2014; Nydal and Banerjee, 1996; Ouyang and Aziz,

2000). For homogeneous models, a single set of the mass-momentum-energy conservation equations depend

on the liquid-gas mixture as a pseudo-single phase. For drift-flux models, the conservation of momentum

depends on the relative phase velocities whilst other conservations may be derived for individual fluids. For

two-fluid models, the full conservations are required, accounting for different physical fluid properties. If the

conservation equations are assumed to be time-invariant, the considered two-fluid flows are steady; otherwise,

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they are transient, describing the space-time varying quantities. Montoya-Hernández et al. (2014) used the

homogeneous model to demonstrate that natural frequencies of a vertical riser transporting the upward gas-oil-

water flows decrease with the increasing mixture velocity. This agrees with a general free vibration trend of a

tensioned straight beam with internal flows (Paidoussis, 2014). Based on a mechanistic model of Taitel and

Barnea (1990), Chatjigeorgiou (2017) investigated planar dynamics of a catenary riser transporting steady slug

oil-gas flows and subject to a top-end harmonic excitation. Using a frequency domain approach, it is found

that slug flows amplify dynamic responses tensions. For a floating free-hanging pipe transporting air-water

flows, Vieiro et al. (2015) simulated the fluid-pipe coupling by using a two-fluid model and solving the

transient fluid equations with a combined slug capturing-tracking approach (Nydal, 2012). They predicted a

severe slugging phenomenon with space-time phase fraction variations and a slug frequency in good agreement

with in-house experiments. Recently, based on a transient slug tracking approach, Ortega et al. (2018) studied

SIV a catenary riser subject to regular waves. Numerical results showed the increased static drifts caused by

the slug weight and the amplified dynamic responses due to combined wave and SIV.

Laboratory experiments of air-water flows in small-diameter pipes have been carried out, providing some

insightful observations and validating numerical models. Liu et al. (2012) measured the air-water flow-induced

fluctuating forces on a rigid pipe bend and reported the predominant forcing frequency being peaked in the

slug flow regime. The impact force was found to be significant in determining the slug flow-induced forces.

For horizontal straight pipes, Ortiz-Vidal et al. (2017) measured pipe responses subject to several flow regimes

of bubbly, dispersed and slug flows, and noticed that the pipe vibration increases with the increased mixture

velocity and the peak frequency strongly depends on the void fraction. Wang et al. (2018) revealed that the

horizontal pipe vibration is enhanced by increasing the slug unit velocity and liquid slug length since these

properties affect the rate of change of system stiffness, mass, damping and loading. For a straight pipe with

multiple supports and carrying unsteady slug flows, Liu and Wang (2018) examined the time-varying natural

frequencies with superficial velocities. They observed a critical gas velocity which may change qualitatively

the trend of the average frequency depending on the liquid velocity, pipe stiffness and span length. Recently,

SIV of a catenary pipe has been studied by Zhu et al. (2019b) who reported the predominantly fundamental

mode responses associated with the characteristic frequencies of pressure fluctuations.

Table 1 summarizes some relevant studies related to SIV modelling and analysis of flexible pipes/risers,

by distinguishing the slug modelling concepts, pipe geometries, two-phase parameters and observed features.

Table 1 and the above literature review reveal that most of the previous studies have focused on a forced

vibration problem of the slug-transported catenary pipe subject to waves, vortex-shedding or support excitation.

An idealised rectangular pulse train modelling a slug unit has frequently been assumed. SIV features of

horizontal pipes have been studied experimentally. Nevertheless, intrinsic features of pure SIV and the effects

of slug characteristics have not been well addressed for a catenary riser conveying slug flows. These will be

numerically investigated in the present study through a generic model of flexible riser and slug hydrodynamics.

This research paper is structured as follows. Section 2 presents a mechanical model for two-dimensional

vibrations of a catenary riser carrying slug flows. In Section 3, a mechanistic model of slug liquid-gas flow is

described. Pipe properties and slug flow characteristics are discussed in Section 4. Parametric studies are

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carried out in Section 5 highlighting several riser SIV features. Validation and sensitivity analysis are presented

in Section 6 with numerical-experimental comparisons. The paper ends with conclusions in Section 7.

2. Model of Flexible Riser Carrying Slug Flows

In a fixed global Cartesian (X-Y) system, Fig. 1 displays a schematic model of an inclined curved flexible riser

carrying slug liquid-gas flows which travel upwardly from the pipe bottom connection (inlet) to the top

stationary support (outlet) as the coordinate origin. Typical pinned-pinned boundary conditions are considered.

Two different states of riser configurations are described using the local Lagrangian (s) coordinate. The initial

equilibrium subject to a static loading is described by x and y coordinates whereas the dynamic configuration

subject to travelling slug flows is described by u and v components measured from the static equilibrium in

the horizontal and vertical direction, respectively. For an infinitesimal riser element at the equilibrium (ds) and

dynamic (dsr) states with associated displacement variations (dx, dy, du, dv), their coordinate relationships are

2 2ds dx dy and 2 2( ) ( ) ,rds dx du dy dv respectively. The present generic model of flexible catenary

riser transporting slug flows is based on some assumptions which are summarized as follows.

2.1 Main Assumptions

Pipe materials are linearly elastic, with spatially uniform properties including the mass per unit length,

smooth-surface outer/inner diameters and cross-sectional areas, moment of inertia, the Young’s modulus,

bending and axial stiffness. Owing to the longitudinal internal flow loading and inherent slenderness of

the long flexible pipe with a high aspect (length-to-diameter) ratio (order of 102-103), shear and torsional

rigidities are disregarded based on the Euler-Bernoulli theory. Therefore, the riser pipe cross section under

axial loading remains circular with a plane perpendicular to its longitudinal axis.

Mean configurations of the loaded pipe are established in two consecutive stages. In the first static stage,

the pipe forms into an inextensible catenary due to its own weight, buoyancy and hydrostatic pressure,

with a negligible static strain compared with the unity. For a specified top tension or pipe length, a closed-

form hyperbolic expression describing a catenary profile as a function of s can be simply obtained (Srinil

et al., 2009). In the second stage, the formed catenary is further subject to additional weights of the moving

slugs whose masses and velocities concurrently induce the flow momenta leading to pipe oscillations.

Therefore, riser displacements and extensional strains comprise both static and dynamic components. A

steady-state dynamic response of the riser is described about this second-stage mean configuration.

In practice, two-phase flows may evolve into different flow patterns along a long riser depending on the

fluid-pipe properties. For an oscillating and flexible pipe, the flow pattern may be further modified. Based

on experiments of an inclined curved flexible riser transporting air-water flows, Bordalo et al. (2008)

observed slug, intermittent (churn) and annular flow regimes whereas Zhu et al. (2019b) only reported

slug flow occurrences depending on the superficial velocities. Bordalo et al. (2008) also suggested a key

interrelationship of liquid-gas flow rates, flow patterns, riser vibration amplitudes and frequencies. When

the slug flow occurred, the riser response was reported with a maximum amplitude of about 6 diameters.

Greater amplitudes of 9 and 15 diameters were found when the slug flow becomes an intermittent (churn)

flow, and when the churn flow becomes an annular flow, respectively. Recently, Silva and Bordalo (2018)

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experimentally reported that a vertical flexible pipe oscillating in a low frequency range produces an air-

water flow pattern map almost identical to that of the static pipe. In this study, attention is placed on the

slug flow regime leading to a small-amplitude (less than a diameter) SIV response such that the slug

properties are assumed to be undisturbed by small-amplitude pipe oscillations.

Slug flows are inherently transient and unsteady. Zhu et al. (2019b) experimentally reported that, despite

the pressure fluctuations induced by unsteady slug flows, a predominant single-mode SIV response of a

catenary riser is found. This suggests that randomness of the slug flow travelling in a flexible pipe may

be approximated by the average properties. Earlier experiments of slug flow evolution in a long pipe by

Fabre et al. (1993) also evidenced that the probability density distributions of bubble and slug velocities

are narrowly distributed about their average values. Such feature has led several researchers to transform

an unsteady problem into a steady one. In this study, the hydrodynamic slug flow is assumed to be stable,

steadily time-varying or, briefly, steady, so that the slug transient behaviour may be approximated by a

time-averaged solution. This implies that the liquid is shed from the back of the slug at the same rate that

the liquid is picked up at the front; therefore, the slug length stays constant as each slug travels along the

pipe (Fabre, 1994). This steady flow assumption for SIV analyses has been considered by some recent

studies (see Table 1) and incorporated into industrial software (e.g. OLGA, LedaFlow) as a preliminary

and complimentary model for flow assurance analysis. For steady slug flow, a time-averaged mass flow

rate of liquid and gas over the period of a slug cycle is constant (Taitel and Barnea 1990).

Slug flow is assumed to be fully developed as described in Section 3 and modelled as a sequence of slug

units travelling through the flexible pipe. In Fig. 1, an idealised slug unit cell is displayed, according to

Taitel and Barnea (1990). Each slug unit comprises (i) a dominant liquid phase (liquid slug) containing

small gas bubbles and (ii) a non-uniform gas phase (elongated gas bubble) above a thin liquid film,

travelling with a translational unit velocity relative to the pipe. To the best of our knowledge, there is

presently no evidence in the open literature reporting actual slug flow unit features in a long inclined

flexible riser with a large diameter in subsea applications. The generic schematic slug unit is therefore an

idealisation following the pioneering work of Wallis (1969) and other researchers who apply mechanistic

slug flow models. It has mostly been assumed that such model is valid for an arbitrarily inclined pipe

transporting liquid-gas flows. Based on a recent laboratory test of a catenary riser carrying air-water flows,

Zhu et al. (2019b) reported some video snapshots of slug flow pattern comprising two main components

(liquid slug and elongated bubble) as in Fig.1. However, due to a very small diameter (4 mm) of the pipe,

small dispersed bubbles within the liquid slug or thin film cannot be directly captured from their figures.

This study is limited to a two-dimensional SIV problem. This scenario is plausible as recent experiments

of a catenary pipe SIV revealed negligible out-of-plane responses (Zhu et al., 2019b). Without an external

excitation, the flexible pipe subject to only internal slug flows tends to response in two dimensions due to

the predominating effect of gravity aligned with the pipe’s vibration plane, and due to the low phase

velocities far from the critical ones which may trigger an out-of-plane motion through a geometrically

nonlinear coupling. With external excitations such as waves or currents, a three-dimensional response can

take place and amplify the combined in-plane and out-of-plane responses as recently observed in Zhu et

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al. (2019a) for a catenary riser subject to external shear flows. Overall large-amplitude three-dimensional

displacements in coupled internal and external flow cases can affect the slug flow pattern evolution along

the riser span and time, modify the riser mean configuration and natural frequencies, and strengthen the

system fluid-structure interactions as a fully complex problem.

2.2 Equations of Riser Motion

Several riser mechanical models have been derived based on a Hamiltonian or Newtonian formulation (Chen,

1985; Lee and Chung, 2002; Paidoussis, 2014). By following Chucheepsakul et al. (2003) and Srinil et al.

(2004, 2007), the variational principles are herein applied to account for the potential or strain energy due to

the static tension, axial dynamic stretching and bending, the work done by the weight, pipe inertia, viscous

damping and internal flow inertia accounting for the absolute flow velocity in the Cartesian coordinate system.

Based on the above-mentioned assumptions and by describing the riser motions with respect to the initial static

configuration, the linear partial-differential equations of motions of a flexible pipe transporting two-phase slug

flows may be expressed in a dimensional form as

0,2 2

2 2

1 1

22 2

1

( ) 2 ( ) 2 (1)

( ) 2 ( )

t i o o o r i i i ii i

t i o o o r i ii

m m u cu EIu T P A PA u EA x u x y v m U x u m U u

m m v cv EIv T P A PA v EA x y u y v m U y

2 2

1 1

2 ,i i ii i

v m U v m g

(2) in which a dot denotes the partial differentiation with respect to the time t and a prime denotes the partial

differentiation with respect to the arclength coordinate s. For static analysis, only a prime is used to denote the

ordinary differentiation with respect to s. Slug properties include the phase mass per unit length mi(s, t), phase

velocity Ui(s, t), where a subscript i = 1 and 2 denotes the coexisting liquid and gas, respectively, and the

internal flow-induced pressure P(s, t) subject to the interfacial shear stresses and the gravitational acceleration

g. With respect to the riser static configuration with x and y coordinates, x and y are the associated slopes

while x and y are the associated curvatures. The riser horizontal (u) and vertical (v) dynamic displacements

govern the translational velocities ( , ),u v accelerations ( , ),u v gradients ( , ),u v curvatures ( , )u v and angular

velocities ( , ).u v System parameters include the Young’s modulus E, the moment of inertia I, the pipe cross-

sectional (Ar), inner (A) and outer (Ao) areas, the pipe pretension To(s), the external hydrostatic pressure Po(s),

the total mass per unit length mt = mr + ma, in which mr is the riser mass and ma is the still-water added mass

equal to wAo, with w being the external water density. For a submerged pipe, the effect of the Poisson’s ratio

() is accounted for through the so-called apparent tension concept (Sparks, 1984). A viscous damping

coefficient c may be identified in terms of a modal damping ratio . A linearized added damping due to a fluid

drag in still water may be accounted for through as a combined structure-fluid damping (Srinil, 2010).

