Hydroelastic response of marine risers subjected to internal slug-flow

Applied Ocean Research 62 (2017) 1–17

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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor

Hydroelastic response of marine risers subjected to internal slug-flow Ioannis K. Chatjigeorgiou a,b a b

School of Naval Architecture and Marine Engineering, National Technical University of Athens, Greece School of Mathematics, University of East Anglia, Norwich, UK

a r t i c l e

i n f o

Article history: Received 19 May 2016 Received in revised form 9 October 2016 Accepted 25 November 2016 Keywords: Pipelines Steel marine risers Slug-flow Linear dynamics Frequency domain

a b s t r a c t It is the purpose of this study to investigate the dynamic behaviour of catenary pipelines for marine applications, assuming the combined effect of harmonic motions imposed at the top, and the internal slug-flow. The analysis is based on the assumption of a steady slug-flow inside the pipe that results in a relatively simplified model for the formulation of the internal flow. The slug-flow model is described using several assumptions and empirical correlations which attempt to reveal the ill-understood and concealed properties of the slug-flow. The pipeline dynamics are investigated in the two dimensional space omitting the out-of-plane vibrations. The system of differential equations is generic and accounts for the steady effect of the internal liquid as is conveyed through the structure. The two models, those of the internal slug-flow and the pipeline’s dynamical model, are properly combined through the internal flow terms of the dynamic equilibrium system. The solution provided is achieved using a frequency domain technique which is applied to the linearized governing set. The effect of the slug-flow is assessed through comparative computations with and without internal flow effects. The conclusions are drawn having the structure excited under axial and normal motions paying particular attention to the variation of the dynamic components along the complete length of the pipeline. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Marine risers are long pipelines which are used to convey liquids from the source located at the sea bottom, up to the floating host facility. Due to the hostility of the environment, in which they operate, they should be considered continuously moving. The dynamic behaviour is a major concern for their structural integrity and it is influenced by i) the motions induced at their top due to the displacements of the floating structure, which is continuously subjected to external excitations (wind, waves and currents), ii) wave and current loading on the immersed pipelines, iii) strongly nonlinear phenomena such as high frequency oscillations induced by vortex shedding effects or compression loading at the touch down zone and iv) the internal flow effects. For long pipelines for marine applications, such as risers, the internal flow is typically formulated using the so-called plug-flow model, which considers that the velocity profile of the conveyed liquid is constant throughout the pipe [1,2]. When the subject of investigation is the coupled dynamic behaviour of the structure subjected to imposed motions and the effects of the internal flow, the plug-flow concept significantly simplifies the combined formu-

E-mail address: [email protected] http://dx.doi.org/10.1016/j.apor.2016.11.008 0141-1187/© 2016 Elsevier Ltd. All rights reserved.

lation as the internal flow terms are directly incorporated into the structural dynamic model [3,4]. Aside from the very simplified plug-flow model, there have been studies that complicate the problem assuming more sophisticated formulations to describe the internal stream, such as assuming a flow that relies on the potential theory [4–8] or a fully turbulent behaviour [9]. Chatjigeorgiou [4] showed that the effects of a potential flow model that assumes inviscid, incompressible fluid and irrotational flow are literarily similar to those due to the plugflow approximation. The same holds for the turbulent model that assumes fully turbulent behaviour inside the pipe as well [9]. Hence, it could be safely claimed that the plug-flow approach is a good and robust engineering approximation, which, in addition, allows fast computations. Nevertheless, it should be noted that the aforementioned remarks assume a single phase flow, that of the liquid, that occupies the whole internal space of the pipeline throughout its length. Under this condition, the effect of the internal flow on typical riser configurations is limited as it does not change significantly the particulars of the response when the structure is subjected to top imposed excitations. In fact, the major source that amplifies the internal loading components comes from the forced motions. Chatjigeorgiou [4] reported that the dynamic behaviour in the plane of reference (in-plane vibrations) remain literarily unchanged, while only the out-of-plane behaviour is practically

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I.K. Chatjigeorgiou / Applied Ocean Research 62 (2017) 1–17

Fig. 1. The geometry of the slug-flow.

