1 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: [email protected]; Tel. +44 191 208 6499; Fax +44 191 208 5491
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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) 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.
Zhang, H.-Q., Wang, Q., Sarica, C., Brill, J.P., 2003. A unified mechanistic model for slug liquid holdup and
transition between slug and dispersed bubble flows. International Journal of Multiphase Flow 29, 97-107.
Zhu, H., Gao, Y., Zhao, H., 2019a. Coupling vibration response of a curved flexible riser under the combination
of internal slug flow and external shear current. Journal of Fluids and Structures 91, 102724.
Zhu, H., Gao, Y., Zhao, H., 2019b. Experimental investigation of slug flow-induced vibration of a flexible
riser. Ocean Engineering 189, 106370.
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
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)