Riser oscillations are subject to the slug distributed parameters (weights, velocities, pressure) varying in

space and time, depending on the specified slug unit velocity and length. Slug flow-induced displacements are

generally small when compared with the riser length such that the effects of geometric nonlinearities may be

negligible (Srinil, 2010). Hence, the riser linear damped free vibration (i.e. self-excitation without an explicit

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time-dependent loading) is subject to the travelling slug flows influencing natural frequencies and modal

characteristics. The internal flow-related terms in Eqs. (1) and (2) may be approximated by accounting for a

summation of individual liquid and gas phase quantities, as in Monette and Pettigrew (2004), without a phase

interaction which could be considered through a more rigorous transient two-fluid model (Issa and Kempf,

2003). Ui have a negative sign for upward flows (Païdoussis and Luu, 1985) because of taking the opposite

direction to the s coordinate (Fig. 1). For steadily time-varying flows, 0i i i iU m U m have been applied.

Equations (1) and (2) contain the flow-induced momentum forces due to the gyroscopic Coriolis effect ( )i imU

and centrifugal acceleration 2( )i im U depending on the pipe curvatures and contributing to the pipe axial

tension variation. A static bending effect has been discarded for a small-sagged catenary. The space-time

varying weight term (mig) in Eq. (2) creates initial conditions required for a free vibration simulation. Due to

lack of an analytical solution for such a gyroscopically conservative system with travelling and non-uniformly

distributed parameters, Eqs. (1) and (2) are herein numerically solved by using a second-order finite difference

discretization in space combined with a 4th-order Runge-Kutta time integration as performed, e.g., in Cabrera-

Miranda and Paik (2019).

Convergence of numerical simulations has been checked for which validation of the present riser model

is shown in Table 2 based on a comparison of fundamental natural frequencies (fn1). In the context of instability

analysis of the pipe transporting a single-phase liquid flow with increasing velocity U, a vertical fully-

submerged production steel riser of Moe and Chucheepsakul (1988) is considered whose E = 2.071011, outer

diameter D = 0.26 m, inner diameter d = 0.20 m, fluid density of 998 kg/m3, riser length L = 300 m, = 0.5

and top tension of 476.2 kN. Good agreement with less than 3% differences is seen in Table 2 based on the

present finite difference simulation vs. the finite element method in Monprapussorn et al. (2007). The

fundamental mode experiences a divergence instability, where the total axial stiffness becomes negative as U

is increased, at the critical U 38 m/s in the case of bending-neglected riser whereas its stability is maintained

to a higher U if the bending and extensibility rigidity are both accounted for. In the latter case, the critical U

64 m/s. This validation enables us to apply the present riser model and numerical approach to SIV simulations

in Section 5. Overall, a time step of t = 0.001 s and a pipe discretized segment s = 5 m, ensuring the

converged simulations of transient and steady-state responses, are applied.

3. Mechanistic Model of Slug Liquid-Gas Flows

The concept of a slug unit cell, first introduced by Wallis (1969) and later considered by several investigators,

is herein applied to a small pipe segment as depicted in Fig. 1. This concept assumes that the slug flow is fully

developed and there exists a frame with a slug unit translational velocity (Fabre and Line, 1992). Each slug

unit cell is further subdivided into the liquid slug and film zones. In the stratified film zone, a non-uniformly

elongated gas bubble is located at the upper part and surrounded by a bottom thin liquid film. The travelling

bubble and liquid film are subject to the gas-wall, liquid-wall or liquid-gas interfacial shear stresses. As for the

liquid slug zone separating the two consecutive films, the bubbly liquid phase fills the whole cross section and

contains small dispersed gas bubbles. The travelling liquid slug is solely subject to the liquid-wall shear stress.

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3.1 Governing Equations

To determine the spatial distributions of liquid-gas masses, velocities and pressure gradient as input parameters

for the riser pipe, a one-dimensional slug flow model of Taitel and Barnea (1990) is employed, assuming the

incompressible liquid and gas. By defining z as the spatial coordinate attached to the moving reference frame

of each slug unit cell as shown in Fig. 1, the spatial profile of the hydrodynamic film height hf as a function of

z can be obtained based the mass and momentum conservations of gas-liquid phases over a slug unit. For a

total film length, the ordinary differentiation of hf with respect to z is expressed as

2 2

1 1( ) sin

d,

dzd1

( ) cosd1

f f g gi i l g

f g f gf

ft l s t b sl g l f g g

ff f

S SS g

A A A Ah

RU U R U U Rg V V

hR R

(3)

in which β is the local inclination angle of the pipe, ρg (ρl) is the gas (liquid) density, Vg (Vf) is the relative gas

(liquid film) velocity with respect to the slug unit translational velocity Ut, Ag (Af) is the cross-sectional gas

(liquid film) area, Sg (Sf) is the gas (liquid film) wetted wall perimeter, Si is the interfacial width, Ub is the

average axial velocity of small bubbles in the liquid slug zone, Ul is the average liquid slug velocity, Rs is the

slug liquid holdup, Rf is the liquid film holdup, dRf /dhf is the ordinary differentiation of Rf with respect to hf,

and τg, τf and τi are the gas-wall, liquid-wall and liquid-gas interfacial shear stresses, respectively. In the film

zone, | | /2,f f l f ff U U | | /2g g g g gf U U and ( ) | | /2,i i g g f g ff U U U U in which Uf (Ug) is the

liquid film (gas bubble) velocity, and ff, fg and fi are the associated friction factors (Spedding and Hand, 1997;

Taitel and Dukler, 1976). In the slug liquid zone, | | /2.f f l l lf U U The pipe cross section in the film zone

has a geometric angle with respect to the diameter centre = 2cos-1(1-2hf/d) and properties Af = d2( - sin)/8,

Ag = A-Af, Sf = d/2, Si = d(2-2cos)1/2/2, and Sg=d-Sf, where d is the internal diameter and A is the inner area.

For a specific Ut, Vf and Vg are given by

( ) ,f t f t l s fV U U U U R R (4)

( ) ,g t g t b s fV U U U U (5)

in which αs and αf are the gas void fraction in the liquid slug and film zone, respectively. A sign of actual

velocities (Ug, Uf, Ul, Ub) is meaningful as it is positive in the downstream direction but negative with respect

to the opposite s coordinate. Note also that Ul and Ub are uniform within the liquid slug zone whereas Ug and

Uf are spatially varied along the film zone, depending on the film hydrodynamics. A geometric relationship

between Rf and hf for a stratified film flow is defined by

22 2 21 1cos 1 1 1 1 .

h h hf f fR f d d d

(6)

Accordingly, dRf /dhf = (4/πd)[1-(2hf /d-1)2]0.5 which can be then substituted into Eq. (3). In numerically

integrating Eq. (3) with respect to z (from z = 0 to the total film length), a certain criterion is imposed for a

given liquid and gas flow rate. From Taitel and Barnea (1990), the liquid mass balance over a slug unit entails

0

1 ,fLls l s t s f u t u fu U R U R L L U L dz (7)

11

 

where uls is the superficial liquid velocity, Lu is the slug unit length and Lf is the film length. All Eqs. (4)-(7)

are combined to evaluate the film zone feature. As the liquid slug is regarded as a bubbly flow, the associated

properties may be estimated through empirical correlations as described in Section 3.2. Ultimately, a pressure

drop over a slug unit length (∆Pu) may be evaluated through

0

sin ,f

s

Lf f f g g

u u ud S S

P L g L dzA A

(8)

where ρu is the average density, ρu=αuρg+(1−αu)ρl, αu is the average void fraction, αu=(ugs −Ubαs+Utαs)/Ut =

(-uls +UlRs+Utαs)/Ut (Barnea, 1990), uls is the superficial liquid velocity, ugs is the superficial gas velocity, and

the slug unit length Lu = Ls + Lf, in which Ls (Lf) is the liquid slug (film) length. For a constant ρg and ρl, the

volumetric flow rate through any cross section is constant such that the mixture velocity Us = uls + ugs = UlRs

+ Ubs in the liquid slug zone or UfRf + Ugf in the film zone. Note that Pu consists of three main components

associated with the gravitational contribution, frictional effects in the liquid slug and film zones. As αu is

independent of the film height, bubble and slug lengths, the first term in Eq. (8) may be assessed independently

of the detailed slug structure (Taitel and Barnea, 1990).

3.2 Slug Flow Variables and Solution Steps

The film hydrodynamics is first analysed to identify a spatial distribution of hf, establishing a nonlinear shape

of the elongated bubble. The so-called closure or empirical correlations for Rs, Ut and Ub are required, rendering

an empirical αs and Ul since αs=1-Rs and Ul = (Us - Ubs)/Rs. Several Rs correlations are available in the literature.

For vertical pipes, Fernandes et al. (1983) suggested Rs = 0.75 (αs =0.25). For inclined pipes, Rs should be

variable as in, e.g., Greskovich and Shrier (1971) who reported 0.5<Rs<1. Herein, a widely used correlation of

Gregory et al. (1978) in which Rs = 1/[1+(Us/8.66)1.39] is considered.

As for Ut, a typical expression Ut = CUs + Ud may be employed, describing a linear superposition of the

drift velocity of the elongated bubble in stagnant liquid Ud and the flow mixture contribution Us multiplied by

a factor C being the ratio of maximum to mean velocities of the liquid slug (C >1). For a good approximation,

C = 1.2 is suggested for turbulent flow (Taitel and Barnea, 1990) whereas Ud = 0.54(gd)0.5cosβ +0.35(gd)0.5sinβ

proposed by Bendiksen (1984) may be used. As for Ub, the closure may be defined in a similar manner to Ut

as Ub = BUs + U0, where B is the distribution parameter and U0 is the average drift velocity of dispersed bubbles.

Following Bendiksen (1984), B(β)=B(0̊)+[B(90̊)- B(0̊)]sin2β where B is bounded between 1 (horizontal pipe)

and 1.2 (vertical pipe). U0 may be obtained through U0=U(1−αs) j, where the modification factor j0 (U0 U)

for the liquid slug (Taitel and Barnea, 1990), and the bubble rising velocity U = 1.54[σg(ρl − ρg)/ρl2]0.25 with

being the interfacial surface tension (Harmathy, 1960). For turbulent slug flow with a low σ, the surface

tension parameter σ/[(ρl − ρg)(d/2)2g] may be taken approximately as 0.001 (Chatjigeorgiou, 2017).

For a smooth surface pipe, the Blasius correlation may be used for the friction factors. In the film zone,

ff = Cfg(DhfUf/l)n and fg = Cfg(DhgUg/g)n in which Dhf = 4Af/Sf or Dhg = 4Ag/(Sg + Si) is the liquid or gas hydraulic

diameter, DhfUf/l or DhgUg/g is the Reynolds number (Re), and l (g) the liquid (gas) kinematic viscosity.

In the liquid zone, ff = Cfg(dUl/l)n. For turbulent flow with Re > 3000, Cfg = 0.046 and n = -0.2 (Taitel and

Dukler, 1976). For an inclined pipe with a stratified flow feature in the liquid film zone, it is assumed that the

12

 

interfacial friction factor fi 0.014 (Cohen and Hanratty, 1968). These coefficients (Cfg, n, fi) have been widely

used in both steady-state (Zhang et al., 2003) and transient (Bonizzi et al., 2009) slug models. Overall, the

above-identified input and output variables may be classified into five groups as follows.

a) Specified slug liquid-gas flow properties: ρg, ρl, g, l, ugs, uls, Us, Lu,

b) Specified pipe geometric variables: β, d, A,

c) Fluid variables through empirical ‘closure’ correlations: Rs, Ub, Ut, fg, ff, fi, Ud, U0, U, σ,

d) Empirical coefficients associated with functions in c): C, B, Cfg, n, j,

e) Numerically obtained variables: ϕ, Ag, Af, Sg, Sf, Si, hf, Lf, αf, Rf, Ls, Ug, Uf, αu, ρu, Pu.

In the literature, there is presently a lack of empirical correlations specifically developed for an inclined,

curved and flexible pipe with a large diameter or high aspect ratio. Most of the correlations have been derived

based on some certain sets of liquid-gas (typically air-water) flows and (small-diameter, straight horizontal or

inclined) pipe conditions. The liquid holdup correlation of Gregory et al. (1978) is considered as this original

model has recently been verified by Pereyra et al. (2012) to be one of the most reliable and simple models for

slug flow. The bubble drift velocity correlation of Bendiksen (1984) is considered as this model has been

shown by several researchers to provide a good approximation for arbitrarily inclined pipes with 0o < β < 90o

(Kaya et al., 1999; Xiao et al., 1990) applicable to the present catenary riser whose inclination is in the range

of 0o < β < 52o. For the same reason, the correlation and associated coefficients for the average propagating

velocity of small dispersed bubbles suggested by the same researcher (Bendiksen, 1984) are considered. These

correlations of gas velocities have been applied to several mechanistic fluid models of slug flows (Petalas and

Aziz, 1998; Xiao et al., 1990; Zhang et al., 2000). For the Blasius correlation coefficients, empirical values

relevant to turbulent slug flows proposed by Taitel and Dukler (1976) are considered as these have recently

been suggested by Zhang et al. (2003) and Taitel and Barnea (1990) who developed a unified mechanistic slug

model for inclined pipes. To further demonstrate the effect of empirical correlations on the SIV prediction, a

sensitivity analysis is presented and discussed in Section 6 with experimental comparisons.

With the selected closures and empirical coefficients, the numerical solution starts by letting Rf = Rs as an

initial condition at z = 0 for each slug unit. This assumption is reasonable as a head section of the elongated

bubble (z = 0) and a tail part of the liquid slug (z = Lu) of consecutive slug units share the same location (Fig.

1). With Rf = Rs, Eq. (6) is solved for the initial hf = hs which is then compared with the critical film height hc

obtained from the zeroing denominator in Eq. (3). The hf value at z = 0 is assigned to be equal to the lower one

of hs versus hc (Taitel and Barnea, 1990). With the deduced hf, values of Rf, f, Uf, Ug, Ag, Af, Sg, Sl and Si are

computed, respectively. Subsequently, Eq. (3) is numerically integrated using a Runge-Kutta scheme with a

small increment z of 0.001 m, during which all related parameters are updated for every station to obtain the

complete spatial hf profile. This process is carried out until the mass balance in Eq. (7) is fulfilled so that Lf,

and subsequently Ls, is identified: Lu = Lf +Ls. The pressure drop is then evaluated through Eq. (8).

For a uniform section A containing liquid (i=1) and gas (i=2) phases, mi = (Rsl +sg)A within a liquid

slug zone while mi = (Rfl +fg)A within a film zone. Depending on a specified bottom (inlet) or top (outlet)

pressure, a variation of the flow-induced mean pressure Pm(s) can be identified through the pressure drop

calculation in Eq. (8). For a given Ut, discretization spacing s and time step t, a function of the space-time

13

 

varying slug flow properties (mi, Ui, P), contributing to Eqs. (1) and (2), may be expressed as

( , ) ( , ),

( , ) ( , ),i i t

i i t

m s t m s U t t t

U s t U s U t t t

( , ) ( ) 1 ( , ) ,m t p tP s t P s U t k R s U t t t (9)

where kp is the percentage of pressure fluctuation assumed to have a space-time varying profile similar to that

of the liquid holdup (Hara, 1977) on account of their interdependence. Herein, the normalized holdup profile

( R ) with a maximum (minimum) amplitude of 1 (-1) is introduced so that the fluctuation occurs around Pm. In

the following, the slug liquid-gas flow features depending on the system flow-pipe parameters are discussed.

4. Riser Properties and Slug Flow Characteristics

Prior to investigating the planar dynamics of catenary pipe transporting slug flows and the effects of slug flow-

pipe parameters on SIV, it is first important to gain some fundamental insights into the pipe statics (geometric

profile, pre-tension variation) and modal (natural frequencies, modes) properties, and the combined liquid-gas

flow features (hf, uL, uG, Rs, Rf ,Pm) contributing to the mass, velocity and pressure variables in Eqs. (1) and (2).

By way of examples applicable to a deep-water application, we consider a very long catenary riser pipe made

of steel with E = 207 GPa, L = 2025 m, d = 0.385 m and D = 0.429 m (L/D 4720). The variable-inclination

riser is fully submerged in a 1000 m depth of quiescent sea waters. The pipe materials have an effective weight

of 699 N/m and initial top tension of 1860 kN, transporting upwardly the combined oil (ρl = 790 kg/m3) and

methane gas (ρg = 0.675 kg/m3) flows (Chatjigeorgiou, 2017). Based on the stability analysis of Bakis and

Srinil (2019), this considered pinned-pinned riser is free from an internal flow-induced divergence instability

if the phase flow velocity is less than the critical value of about 80 m/s.

A referenced static profile of a submerged catenary pipe subject to the combined effective weight and

external hydrostatic pressure is displayed in Fig. 2 including (a) x-y coordinates, (b) local inclination angles

(0 < < 90o) measured clockwise from the x axis, (c) static tensions T and (d) curvatures along s from the

top (x=y=s=0) to the bottom (x=1688m, y=1000m, s=2025m). In Fig. 2a, the maximum sag measured

perpendicularly from the inclined chord, per the chord span (dashed line), is about 0.11 at s 1096 m (more

than a half of L at s 1012.5 m). This signifies the pipe asymmetric geometry which would influence on the

associated modal characteristics of SIV responses. In Fig. 2b, the maximum 51o occurs at the top whereas

the minimum 2o occurs at the bottom, the latter reflecting a nearly horizontal laid-down pipe. By averaging

all nodal (with s = 1m), the mean 30o is coincidentally equal to the chord inclination (tan-1(1000/1688)).

In Fig. 2c, the variable pre-tension experiences a considerable reduction (37.4%) from the top (1860 kN) to

the bottom (1164 kN), caused by the distributed gravity and static pressure loads. Such tension and asymmetric

profile changes affect the pipe stiffness properties. In Fig. 2d, values are as small as order of 10-4, albeit

nonlinearly varying. The maximum (minimum) is near the bottom (top) before the pinned end with =0.

Additional travelling slug flow masses would further modify such initial geometry and tensions along the span

due to the moving weight effect and the internal pressure drop change. Based on the finite element method of

free vibrations (Safrendyo and Srinil, 2018), natural frequencies of the submerged and empty pipe (fn) in Fig.

2 are found to be in the range of 0.03-0.71 Hz for the first 40 planar modes, suggesting a tension-dominated

14

 

flexible riser due to such closely spaced low frequencies (Srinil et al., 2009).

As for slug flows consisting of a series of slug unit trains, a fixed normalized unit length Lu/d is herein

considered so that the liquid-gas mass balances of all slug units travelling along the pipe span are consistent

with the assumed steady-state flow model described in Section 3. As each slug unit propagates upwardly, part

of the liquid film may move backward with a negative Ul relative to the gas-liquid interface. Nevertheless, the

amount of this part is picked up by the following slug liquid. This enables a constant flow rate at a fixed cross

section over the time of the slug unit passage (Taitel and Barnea, 1990). Based on the experimental liquid-gas

flow maps of inclined pipes with 0o< < 90o (Spedding and Nguyen, 1980), a volumetric flow rate is chosen

such that uls and ugs are associated with a slug flow regime whose likelihood of occurrence increases with

(Spedding and Nguyen, 1980). The main slug unit outputs may be described as a function of β, Lu/d, ugs and

uls, where the associated slug unit velocity Ut depends on the mixture velocity Us = uls + ugs. For steady flows,

a characteristic or primary slug frequency (fs) may be approximated as fs Ut/Lu (Hz), depending on the

specified Lu/d. In the literature, the suggested Lu/d values for subsea applications are variable, depending on

several practical and operational factors as exemplified in Table 1. In this study, Lu/d is assumed such that a

unique real-valued numerical solution of the Taitel-Barnea model can be found.

By assigning uls = 2 m/s, Table 3 presents case studies with specified β, Lu/d, ugs, Ut, and the associated fs.

Note that β is based on the maximum, minimum and average β from Fig. 2b. There are two parametric (Lu/d,

Ut) pairs yielding a comparable fs: (80, 6) vs. (208, 16) and (63, 16) vs. (80, 20). These cases will be elaborated

in Section 5 to verify if there is a dependence of SIV on individual Lu/d or Ut, regardless of fs.

The effect of varying ugs (Ut) on slug flow features is first analysed. By considering β=30̊ and Lu/d 80,

and varying ugs = 2, 4.5, 7, 10.3 and 14.0 (Ut 6, 9, 12, 16 and 20) m/s, Fig. 3 illustrates spatial profiles of slug

flow properties in the liquid and gas zones including the normalized film thickness hf/d (Fig. 3a), liquid holdup

R (Fig. 3b), liquid velocity uL,(Fig. 3c) and gas velocity uG (Fig. 3d), along Lu/d. Actual values of Lu, film

length (Lf), liquid slug length (Ls) and Lf/Ls are reported in Table 4a. Note that, in Fig. 3 (and subsequent Figs.

4-5), the flow direction is from left to right; uG = Ub, uL = Ul and hf/d =1 (i.e. fully occupied with liquid) in the

slug liquid zone whereas uG = Ug and uL = Uf in the film zone. As ugs is increased from 2 to 14 m/s, Fig. 3a

shows that hf/d decreases whereas Lf/d (Ls/d) elongates (shrinks), rendering a wider, longer and more uniform

gas bubble shape. This is justified in Table 4a with the increasing Lf/Ls, Lf/Lu and gas volumetric fraction per

slug unit vol. Correspondingly, Rf decreases from the bubble nose to its tail as in Fig. 3b. A spatial variation

in hf/d and Rf is more nonlinear in the lower ugs case entailing a greater liquid mass variation in the film zone.

Hence, an assumption of a rectangular slug unit pulse employed in some previous studies (Table 1) might not

be justified at a low ugs. For the highest ugs, the gas phase dominates the slug unit with the maximum vol. The

uL profile in Fig. 3c experiences a transition from being positive at the bubble nose to negative at the tail for

low ugs = 2.0, 4.5 and 7 m/s. Such a negative uL is influenced by low Ut and Rf (Eq. 4), signifying the prevailing

gravity effect. On the other hand, uG in Fig. 3d is always positive because of the greater f (Eq. 5): this is

typical for the gas buoyancy feature. Nevertheless, both uL and uG increase with ugs, with the maximum uL (uG)

at the bubble nose (tail). In all cases, absolute values of uG > uL and uG profiles are relatively uniform with a

smaller percentage change along Lu/d. There is a discontinuity (vertical dashed lines) in the obtained outputs,

15

 

clearly distinguishing the liquid slug from the film zones. This essentially emphasizes a rapid change in slug

flow properties which, in turn, would influence the pipe distributed mass and stiffness. Based on Fig. 3 and

Table 4a, the pipe SIV would be subject to a greater flow velocity effect in the case of higher ugs (i.e. Ut) and

to a greater liquid mass (density) fluctuation in the case of lower ugs.

The effect of varying Lu/d is next discussed through Fig. 4 and Table 4b. By assigning β=30 ̊and ugs =

10.3 (Ut 16) m/s, variations of hf/d (Fig. 4a), R (Fig. 4b), uL,(Fig. 4c) and uG (Fig. 4d) are displayed in the

case of increasing Lu/d from 63 to 208. Both Lf and Ls are seen to increase such that a conservation of unit mass

is satisfied for a fixed uls and ugs; i.e., vol = 0.749 is constant as justified in Table 4b. For this fixed ugs, Lf/Ls

decreases as Lu/d increases, contrary to the increasing Lf/Ls trend in Table 4a. The variable ranges of 0.5 < Lf/Lu

< 0.7, 0.2 < hf/d < 0.4 and 0.1< R< 0.38 in the varying Lu/d case are smaller than those (0.25 < Lf/Lu < 0.7, 0.2

< hf/d < 0.7 and 0.1< R< 0.75) in the varying ugs case in Fig. 3. This implies a greater effect of varying Ut, than

varying Lu/d, on the slug profile modification. As Lu/d is increased, hf/d and Rf in the film zone decrease, the

latter rendering the decreasing and eventually negative uL (Eq. 4). On the contrary, uG slightly increases owing

to the increasing f (Eq. 5). Parameters within Ls remain unchanged since uls and ugs remain the same. The

change in Lu affects Eq. (7) for determining Lf such that a greater Lu would result in a lower hf and Rf (Eq. 6).

The slug flow features are also dependent on which affects some empirical input variables (Ut, Ub).

Understanding of this effect is meaningful for a catenary pipe with variable inclinations. By specifying ugs =

4.5 m/s and Lu/d 80, variations of hf/d (Fig. 5a), R (Fig. 5b), uL,(Fig. 5c) and uG (Fig. 5d) are plotted for β=

2o, 15o, 30o and 51o, together with outputs in Table 4c. As β is increased, Fig. 5a shows that Lf/d (Ls/d) is

shortened (elongated), enabling the decreasing Lf/Ls, while hf/d is slightly reduced. Hence, the associated Rf is

reduced as shown in Fig. 5b. These indicate that the gas bubble shape becomes shorter and wider, increasingly

occupying the cross section as the pipe is more inclined towards a vertical 90o where a so-called Taylor bubble

would be developed (Fernandes et al., 1983). At higher , uG becomes faster due to the buoyancy effect (Fig.

5d) whereas uL becomes slower (Fig. 5c) due to the gravity effect. For such a low uls = 2 m/s and ugs = 4.5 m/s,

uL < 0 appears at a relatively large β, suggesting a falling liquid film near the tail region. Because of the fixed

ugs and Lu/d, vol slightly decreases as is increased. This implies that for a system with variable inclination

under steady-state flows, the changes in the liquid-gas velocities predominate over a volumetric mass variation.

For a fixed β = 30o, Table 4 summarizes the slug outputs for the two parametric pairs (ugs, Lu/d) having a

comparable fs (Table 3) whose slug profiles have been displayed as part of Figs. 3 and 4. For the first pair (2,

80) vs. (10.3, 208), there is a considerable difference in both Lf/Ls (0.41 vs. 1.18) and vol (0.393 vs. 0.749),

apart from the noticeable different ranges of uL and uG in Figs. 3 and 4. As for the second pair (14, 80) vs.

(10.3, 63), Lf/Ls (2.38 vs. 2.27) and vol (0.797 vs. 0.749) are comparable, as are uL and uG. These results suggest

that the pipe SIV, especially for the first parametric pair case, could be somewhat different despite having a

similarly potential resonance with the comparable fs. This point will be discussed again in Section 5.

Once a fully developed slug flow profile is established as in Figs. 3-5 with Re in the approximate range

of 4104 to 1.5106, it is then feasible to estimate the total pressure drop over a slug unit (Pu). By accounting

for all the travelling slug units fully occupying the pipe span, the resulting mean-valued tension in the pipe

16

 

wall Tm = T-2PmA, where Pm is the mean pressure spatially varied along the pipe. From Eq. (8), Pu consists

of the gravitational effect (Pg), the frictional effect in the slug liquid zone (Ps) and the frictional effect in

the film zone (Pf). By assuming that the top outlet mean pressure (Pt) is equivalent to the atmospheric pressure

of 101.3 kPa as in Ortega et al. (2018), Table 5 summarizes Pg, Ps, Pf, Pu, the integer number of full slug

units occupying the pipe (Ns L/Lu) over a constant slugging period (1/fs), the bottom inlet mean pressure (Pb),

the percentage of the mean pressure drop from the bottom to the top [Pdp = (Pb-Pt)×100/Pb], the resulting mean

tension at the bottom (Tb), and the percent reduction of mean tension (Tdp) from the top (Tm = 1855.5 kN) to

Tb. Results in cases of varying (5a) Ut, (5b) Lu/d and (5c) β, including Ut-Lu/d pairs for a fixed fs, are presented.

As Ut is increased for a given Lu/d = 80 and β = 30o, Table 5a shows that Pg decreases while both Ps

and Pf (Ps > Pf) increase. A reduction in Pg is due to an increment of u, hence, a reduction in u in Eq.

(8) (i.e. having less liquid and more vol) as ugs is increased. On the contrary, Ps increases due to the increasing

liquid slug velocity (Fig. 3c) where a squared value amplifies the shear stress f notwithstanding the decreasing

Ls (Table 4a). Likewise, Pf increases due to the increasing Lf (Table 4a) and amplification of f and g as a

result of increasing liquid-gas velocities (Figs. 3c and 3d). By summing up all the pressure drop components

(Pu) and accumulating that along the pipe, Pu and Pb decrease as ugs is increased, mostly following the Pg

trend. Note that, at ugs = 10.3 and 14 m/s, Pu and Pb in both cases are comparable with a similar Pdp and Tdp,

regardless of the increasing (decreasing) trend of Pf and Ps (Pg). For a constant Ns, the lowest ugs case

experiences the maximum Tdp 57%.

As Lu/d is increased for a given ugs = 10.3 m/s and β = 30o, Table 5b shows that both Pg and Ps (Ps >

Pf) increase, contributing to the increasing Pu. The increment of Pg is due to the increasing Lu since vol is

constant (Table 4b) whereas the increment of Ps is due to the increasing Ls since the liquid slug velocity

associated with the fixed ugs is unchanged. Because of a constant vol, Pf is nearly unaltered as Lu/d is varied:

this justifies a conservation of mass and momentum within a slug unit with a constant uls and ugs. The minimum

Pb is associated with the lowest Lu/d case having the highest Ns whereas the minimum Tb is associated with the

greatest Lu/d case. Nevertheless, both Tables 5a (varying ugs) and 5b (varying Lu/d) reveal similar values of

nearly 97% of Pdp and 50% of Tdp in comparison with 37% of the static tension reduction in Fig. 2a. This slug-

induced tension variation would influence the pipe axial stiffness, associated natural frequencies and modal

properties, which, in turn, could modify the pipe resonant SIV during the slug transportation.

For a given ugs = 4.5 m/s and Lu/d 80, Table 5c shows that both Pg and Ps increase, contributing to

the increasing Pu, as β is increased. This is similar to the increasing Lu/d case in Table 5b, but due to a different

reason. Since ugs and Lu/d are constant and vol is nearly unchanged (Table 4c), the increasing Pg is now due

to the inclination effect (i.e. increasing sin in Eq. 8). The increasing Ps is due to the increasing Ls. On the

other hand, despite the decreasing Lf, Pf tends to decrease as the pipe is more inclined. This is attributed to

the increasing liquid-gas velocities moving in the opposite directions within the liquid film (Figs. 5c vs. 5d),

enabling the sign differences of the associated shear stress terms and, hence, the ensuing reduction of integrated

values in Eq. (8). It is also interesting to remark for the pipe with highest β = 51o that Pb is maximum while

Tb is minimum, the latter reducing the pipe stiffness, potentially increasing the pipe sagging displacement and

17

 

hence reducing the pipe natural frequencies (Srinil et al., 2003).

In accordance with Table 4, Table 5 also compares the pressure change components, resulting Pb and Tb,

Pdp and Tdp, for the two parametric pairs (ugs, Lu/d) having a comparable fs (Table 3). It is seen that the first pair

(2, 80) vs. (10.3, 208) experiences a greater difference in both Pb (5174.5 vs. 3384.3 kPa) and Tb (802.19 vs.

927.23 kN) than the other pair (14, 80) vs. (10.3, 63) with comparable Pb (3212.7 vs. 3126.6 kPa) and Tb

(939.21 vs. 945.23 kN). These are due to the greater differences in Lf, Ls, Lu, vol, the associated liquid-gas

velocities (Figs. 3 and 4) and the resulting shear stresses in the former case. Overall variations in the input

flow-pipe parameters will influence the planar static and dynamic responses (amplitudes, modes, oscillation

frequencies, resonances) of the slug-transported pipe. These are analysed and discussed in the following.

5. Dynamic Characteristics of Catenary Riser Carrying Slug Flows

Mechanisms of a long inclined curved flexible riser transporting slug flows are now investigated based on the

slug flow-pipe properties in Section 4. In this study, multiple slug unit trains travelling along the pipe have the

same features according to the steady-state flow assumption. As the mean pipe inclination and the chord angle

are both about 30o, the slug flow profile with = 30o is specified, unless stated otherwise, as a representative

slug unit cell. From Eq. (9), a small local fluctuation amplitude (kp) of the internal pressure about a mean value

is taken as 5%, following recent experimental-numerical works of Ortega (2015) which suggested kp 5.5%.

Attention is placed on investigating the effect of varying Ut and Lu/d primarily governing fs = Ut/Lu. The slug

flow properties (Figs. 3-5), pressure and tension variations (Tables 3-5) are incorporated as the space-time

varying inputs into the pipe model through the slug flow-induced momentum and gravity loading terms. In the

following, features of slug-induced fluctuation frequencies, transient drifts, amplified mean displacements,

SIV responses and bending/axial dynamic stresses are presented and discussed.

5.1 Slug Fluctuation Frequencies

With Lu/d = 80 and Ut =16 m/s, the space-time varying profiles of the four main properties of slug flows are

illustrated in Fig. 6, including (a) R, (b) uL, (c) uG and (d) P, within the first 20 s and 200 m of the pipe’s bottom

part starting from the inlet (1825 < s < 2025 m). For this Lu/d = 80 and Ut =16 m/s, it takes about 126 s (i.e. the

transient period) for the pipe to be fully occupied by about 65 slug units. Note that, for a demonstration purpose,

the incremental profiles in Fig. 6 are plotted for every second, despite the actual t = 0.001 s, with the flow

direction being from right to left as in Fig. 1. Based on Fig. 6, time histories of input variables at a specific

pipe location can be recorded, showing a certain periodicity. The associated fluctuation frequencies may be

approximated using the Fast Fourier Transform (FFT). Despite having different profiles of R, uL, uG and P,

their frequency contents are the same based on the unique space-time variation.

In the case of varying Ut = 6, 9, 16 and 20 m/s, Fig. 7 illustrates FFT plots in which the peaks have a

normalized spectral density amplitude of unity. It is seen that, in consistent with results in Table 3, the dominant

slug frequency increases as Ut is consecutively increased, confirming that fs = 0.20, 0.29, 0.52 and 0.66 Hz

according to the relation fs = Ut/Lu. In addition to fs, several secondary peaks with smaller amplitudes can be

remarked whose frequencies are approximately about (N+1)fs where N = 1, 2, 3…. Such a multi-harmonic

feature has recently been shown by Safrendyo and Srinil (2018) based on a rectangular pulse train model of

18

 

slug flows. Within the displayed 3 Hz range, the minimum (maximum) Ut =6 m/s (20 m/s) case experiences

the highest (lowest) number of 15 (4) frequency contents in Fig. 7.

By assigning Ut = 16 m/s, Fig. 8 displays FFT plots in the case of varying Lu/d = 63, 120, 140 and 208,

confirming fs values reported in Table 3. As expected from fs = Ut/Lu and in contrast to the varying Ut case, the

minimum (maximum) Lu/d = 63 (208) case experiences the lowest (highest) number of 4 (14) frequency

contents within the displayed 3 Hz range. For a fixed Ut = 9 m/s and Lu/d = 80, FFT plots in the case of varying

= 2o, 15o, 30o and 51o are shown in Fig. 9. Their frequency contents in all cases appear to be similar with

identical fs = 0.29 Hz. There are 9 secondary peaks with very small moderations of amplitudes and frequencies.

This feature may justify our assumption in neglecting the variation along the inclined pipe for steady slug

flows. Overall, based on the observed fluctuation frequencies, it is expected that the pipe SIV in the lower Ut

or greater Lu/d case might be subject to a greater number of excited vibrational modes. These will be examined

in Sections 5.3 and 5.4, by also accounting for other key factors related to the pipe initial static profile, fluid

propagating masses, velocities and pressure drop changes.

5.2 Transient Drifts

Figure 10 displays the space-time variations of pipe responses in u and v directions, capturing the slug flow-

induced initial transient (quasi-static) and steady-state (dynamic) stages, in the case of varying Ut (Fig. 10a, b),

Lu/d (Fig. 10c, d) and β (Fig. 10e, f). As slug flows gradually propagate into the pipe and fully occupy it, Fig.

10 reveals that, during t < 250 s, the pipe experiences a considerable excursion or transient drift in all directions

and parametric cases. Such u and v drift profiles are asymmetric with respect to the middle span, having an

inflection point around s/L 0.6 which exhibits a sign change in the pipe curvatures between the bottom and

top parts. This feature agrees with an observation in Chatjigeorgiou (2017). The u drifts in the lower part are

negative and nearly zero because of a small local pipe inclination (nearly horizontal) whereas those in the

upper more-inclined part are positive with greater values. On the other hand, v drifts in the lower part are

positive because of the dominant gravity effect in the Y direction whereas they are negative in the upper part

through an inflection point. These features, which are triggered by a sudden entrance of fluctuating slug flows

unbalancing the space-time varying liquid holdups in conjunction with the variable initial tensions and

geometric profiles (Fig. 2), make the lower and upper parts bend outward (i.e. far away from the curvature

centre) and inward about an inflection point, respectively. Similar large excursions of a catenary riser owing

to the instantaneous passage of slug flows have been reported by Ortega (2015), Meléndez and Julca (2019).

In association with Fig. 10, Table 6 compares maximum values of mean u and v drifts (um, vm) occurring

at s/L 0.3 and 0.8, respectively, in all cases. These nearly quarter-span locations and the associated drift

profiles are reminiscent of the fundamental planar mode of a catenary pipe Srinil (2010). With increasing Ut,

it is seen in Figs. 10a and 10b that the excursion duration decreases as Ut is increased from 6 to 20 m/s. This

is expected since slug flows with a higher Ut requires less time to reach the pipe outlet. For instance, the pipe

conveying slug flows at Ut = 6 and 16 m/s switches to the perfectly steady states after t 337 and 126 s,

respectively. However, the maximum (minimum) transient effect is observed at the lowest (highest) Ut,

attaining the maximum (minimum) um of about 47d (17d) and vm of about 56d (22d). Such considerable drifts

19

 

are caused by the greater R or lower vol (Table 4a), giving rise to a greater gravity effect which is further

imbalanced between the two asymmetric pipe parts. In the case of varying Lu/d, results in Figs. 10c and 10d

reveal similar behaviours of planar drifts and excursion durations, with maximum um 20d and vm 24d. This

is expected since the transient duration depends on Ut being fixed as 16 m/s whereas the maximum drifts are

proportional to vol which is nearly unchanged as about 0.75 (Table 4b). These rules may also be applied to

the case of varying β as shown in Figs. 10e and 10f. Although the excitation durations are comparable in all

cases with the same Ut, the maximum transient um and vm increase with β. By comparing the cases of β =51°

and 2°, the former experiences um 35d and vm 41d, being slightly greater than 31d and 36d in the latter case,

respectively. Again, a small increment of maximum transient drifts with increasing β is attributed to the

associated decreasing αvol as in Table 4c.

5.3 Amplified Mean Displacements

Once the pipe is fully occupied by the slug flows, the transient responses decay and eventually die out due to

the system damping. Consequently, a steady-state SIV response occurs after t > 300s and the pipe experiences

an oscillation around a new stabilized static equilibrium or mean displacement profile. Figure 11 illustrates

contour plots of space-time varying responses in the case of varying Ut for 2990 < t < 3000s. Some interesting

features are noticed. Because of the gravity and flow momenta, the total u and v responses – inclusive of mean

and oscillatory components – decrease as Ut is increased. For the low Ut = 6 and 9 m/s, contour plots appear

to contain considerable mean components with some moderate oscillations along the pipe. For the intermediate

Ut = 16 m/s, larger moderating oscillations become visible. Both mean and oscillation components decrease at

the high Ut = 20 m/s. Figure 12 depicts um and vm envelopes during the steady state and associated with the

case of varying Ut (Fig. 12a, b), Lu/d (Fig. 12c, d), β (Fig. 12e, f) and for the two parametric pairs (Fig. 12g,

h). Table 6 compares the maximum um and vm in the steady-state vs. transient cases. Table 7 compares the top

and bottom mean tensions accounting for the additional stretching during the steady-state response.

Different from the transient excursions reported in Section 5.2, the plots in Fig. 12 reveal that the pipe is

quasi-statically forced to bend outward in both u and v directions, exhibiting a new asymmetric and larger-

sagged catenary profile with maximum displacements being less than 2d and locating near s/L 0.5 for u and

0.6 for v. In the case of varying Ut (Fig. 12a, b), mean values decrease from 0.68d (u) and 1.62d (v) to 0.24d

(u) and 0.55d (v) as Ut is increased from 6 to 20 m/s, respectively. Again, such difference is attributed to the

associated vol (Table 4a) and Tb (Table 5a) entailing the different slug weight effects. In the case of varying

Lu/d (Fig. 12c, d), the mean displacement profiles and their maximum values are slightly varied (about 0.3 for

u and 0.7 for v) because of a small variation in vol (Table 4b) and Tb (Table 5b). In the case of varying β (Fig.

12e, f), the mean displacements generally and slightly increase with β due to the decreasing vol (Table 4c) and

Tb (Table 5c). In comparison with the initial pre-tension results Fig. 2 and Table 5, the overall end tensions in

Table 7 are further increased due to the mean displacement amplifications subject to the steady static weights

of the moving slugs. In the case of the two parametric pairs (Fig. 12g, h), the prevailing effects from vol (Table

4) and Tb (Table 5) are revealed regardless of fs, in which the case (6, 80) shows distinctively larger mean

displacements than the other three cases. Overall, such modifications in the mean displacement profiles and

20

 

associated tensions affect geometrically the pipe restoring forces and natural frequencies. These, in turn,

influence the SIV features as discussed in the following.

5.4 Oscillation Amplitudes and Frequencies

By removing the mean components from the total steady-state responses (e.g. Fig. 11), Fig. 13 displays contour

plots of the space-time variations of coexisting u (a, c, e, g) and v (b, d, f, h) oscillations (units in mm) in the

case of varying Ut for a given Lu/d = 80, within 2990 < t < 3000 s. The associated spanwise FFT plots along

the pipe are shown in Fig. 14 for the 3 Hz range of oscillation frequencies (fo) versus the slug fluctuations in

Fig. 7. Recall that fs increases as Ut is increased or Lu/d is decreased. At low Ut = 6 (Fig. 13a, b) and 9 (Fig.

13c, d) m/s, high modulations of response amplitudes are evidently captured. Accordingly, FFT plots reveal

multi harmonics in which the dominant fo (with a highest normalized spectral density) is about 0.20 Hz (Fig.

14a, b) for Ut = 6 m/s and 0.29 Hz (Fig. 14c, d) for Ut = 9 m/s. These dominant frequencies are the same as the

reported fs (Fig. 7a, b) caused by the travelling slug excitations. Higher frequency contents with smaller

amplitudes come into play in both Ut cases, implying the increased likelihood of dynamic stresses and fatigue-

related issue. Nevertheless, as Ut becomes higher at 16 (Fig. 13e, f) and 20 (Fig. 13g, h) m/s, overall u and v

responses exhibit a periodic, unimodal and standing-wave feature with distinctive positions of minimum, zero

(at nodes) and maximum amplitudes. The associated FFT plots reveal a single harmonic peak whose fo is

increased, due to increasing Ut, to 0.52 (Fig. 14e, f) and 0.66 (Fig. 14g, h) Hz, being the same as fs in Fig. 7c

and d, respectively. In such a high Ut range, higher frequencies of the slug fluctuation play a negligible role.

Hence, results in Figs. 13 and 14 highlight the effect of increasing Ut leading to a transition from being

multimodal to unimodal SIV.

The effect of varying Lu/d is next discussed. In the case of high Ut = 16 m/s, contour plots of coexisting u

and v oscillations (units in mm) are displayed in Fig. 15a, b for Lu/d = 120 and in Fig. 15c, d for Lu/d = 208.

The associated FFT plots are shown in Fig. 16a-d. In comparison with Fig. 13e, f and Fig. 14e, f with the same

Ut = 16 m/s but lower Lu/d = 80, the main standing-wave feature is maintained in Fig. 15a-d with a greater

Lu/d. With decreasing fs, a lower vibration mode is excited with the dominant fo 0.35 Hz for Lu/d = 120 and

0.2 Hz for Lu/d = 208. These results signify the unimodal SIV at high Ut, regardless of Lu/d. By comparing Fig.

13a and b with Fig. 15c, d, their u and v contour plots are clearly distinctive, despite being subject to the same

fs. These results further emphasize a different role played by individual Ut or Lu/d. The case with higher Lu/d =

208 and faster Ut = 16 entails greater u and v responses potentially due to a greater flow momentum effect.

In the case of lower Ut = 6 m/s, contour plots of u and v oscillations are displayed in Fig. 15e, f for Lu/d =

120 and Fig. 15g, h for Lu/d = 208. The associated FFT plots are shown in Fig. 16e to h. These results should

be compared with those in Fig. 13a, b and Fig.14a, b with the same Ut = 6 m/s but lower Lu/d = 80. In the case

of Lu/d = 120, the modulation feature is still visible in Fig. 15e, f showing two dominant lower/higher modes

intermittently switching among themselves over time. This mode-switching occurrence is justified by FFT

plots with two outstanding and comparable fo peaks at 0.13 and 1.31 Hz. Figure 17 exemplifies u (a, c) and v

(b, d) spatial modal profiles inclusive of mean displacements at different time instants where the lower (a, b)

or higher (c, d) mode prevails the riser dynamics with large drifts and variable curvatures along the span. Some

local modulations are observed (see the red dotted circles in Fig. 17b), suggesting a multi-modal interaction

21

 

between higher/lower modes during SIV. Nevertheless, as Lu/d is further increased to 208, contour plots of u

(Fig. 15g) and v (Fig. 15h) responses, together with FFT (Fig. 16g, h), appear to become unimodal as in the

higher Ut range. This feature now suggests a greater effect of increasing Lu/d with respect to a multi-to-single

modal transition in a low Ut range.

Some selected phase plane trajectories associated with the spatially maximum root-mean-squared u and

v oscillations are displayed in Fig. 18 for (a, b) Lu/d = 80 and Ut = 6 m/s, (c, d) Lu/d = 80 and Ut = 16 m/s, (e,

f) Lu/d = 208 and Ut = 16 m/s, and (g, h) Lu/d = 120 and Ut = 6 m/s. In the low Lu/d and low Ut case (a, b),

trajectories appear rather complicated, modulated and non-repetitive for each cycle due to the multi-harmonic

effect. In the low Lu/d but higher Ut case (c, d), trajectories become dominated by a single harmonic, yet

exhibiting a peculiar closed-loop pattern with a high level of repetitive cycles. In the high Lu/d and high Ut

case (e, f), a nearly perfect circular trajectory associated with a single harmonic, unimodal and periodic motion

takes place. Finally, in the intermediate Lu/d and low Ut case (g, h) exhibiting a two-dominant mode response

(Figs. 15e, f, 16e, f and 17), trajectories reveal a switching pattern between outer and inner closed-loop orbits

with some modulations as a result of modal transition.

In summary, SIV features are found to be dependent on the individual effect of Lu/d and Ut, rather than fs.

In order to determine the trends and most critical responses, Fig. 19 plots the spatially maximum values of the

root-mean-squared u and v amplitudes (urms, vrms) in the case of varying Ut for a fixed Lu/d = 80 (a, b) and in

the case of varying Lu/d for a fixed Ut = 6 m/s (c, d) and 16 m/s (e, f). It is found that the peak u and v responses

occur when Lu/d = 80 and Ut = 16 m/s (Fig. 19a, b), Lu/d = 120 and 200 for Ut = 6 m/s (Fig. 19c, d), and Lu/d

= 80, 108 and 208 for Ut = 16 m/s (Fig. 19e, f). Such multi-peak occurrences as Ut or Lu/d is varied suggest a

dynamic resonance condition between the travelling slug flows and the pipe’s dominant oscillation frequencies.

Overall, the case of Lu/d = 208 and Ut = 16 m/s appears to be the worst scenario producing largest u and v

amplitudes for the present parametric case studies.

5.5 Pipe Bending and Axial Stresses

It is of practical importance to examine bending and axial stresses caused by SIV. For the space-time varying

maximum bending stresses in X and Y directions, / 2u DEu and / 2v DEv , respectively, where + (−)

denotes the tensile (compressive) stress depending on modal curvatures ( ,u v ). Correspondingly, the space-

time varying axial stress can be evaluated through a E x u y v where the terms in the parenthesis are the

extensional dynamic strain proportional to the pipe displacement gradients (Srinil et al., 2007). The total

stresses (σt) can be finally computed based on the summation of static, dynamic bending and axial stresses. In

the following, stress values are reported in MPa.

Figure 20 presents contour plots of σu and σv inclusive of mean and oscillatory components along the pipe

span and time (2980 < t < 3000 s) by comparing the four chosen cases of (a, b) Lu/d = 80 and Ut = 6 m/s, (c, d)

Lu/d = 120 and Ut = 6 m/s, (e, f) Lu/d = 80 and Ut = 16 m/s, and (g, h) Lu/d = 208 and Ut = 16 m/s. Note that

the first and fourth cases are based on the same fs = 0.20 Hz. Due to small curvature variations, patterns and

values of σu and σv exclusive of mean components are similar and comparable to those plotted in Fig. 20. This

is, however, not the case for the associated axial stresses due to their lower-order displacement derivatives.

22

 

For σa evaluations, the effects of mean components are certainly meaningful as noted in Zanganeh and Srinil

(2016). For the above chosen cases, contour plots of σa are displayed in Fig. 21 by comparing between the total

(mean and oscillatory) σa (Fig. 21a, c, e, g) and mean-free σa (Fig. 21b, d, f, h). For the cases of fs 0.20 Hz,

contour plots of σt are shown in Fig. 22 in both x and y directions. Table 8 summarizes the spatially and

temporally maximum (+) and minimum (-) σu and σv associated with Fig. 20 and σa (σt) associated with Fig. 21

(Fig. 22). In all cases, maximum and minimum bending stresses occur in the v direction, and they are below

the typical yield strengths ( 200-500 MPa) for a standard steel pipe used in oil and gas applications.

For a low Ut = 6 m/s and Lu/d = 80, high oscillatory modulations in σu (Fig. 20a), σv (Fig. 20b) and σa (Fig.

21b) contours are clearly observed, as in the displacement plots (Fig. 13a, b), accompanied by large mean

stresses (Fig. 21a) associated with large mean displacements (Fig. 12a, b). By increasing either Lu/d to 120

(Fig. 20c, d) or Ut to 16 m/s (Fig. 20e, f), overall σu and σv contours reveal standing-wave patterns, as in the

associated displacements, whose values are positively and negatively amplified in Table 8, with the much

worst scenario being associated with the former case. This highlights the important effect of increasing slug

unit length on the pipe SIV bending stresses. Nevertheless, the variation of Ut plays a greater role than that of

Lu/d in evaluating the total σa as shown in Fig. 21e and f vs. Fig. 21c and d. As reported in Table 8, the mean

σa values are considerably decreased with Ut = 16 m/s and Lu/d = 80 due to the decreased mean displacements

(Fig. 12). The trend of oscillatory σa behaves similarly to that of the total σu and σv according to the same

excited vibration modes. By comparing the case of Ut = 6 m/s and Lu/d = 80 vs. Ut = 16 m/s and Lu/d = 208,

both having similar fs, the latter exhibit greater σu, σv and oscillatory σa and whereas the former exhibits the

maximum mean σa. These emphasizes the individual role played by Lu/d or Ut, regardless of fs. Comparisons

in Fig. 22 and Table 8 also suggest the worst σt scenario (i.e. tensile type) associated with the case subject to

the maximum slug gravity weight effect under Ut = 6 m/s and Lu/d = 80. Nevertheless, since σt decreases from

the riser top to bottom, the case with higher Ut = 16 m/s and Lu/d = 208 may pose a critical challenge associated

with the diminishing σt leading to a greater likelihood dynamic buckling with a negative σt.

6. Validation and Sensitivity Analysis

To validate the present combined riser-slug models, comparisons of numerical and experimental results as well

as sensitivity studies are performed by considering a small-scale catenary riser carrying air-water slug flows

recently tested by Zhu et al. (2019b). For this flexible pipe, the end supports are fixed; the mixed air and water

enter the pipe bottom inlet, being released to the atmosphere and water tank from the top outlet, respectively.

Pipe properties include L = 0.95 m, d (D) =4 (6) mm, E = 7.15106 N/m2, mr=0.0164 kg/m, = 0.48 and

2%. The air-water flows have l = 998 kg/m3, g = 1.225 kg/m3, l 110-6 and g 1.510-5 m2/s. A mean

catenary configuration is numerically formed by incorporating the effects of pipe weight and average internal

flow pressure (T-2PmA) such that a resulting top and bottom tension is about 0.17 and 0.07 N, respectively.

From a free-vibration simulation, this numerical catenary shape yields the fundamental natural frequency of

about 3.16 Hz matching 3.163 Hz from a free decay test of the water-filled pipe. The experimental case with

ugs =1.061 and uls=0.35 m/s, yielding the maximum SIV predominant in its first mode, is considered where the

inlet average pressure is about 14 kPa with a fluctuation of about 2%. Due to a lack of information in Zhu et

al. (2019b) on the pipe’s dynamic bending strains and tensions as well as the statistical properties of slug

23

 

formations (e.g. the probability density distributions of liquid slug and bubble lengths), the average Lu/d is

parametrically varied for an assigned Ut (based on experimental ugs and uls), assuming the steadily time-varying

slug flow such that the dominant first-mode SIV is captured in numerical simulations with the converged z =

0.0001 m, s = 0.01 m and t = 0.001 s. This occurs when Lu/d 136 entailing the riser dominant oscillation

frequency of about 3.3 Hz comparable to the experimentally reported 3.41 Hz.

For sensitivity studies, three sets of four empirical correlations of the slug liquid holdup (Rs), drift velocity

(Ud) and Blasius constants (Cfg, n) for friction coefficients ff and fg in turbulent flow are considered to exemplify

uncertainty in numerical SIV prediction. These variables are considered as they play a considerable role in the

slug flow-induced fluctuation and momentum forces. Other empirical variables related to the gas have been

found to produce a minor effect on SIV due to the low gas density and mass. Expressions and references of

the considered 12 correlations are summarized in the Appendix, and they are referred to as model [1]-[4] for

Rs, [5]-[8] for Ud, and [9]-[12] for Cfg and n. Note that models [1], [5] and [9] have been employed in previous

Sections 4 and 5; hence, the combined use of these correlations is a baseline case in the sensitivity analysis.

Comparisons of numerical and experimental results in terms of the root-mean-squared response profiles

associated with the fundamental mode SIV are presented in Fig. 23 for horizontal (urms/d) and vertical (vrms/d)

components which are plotted versus the vertical coordinate normalized by the vertical riser span (y*). Initial

transients and amplified mean drifts have been discarded from the steady-state dynamics. The sensitivity of

SIV prediction to Rs, Ud and Blasius coefficients is presented in Figs. 23a-b, c-d, and e-f, respectively. Overall,

a good agreement of numerical-experimental responses in terms of urms/d and vrms/d profiles of the excited riser

fundamental mode is noticed. To identify the output uncertainty, a relative error with the percentage difference

is calculated, e.g., as |urms(numerical)-urms(experimental)|100/urms(experimental). The spatially maximum

errors can then be compared for different correlation models. For the baseline case with models [1], [5] and

[9], the maximum urms/d and vrms/d errors are found to be about 1.8% and 4.0%, respectively.

By varying Rs, Figs. 23a and b entail the maximum errors of about 21.8% for urms/d and 37.2% for vrms/d

associated with model [2] overestimating the SIV response. This is because the predicted Rs 0.594 of model

[2] is the lowest and much lower than Rs 0.926, 0.972 and 0.846 of model [1], [3] and [4] by about 35.8%,

38.9% and 29.8%, respectively. Such an underestimated Rs considerably affects the slug liquid mass, weight,

film hydrodynamics and momentum forces. By varying Ud, Figs. 23c and d entail the maximum errors of about

13.1% for urms/d and 18.1% for vrms/d associated with model [6] overestimating the SIV response like model

[2]. This is because the predicted Ud 0.109 m/s of model [6] is the lowest, being lower than the baseline Ud

0.127 m/s of model [5] by about 14.5%, and directly affecting the slug unit translational velocity. This trend

of increasing urms/d and vrms/d as Ud is decreased is in qualitative agreement with results in Figs. 19a and b.

Finally, by varying the Blasius coefficients Cfg and n for ff and fg, which in turn affect τf and τg, results in Figs.

23e and f reveal a less sensitive output than that in Figs. 23a-d. This observation supports Chatjigeorgiou (2017)

who remarked that a random change in the interfacial friction factor (fi) has a negligible effect on the riser

dynamics. The associated fg (0.01 < fg < 0.1) and ff (0.008< ff < 0.03) values are lower than Rs and Ud by about

one or two orders of magnitudes. The maximum errors of about 8.6% for urms/d and 12.2% for vrms/d are found

to be associated with model [11] due in part to its maximum Cfg=0.316. Overall, results in Fig. 23 suggest that,

24

 

for this catenary riser carrying air-water slug flows, the combined use of models [3], [5] and [9] may be the

most suitable one producing the lowest errors of 0.88% for urms/d and 1.35% for vrms/d. Nevertheless, other

correlations and combinations should be further considered for minimizing the SIV prediction uncertainty.

7. Conclusions

Slug liquid-gas flow-induced planar vibrations of a long inclined curved flexible riser have been numerically

investigated to understand the mechanical effects of slug flow characteristics including the slug unit length,

translational velocity and fluctuation frequencies. A mechanistic slug unit cell concept combined with the

hydrodynamic liquid film model has been applied to approximate the system phase fractions, local velocities

and pressure changes of steady slug liquid-gas flows travelling upwardly through the flexible cylindrical pipe.

Coupled horizontal and vertical vibrations of the bendable riser with pinned-pinned end conditions are subject

to the space-time varying fluid weights, flow-induced centrifugal momenta and Coriolis forces associated with

the steadily time-varying, travelling slug units. Depending on the pipe diameter and inclination, phase fractions,

superficial velocities and internal pressure changes, parametric studies have been performed in cases of varying

slug unit lengths, translational (mixture) velocities and excitation frequencies. Comparisons with experimental

results and sensitivity studies have been carried out to validate the model and identify the prediction uncertainty.

Several observations on slug flow features and resonant SIV mechanisms are summarized as follows.

For a given pipe inclination, a greater liquid mass or density fluctuation occurs in the case of lower gas

superficial velocity. For the catenary riser with variable inclinations, the changes in the liquid-gas flow

velocities predominate over a volumetric mass variation. The liquid mass distribution is primarily

responsible for the pressure drop and the associated pipe wall tension change. These, in turn, modify the

riser axial extensibility, static equilibrium reconfiguration and SIV resonance condition.

An intrinsic excursion in global riser displacements is noticed during the initial transient slug initiation

leading to a slender pipe flexing into a new double-curvature configuration. This short-period excursion

is diminished due to the system damping after which the riser reconfigures itself into a new static

equilibrium owing to the combined effects of slug gravitational weight and flow momentum. The

associated pipe tensions and axial/bending dynamic stiffness are spatially modified.

For steady-state SIV, riser responses comprise the amplified mean and oscillatory components which

are dynamically coupled as a result of the space-time varying gravity (mostly due to the liquid holdup

with a higher fluid density) and the liquid-gas flow (centrifugal and Coriolis) momenta depending on

the combined two-phase masses and velocities.

Depending on the pipe inclination, slug unit length (Lu/d) and translational velocity (Ut), the slug

characteristic frequencies reveal several harmonic components which may trigger a multi-modal

vibration of the flexible riser. It is found that Ut is a primary governing parameter enabling a transition

from being a multi-mode SIV (low Ut) to a single-mode SIV (high Ut). In the low Ut range, the single-

mode SIV response could take place as Lu/d is increasingly varied. These results suggest that SIV

behaviour converges to a standing-wave response depending on the Lu/d-Ut combination.

Planar SIV is found to be governed by an individual effect of Ut and Lu/d, regardless of the associated

characteristic slug excitation frequency. To determine a critical SIV with maximum amplitudes of

25

 

practical importance, Ut should be parametrically varied for a given Lu/d, and vice versa. In some cases,

a certain Lu/d-Ut combination may lead to a multiple resonant SIV involving two different excited

(lower/higher) modes switching intermittently through the simulation time.

Space-time varying bending, axial and total stresses associated with SIV have been examined to verify

the material strength and identify a potential dynamic buckling occurrence. Both mean and oscillatory

stress components have been evaluated, depending on the pipe dynamic extensibility. By comparing the

two parametric (Lu/d-Ut) cases having an identical slug frequency, greater oscillatory bending and axial

stresses occur in the case of higher Ut and Lu/d, whereas greater mean stress components occur in the

lower Ut case. This emphasizes how Ut or Lu/d individually plays a meaningful role in riser SIV.

Although the present numerical study is limited to the steady slug flows, the mechanical slug-flexible riser

models capture the fundamental SIV mechanism of static drift-oscillation coupling and the effects of pipe-slug

flow properties leading to resonances. Advanced models should account for the unsteadiness/randomness in

slug flows and the two-way fluid-riser interactions. Three-dimensional numerical simulations and experiments

should be further carried out to provide meaningful data for improving empirical correlations as well as SIV

predictions of flexible risers and pipelines with arbitrary diameters, inclinations and operational conditions.

Acknowledgment: The authors are grateful for the funding support from the Engineering & Physical Sciences

Research Council (EPSRC) of the UK Research and Innovation through the “MUltiphase Flow-induced Fluid-

flexible structure InteractioN in Subsea applications (MUFFINS)” project grant EP/P033148/1.

Appendix: Empirical correlations used in Section 6 for sensitivity studies.

Correlations References Expressions Rs

[1] Gregory et al. (1978) 1.391 / 1 ( / 8.66)s sR U

[2] Malnes (1982) 0.251 / [83( / ) ]s s l sR U g U

[3] Marcano et al. (1998) 21 / (1.001 0.0179 0.0011 )s s sR U U

[4] Al-Safran et al. (2013) 20.85 0.075 0.057 2.27sR ,

where 0.2 0.89FrN N ,

( / ) / ( )Fr s l l gN U gd ,

2/ [ ( )]s l l l gN U gd

Ud

[5] Bendiksen (1984) 0.54 cos( ) 0.35 sin( )dU gd gd

[6] Hasan and Kabir (1988) 1.2

sin( )[1 cos( )]d dvU U ,

where 0.35 ( ) /dv l g lU gd .

[7] Jeyachandra et al. (2012) cos( ) sin( )d dh dvU U U ,

where 0.46 0.113.70.53 N Eo

dhU e gd ,

1.5 0.5/ ( )lN d g , 2 /lEo d g ,

2 2

( 4 / 3)[2 / ]

(4 / 9)( / 2) (16 / 9)[(2 ) / ( ) ]

dv l

l

U d

gd d

.

26

 

Note that Fr is the generalized Froude number, Eo is the Eotvos number, and Udv (Udh) denotes the bubble drift

velocity in the vertical (horizontal) pipe. Notation of other symbols is listed in the Nomenclature.

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[8] Moreiras et al. (2014) 0.5 0.5Fr [ ( )]d l l gU gd ,

where 1.2391 1.2315Fr Fr cos( ) Fr sin( )h v Q ,

Fr 0.54 / (1.886 0.01443 )h N N ,

2Fr ( 8 / 3) (2 / 9) / ( ) (64 / 9)

( 2 / 3 0.35) / ( ),

v l l g

l l g

N N

3 0.5[ ( ) ]l l l g lN gd ,

hIf Fr Fr , 0v Q , 0.704If Fr Fr , 2.15(Fr Fr ) sin( )(1 sin( )).v h v hQ

Blasius coefficients

[9] Taitel and Dukler (1976) 0.20.046Ref ff ; 0.20.046Reg gf

[10] Spedding and Hand (1997) 0.1390.0262Ref ff ; 0.250.046Reg gf

[11] Kjeldby and Nydal (2013) 0.250.316Ref ff ; 0.250.316Reg gf

[12] Conte et al. (2014) 0.250.079Ref ff ; 0.250.079Reg gf

27

 

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Highlights

• Planar dynamics of inclined curved flexible riser carrying slug liquid-gas flows is investigated.

• Mechanistic slug flow model is applied to identify liquid-gas flow features within a slug unit cell.

• Slug flow-induced vibration is subject to fluctuations of phase fractions, velocities and pressure.

• Effects of slug unit length, translational velocity and characteristic frequency are investigated.

• Transient and steady-state responses highlight fundamental static drift-oscillation coupling aspect.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 1. A planar dynamic model of an inclined curved flexible riser conveying slug liquid-gas flows.  

 

 

   

 

 

 

 

 

 

 

Fig. 2. Properties associated with initial static profile of catenary riser: (a) x-y coordinates, (b) local inclination angles, (c) tensions, (d) curvatures.

 

 

 

   

(a) (b)

(c) (d)

 

 

 

 

 

 

 

 

 

Fig. 3. Influence of Ut (6, 9, 12, 16, 20 m/s) on (a) hf /d, (b) R, (c) uL and (d) uG for a slug unit with Lu/d = 80, uls=2 m/s and β=30°.

 

   

Ut = 20 m/s

Lu /d Lu /d

Lu /d Lu /d

Ut = 6 m/s Ut = 6 m/s

Ut = 6 m/s

Ut = 20 m/s

Ut = 6 m/s

Ut = 20 m/s

Ut = 20 m/s

(a) (b)

(c) (d)

 

 

 

 

 

 

 

 

 

Fig. 4. Influence of Lu/d (63, 80, 120, 140, 208) on (a) hf /d, (b) R, (c) uL and (d) uG for a slug unit with ugs=10.3 m/s, uls=2 m/s and β=30°.

 

 

   

Lu /d Lu /d

Lu /d Lu /d

(a) (b)

(c) (d)

 

 

 

 

 

 

 

 

    

Fig. 5. Influence of β (2°, 15°, 30°, 51°) on (a) hf /d, (b) R, (c) uL and (d) uG for a slug unit with ugs=4.5 m/s, uls=2 m/s and Lu/d=80.

 

 

   

β = 51° β = 2° β = 2° β = 51°

β = 2°

β = 51°

β = 51°

β = 2°

Lu /d Lu /d

Lu /d Lu /d

(a) (b)

(c) (d)

 

 

 

 

 

 

 

Fig. 6. Illustration of space-time varying profiles (a) R, (b) uL, (c) uG and (d) P based on Lu/d = 80, Ut =16 m/s. Dashed lines in (d) denote Pm.

 

 

   

(a) (b)

(c) (d)

 

 

 

 

 

 

 

 

 

 

Fig. 7. Frequency spectra of slug fluctuations based on Lu/d= 80 and varying Ut.  

 

 

 

 

 

 

 

 

 

 

 

(a)

(b)

(c)

(d)

 

 

 

 

 

 

 

 

 

 

Fig. 8. Frequency spectra of slug fluctuations based on Ut = 16 m/s and varying Lu/d.  

 

   

Lu /d = 208

Lu /d = 63

Lu /d = 120

Lu /d = 140

Lu /d = 208

(a)

(b)

(c)

(d)

 

 

 

 

 

 

 

 

 

 

Fig. 9. Frequency spectra of slug fluctuations based on Ut = 9 m/s and Lu/d = 80 with varying β.  

 

 

 

 

 

 

 

 

 

 

 

(a)

(b)

(c)

(d)

 

Fig. 10. Space-time varying (a, c, e) u/d and (b, d, f) v/d during initial transient slug initiation and subsequent steady state in the case of varying (a, b) Ut, (c, d) Lu/d and (e, f) β.

 

   

 

 

 

 

 

 

Fig. 11. Space-time varying (a, c, e, g) u and (b, d, f, h) v inclusive of mean drifts during steady-state SIV for Lu/d= 80 at (a, b) Ut = 6 m/s, (c, d) 9 m/s, (e, f) 16 m/s and (g, h) 20 m/s.

   

(a) (b)

(c) (d)

(e) (f)

(g) (h)

u v

u

u

u

v

v

v

 

 

    

    

    

      

Fig. 12. Spatial profiles of mean drifts in (a, c, e, g) X and (b, d, f, h) Y directions in the case of varying (a, b) Ut, (c, d) Lu/d, (e, f) β and (g, h) fs.

   

Lu /d

(Ut, Lu /d)

(a) (b)

(c) (d)

(e) (f)

(g) (h)

u m/d

u m

/d

u m/d

u m

/d

v m/d

v m

/d

v m/d

v m

/d

Ut

 

 

 

 

 

Fig. 13. Space-time varying (a, c, e, g) u and (b, d, f, h) v exclusive of mean drifts during steady-state SIV for Lu/d= 80 at (a, b) Ut = 6 m/s, (c, d) 9 m/s, (e, f) 16 m/s, (g, h) 20 m/s.

 

u v

u

u

u

v

v

v

(a) (b)

(c) (d)

(e) (f)

(g) (h)

 

 

 

 

 

 

 

 

Fig. 14. Spatial profiles of oscillation frequencies associated with responses in Fig. 13.

 

 

 

 

 

 

(a) (b)

(c) (d)

(e) (f)

(g) (h)

0.20

1.18 1.37

0.20

1.18

1.37

0.29

1.47 1.76

0.29

1.47 1.76

0.52

0.52

0.66

0.66

 Fig.15. Space-time varying (a, c, e, g) u and (b, d, f, h) v exclusive of mean drifts during steady-state SIV for (a, b & e, f) Lu/d = 120, (c, d & g, h) Lu/d = 208 at (a, b & c, d) Ut = 16 m/s and (e, f & g, h)

Ut = 6 m/s.

 

 

 

 

 

 

 

 

 

 

Fig.16. Spatial profiles of oscillation frequencies associated with responses in Fig. 15.

 

 

 

 

(a) (b)

(c) (d)

(e) (f)

(g) (h)

0.35

0.2

0.35

0.2

0.13 1.31

1.31 0.13

0.07

0.07

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig.17. Illustrative spatial modal profiles in (a, c) X and (b, d) Y directions for Lu/d = 120 at Ut = 6 m/s dominated by lower (a, b) and higher (c, d) modes.

 

 

 

 

 

 

 

(a) (b)

(c) (d)

 

 

 

                    

                   

                     

                      Fig. 18. Phase plane trajectories associated with spatially maximum (a, c, e, g) urms and (b, d, f, h) vrms for (a, b) Lu/d = 80 and Ut = 6 m/s; (c, d) Lu/d = 80 and Ut = 16 m/s; (e, f) Lu/d = 208 and Ut = 16 m/s,

(g, h) Lu/d= 120 and Ut = 6 m/s.

u̇ (m

/s)

u̇ (m

/s)

u̇ (m

/s)

u̇ (m

/s)

v̇ (m

/s)

v̇ (m

/s)

v̇ (m

/s)

v̇ (m

/s)

(a) (b)

(c) (d)

(e) (f)

(g) (h)

 

 

 

 

 

 

           

       Fig. 19. Variations of spatially maximum (a, c, e) urms and (b, d, f) vrms in the case of (a, b) varying

Ut for Lu/d = 80, (c, d) varying Lu/d for Ut = 6 m/s and (e, f) varying Lu/d for Ut = 16 m/s.

 

  

 

 

 

Lu /d Lu /d

Lu /d Lu /d

(a) (b)

(c) (d)

(e) (f)

 

 

 

 

   

Fig. 20. Space-time varying (a, c, e, g) σu and (b, d, f, h) σv inclusive of mean components: a, b (c, d) for Lu/d = 80 (120) at Ut = 6 m/s; e, f (g, h) for Lu/d = 80 (208) at Ut = 16 m/s.

 

 

σu

σu

σu

σu σv

σv

σv

σv (a) (b)

(c) (d)

(e) (f)

(g) (h)

 

   

 

 

 

Fig. 21. Space-time varying σa (a, c, e, g) with and (b, d, f, h) exclusive of mean components: a, b (c, d) for Lu/d = 80 (120) at Ut = 6 m/s; e, f (g, h) for Lu/d = 80 (208) at Ut = 16 m/s. 

 

 

(a) (b)

(c) (d)

(e) (f)

(g) (h)

 

 

 

 

 

 

 

 

 

   

 

Fig. 22. Space-time varying σt in (a, c) X and (b, d) Y directions for (a, b) Lu/d = 80 and Ut = 6 m/s and (c, d) Lu/d = 208 at Ut = 16 m/s.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(a) (b)

(c) (d)

 

Fig. 23. Comparisons of numerical (lines) and experimental (squares) results in terms of the root-mean-squared SIV response profiles in (a, c, e) horizontal and (b, d, f) vertical directions, with [1]-[12] denoting empirical correlation models as provided in the Appendix: sensitivity to (a, b) slug

liquid holdup, (c, d) drift velocity and (e, f) Blasius coefficients. 

 

(a) (b)

(c) (d)

(e) (f)

Table 1 Review of some relevant studies on SIV modelling and analysis with key observations

References Slug Flow Model Pipe Geometry Multiphase Flow Properties

External Excitation

Observed SIV features Type

L (m)

D (d) mm

uls (m/s)

ugs (m/s)

ρl (kg/m3)

ρg (kg/m3)

Ut (m/s)

Lu/d

Hara (1977)

Pulse train H 2.2 30

(23.8) 0.11-0.4

0.19-2.25

998 1.2 2.6-11 11.6-46.2

- SIV response due to a primary resonance.

SIV response due to a parametric resonance.

Patel & Seyed (1989)

Sinusoidal S 3.071 9.2 (6)

- - 998 1.2 0.45 411.7 - Slug flow modifies pressure and pipe tension.

SIV resonance amplifies pipe response/stress.

Chatigeorgiou (2017)

Steady state (Taitel & Barnea 1990)

C 2025 429

(385) 0.4 2 790 0.675 - 20

Top imposed

excitation

Slug flow amplifies dynamic tension/curvature.

Slug resonance affects structural dynamics.

Bordalo & Morooka (2018)

Pulse train

C 3100 212

(165)

- - 1000 0.01 2.4–4 606.1

-

Large-amplitude SIV is found.

SIV is dominated by weight force when pipe curvature and flow speed is small.

LW 4560 - - 1000 0.01 2.1–2.8 842.4

Liu & Wang (2018)

Steady state (Cook & Behnia 2000)

H 15, 20,

30

40 (30) 30 (25.4) 25 (20)

0.8- 1.2

2-20 998 1.2 - - - Increased pipe frequency with gas velocity.

Divergent instability at high gas velocity.

Ortega et al. (2018)

Transient slug tracking (Nydal & Banerjee, 1996)

C 445 275.5

(200.8) 2.1 38.7 998 1.2 - - Waves

Slug flow leads to static or mean drift.

Slug flow induces irregular dynamics, coupling with wave-induced motions.

Wang et al. (2018)

Steady-state model Zhang et al. (2003)

H 3.81 63

(51.4) 0.29 1.39 1000 1.2 1.5-4.5 - -

Amplified SIV exists at higher liquid velocity and slug translational velocity.

SIV amplitudes increase with slug unit lengths.

Safrendyo & Srinil (2018)

Pulse train C 2025 429

(385) - - 790 0.675 2-10 20-50

Vortex shedding

Steady/random SIV amplify pipe responses.

Coupling of internal/external flows is complex.

Cabrera-Miranda & Paik (2019)

Pulse train LW 2641 458.4

(254.1) - - 800 0 1-32

5197, 20787, 41575

- Large-amplitude SIV occurs near riser bottom.

SIV fatigue damage may be crucial.

Vasquez & Avila (2019)

Sinusoidal C 450 400

(360) 0.98-3.94

0.39 998 100-400 - 6.7-29.2

Currents

Higher liquid mass flow rate induces greater top tension amplitudes.

Coriolis force has negligible effects at small slug flow velocities.

Dong & Shiri (2019)

Pulse train C 2333 324

(283) - - 600 100 10, 25 106,

176.7 Waves

Combined effects of SIV and waves are meaningful and greater than individual cases.

Slug flow-induced inertia and momentum effects are important for riser analysis.

H, S, C, LW denote horizontal pipe, S-shaped, catenary and lazy-wave riser, respectively. Notation of variables and symbols can be found in Nomenclature.

Table 2 Validation of the present model and numerical simulation through comparison of fundamental natural frequencies of production riser with previously published results

U (m/s)

fn1 (rad/s) (accounting for bending and tension effects)

fn1 (rad/s) (accounting for only tension effect)

(a) Monprapussorn

et al. (2007)

(b) This study 100a b

a−

× (c)

Monprapussorn et al. (2007)

(d) This study 100c d

c−

×

0 0.3001 0.2992 0.30 0.2891 0.2891 0.00 5 0.2994 0.2985 0.30 0.2881 0.2882 0.03 10 0.2972 0.2962 0.34 0.2853 0.2857 0.14 15 0.2934 0.2923 0.37 0.2804 0.2813 0.32 20 0.2880 0.2874 0.21 0.2731 0.2736 0.18 25 0.2809 0.2799 0.36 0.2627 0.2652 0.95 30 0.2717 0.2711 0.22 0.2478 0.2517 1.57 35 0.2603 0.2592 0.42 0.2224 0.2279 2.47 40 0.2461 0.2446 0.61 Unstable* Unstable* 45 0.2282 0.2255 1.18 50 0.2052 0.2021 1.51 55 0.1738 0.1701 2.13 60 0.1250 0.1215 2.80 65 Unstable** Unstable**

Critical U ≈ 38* and 64** m/s.

Table 3 Specified slug flow parameters for SIV case studies with uls = 2 m/s

β (o) Lu/d ugs (m/s) Ut (m/s) fs (Hz)

30 63, 80, 120, 140, 208 2.0 6 0.25, 0.20*, 0.13, 0.11, 0.08 2, 15, 30, 51 80 4.5 9 0.29 30 80 7.0 12 0.39 30 63, 80, 120, 140, 208 10.3 16 0.66**, 0.52, 0.35, 0.30, 0.20* 30 80 14.0 20 0.66**

* and ** denote cases with similar fs.

Table 4 Slug flow profile properties in cases of varying (a) Ut, (b) Lu/d and (c) β

(a) β = 30° and Lu/d = 80

Ut (m/s) Lf (m) Ls (m) Lu (m) Lf/Ls Lf/Lu αvol 6 8.96 21.84 30.8 0.41 0.29 0.393 9 14.74 16.06 30.8 0.92 0.48 0.579

12 17.8 13 30.8 1.37 0.58 0.676 16 20.2 10.6 30.8 1.90 0.66 0.749 20 21.7 9.1 30.8 2.38 0.70 0.797

(b) β = 30° and Ut = 16 m/s

Lu/d Lf (m) Ls (m) Lu (m) Lf/Ls Lf/Lu αvol 63 16.73 7.37 24.1 2.27 0.69 0.749 80 20.2 10.6 30.8 1.91 0.66 0.749

120 27.79 18.41 46.2 1.51 0.60 0.749 140 31.45 22.45 53.9 1.4 0.58 0.749 208 43.49 36.56 80.05 1.18 0.54 0.749

(c) Ut = 9 m/s and Lu/d = 80

β (o) Lf (m) Ls (m) Lu (m) Lf/Ls Lf/Lu αvol 2 22.61 8.19 30.8 2.76 0.73 0.598

15 16.86 13.94 30.8 1.21 0.55 0.591 30 14.74 16.06 30.8 0.92 0.48 0.579 51 12.84 17.96 30.8 0.71 0.42 0.560

Table 5 Pressure drop components, bottom pressure and resulting tension variations in cases of varying (a) Ut, (b) Lu/d and (c) β

(a) β = 30° and Lu/d = 80

Ut (m/s) Ns ∆Pf

(kPa) ∆Ps

(kPa) ∆Pg

(kPa) ∆Pu

(kPa) Pb

(kPa) Pdp

(%) Tb

(kN) Tdp

(%) 6 65 0.08 4.55 72.51 77.14 5174.5 98.04 802.19 56.77 9 65 0.39 7.68 50.30 58.37 3936.8 97.42 888.64 52.11

12 65 1.07 10.64 38.82 50.53 3420.2 97.04 924.72 50.16 16 65 2.62 14.19 30.14 46.95 3184.2 96.82 941.21 49.27 20 65 5.09 17.71 24.32 47.12 3212.7 96.85 939.21 49.38

(b) β = 30° and Ut = 16 m/s

Lu/d Ns ∆Pf

(kPa) ∆Ps

(kPa) ∆Pg

(kPa) ∆Pu

(kPa) Pb

(kPa) Pdp

(%) Tb

(kN) Tdp

(%) 63 84 2.60 9.87 23.53 36.00 3126.6 96.76 945.23 49.06 80 65 2.62 14.19 30.11 46.92 3184.2 96.82 941.21 49.27

120 43 2.62 24.65 45.12 72.39 3274.3 96.91 934.91 49.61 140 37 2.63 30.10 52.62 85.35 3306.9 96.94 932.64 49.74 208 25 2.65 48.94 78.24 129.83 3384.3 97.01 927.23 50.03

(c) Ut = 9 m/s and Lu/d = 80

β (o) Ns ∆Pf

(kPa) ∆Ps

(kPa) ∆Pg

(kPa) ∆Pu

(kPa) Pb

(kPa) Pdp

(%) Tb

(kN) Tdp

(%) 2 65 2.31 4.17 3.36 9.84 748.4 86.46 1111.3 40.11

15 65 0.71 6.98 25.29 32.98 2269.8 95.54 1005.1 45.83 30 65 0.39 7.68 50.27 58.34 3936.8 97.43 888.6 52.11 51 65 0.25 7.81 81.72 89.78 6004.0 98.31 744.2 59.89

Table 6 Maximum amplified mean drifts during transient and steady states in cases of varying (a) Ut, (b) Lu/d and (c) β

(a) β = 30° and Lu/d = 80

Ut (m/s)

Transient Steady Max um/d Max vm/d Max um/d Max vm/d

6 47.31 56.16 0.68 1.62 9 33.11 38.80 0.50 1.13

12 25.46 30.76 0.37 0.90 16 20.57 24.01 0.26 0.68 20 16.80 21.69 0.24 0.55

(b) β = 30° and Ut = 16 m/s

Lu/d Transient Steady Max um/d Max vm/d Max um/d Max vm/d

63 20.55 23.97 0.32 0.66 80 20.57 24.01 0.26 0.68

120 20.58 24.02 0.29 0.68 140 20.58 24.01 0.29 0.68 208 20.62 24.04 0.29 0.68

(c) Ut = 9 m/s and Lu/d = 80

β (o)

Transient Steady Max um/d Max vm/d Max um/d Max vm/d

2 30.97 36.15 0.48 1.08 15 31.83 37.19 0.48 1.10 30 33.11 38.78 0.50 1.13 51 35.08 41.16 0.50 1.19

Table 7 Modified mean tensions associated with riser equilibrium reconfigurations during SIV

Ut

(m/s) Tm (kN) Lu/d Tm (kN) β

(o) Tm (kN)

Top Bottom Top Bottom Top Bottom 6 3310 1712 63 2460 1324 2 2821 1715 9 2865 1520 80 2460 1320 15 2837 1619 12 2635 1412 120 2460 1314 30 2865 1520 16 2461 1320 140 2460 1312 51 2911 1040 20 2345 1246 208 2460 1306

Table 8 Maximum/minimum bending, axial and total stresses in MPa corresponding to (a) Fig. 20, (b) Fig. 21 and (c) Fig. 22, respectively.

(a)

Ut

(m/s) Lu/d σu σv Max Min Max Min

6 80 3.34 -2.89 4.92 -5.68 6 120 31.68 -30.57 49.51 -49.15

16 80 5.42 -6.00 8.64 -8.59 16 208 4.51 -4.38 10.70 -11.55

(b)

Ut

(m/s) Lu/d σa

(mean & oscillation parts) σa

(oscillation part) Max Min Max Min

6 80 52.08 32.01 0.42 -0.43 6 120 53.62 30.75 1.96 -1.96

16 80 22.06 12.86 0.64 -0.63 16 208 21.94 12.76 0.73 -0.74

(c)

Ut

(m/s) Lu/d σt

(u direction) σt

(v direction) Max Min Max Min

6 80 119.51 60.51 119.20 55.48 16 208 89.58 45.31 89.22 34.68