affected due to the Coriolis effects that introduce an additional damping component into the system. In contrast, it has been reported that a two-phase internal flow model could lead to significant variations as regards the dynamics of marine pipelines. Analogous flow patterns can generate rapid changes of the mass distribution and pressure fluctuations along a riser, which may lead to vibrations. These vibrations combined with vibrations generated by other environmental loads will produce time varying stresses and consequently lead to accumulation of fatigue damage, excessive bending or even local buckling. Moreover, vibrations induced due to the internal flow could start resonating with the vibrations of the structural system itself. Slugflow models are now being incorporated into the features of respected commercial software dealing with the dynamics of risers and pipelines [10]. Different types of two-phase flow in pipes are designated as “plug-flow with bubbles”, “slug-flow”, “churn-flow”, “annular flow” and “wispy annular flow”. The most important type of loading exerted on marine riser type structures due to internal two-phase flow arises from the slug-flow model [11,12]. In the two-phase slugflow, the gas phase exist as elongated bubbles separated by the liquid phase, the slugs. The slugs occupy the whole cross sectional area of the pipe along the slug zone. Typically, the slugs contain bubbles as well in contrast with the liquid film, i.e. the liquid zone below the elongated gas bubble (see Fig. 1). As the liquid is transported through the pipe, the slugs are formed continuously and in the unsteady condition may change location and volume. The two-phase slug-flow is typically correlated with applications involving heat and mass transfer in pipes, although the former is not an issue of concern in the present study. The majority of the existing studies concern fixed and straight pipes which may be inclined. One of the first studies on the subject that assumed a catenary configuration of the pipeline and tried to provide answers to its dynamic behaviour subjected to both external excitations and the internal slug-flow was that due to Patel and Seyed [13]. This study however, relied on an extremely simplified model for the internal slug-flow. The importance of the slug-flow on the dynamics of riser type structures for marine applications has been identified only recently (e.g. [11,12]) and there is an obvious lack on information on the subject. Regarding the way slug-flow is approximated, irrespectively whether the pipe is excited externally or not, one may choose to assume the steady [14] or the unsteady formulation [15–18]. The latter is formulated via the continuity and momentum equations, separately for the liquid and the gas phases, which form a strongly nonlinear system that can be treated only in the time domain. An alternative and adequately accurate method to formulate the associated two-phase flow is to assume a steady slug-flow.

For a thorough analysis on the particulars of the steady slug-flow the reader is referred to the seminal paper of Taitel and Barnea [14]. Regardless the model that is being considered, the problem of the slug-flow inside pipes is very complicated. Many issues are uncertain and therefore both concepts (steady and unsteady) rely practically on the same approximations which are frequently represented by empirical rules. The latter are based on the postprocessing of measurements and observations. Relevant examples of concealed topics include, but not limited to, whether the flow is laminar or fully turbulent, the computation of the friction forces between the liquid and the wall, the gas and the wall and on the interface (the free-surface) between the liquid and the gas, the surface tension on the interface, the containment of bubbles in the slug zone (which accordingly defines the liquid hold-up in the zone) and the velocity of the bubbles, to mention a few. The aforementioned challenges must be properly addressed irrespectively whether the pipe is fixed or moving. Clearly, the situation is much more difficult when the latter condition occurs, such in marine risers, which perform continuous oscillations due to the motions imposed at their top by the host facility. In such a case, the structural dynamical system of the pipeline, which typically has a curved (catenary) configuration, must be combined with the slug-flow model, which constitutes itself a different dynamical system governed by a discrete set of differential equations. Marine risers are subjected to a variety of impacts in their operational environment. Of paramount importance are the forced oscillations due to the displacements of the host facility. The axial components of the excitations in particular may lead to local buckling effects and parametric displacements associated with nonlinear resonances [19]. In this context, the task of the present study is to provide a solution of the combined dynamic problem of a catenary pipeline that conveys a slug-flow and in addition is excited by external factors which are represented by imposed motions applied at its top terminal point. The main goal is to reveal the details of the effect of the internal slug-flow on the dynamics of the pipeline and how it modifies its global dynamic behaviour. The structural dynamics problem is properly linearized in order to allow processing in the frequency domain, assuming harmonic imposed motions and alike responses. The linear dynamic equilibrium system of the pipeline is combined with a steady slug-flow model. To this end, the most complete formulation, that of Taitel and Barnea [14], was employed. In fact, most of the information and the assumptions used in the present were taken from reference [14]. The two models, those of the dynamics of the pipe and the slug-flow, are properly combined through the relevant terms of the inner flow components which are incorporated into the pipe’s dynamical system. The global approach accounts for the variation of the liquid mass and its velocity inside the pipe, both in the liquid film and in the slug zones, as well as for variable inclination of the catenary. The paper is organized as follows: Section 2 describes the steady slug-flow model that has been adopted, while the structural dynamic system of the catenary pipeline is outlined in Section 3. Information on how the two models are combined are given in Section 4. Relevant computations are shown in Section 5 followed by Discussion.

2. The model of the steady slug-flow The geometry of the steady slug-flow is depicted in Fig. 1. The two phases of the flow, the liquid and the gas are propagating inside the curved pipe with constant inner diameter D along the generalized Lagrangian coordinate s that takes values along the curved configuration at static equilibrium. The densities of the two phases are denoted by G and L for the gas and the liquid respectively. In

I.K. Chatjigeorgiou / Applied Ocean Research 62 (2017) 1–17

the general case the flow is considered viscous and accordingly the kinematic viscosities of the two media are G and L . As the two fluids are conveyed along the pipe, slug units are formed. Each slug unit has length lu . The length of the pipe, L, is discretized by N nodes creating N − 1 equal length segments s. The spatial discretization requires that the segments are rectilinear making an angle ˇ with respect to the horizontal. The length of each segment is purposely taken equal with the length of the slug unit, i.e. s = AC = lu (see Fig. 1). According to the employed assumptions, the slug unit is rectilinear as well. The Lagrangian coordinate s coincides with the local coordinate x in each slug unit. The length of the slug unit is subdivided in the length of the liquid film lf and the length of the slug ls . The slug unit propagates with steady translational velocity ut . The corresponding velocities of the liquid film and the gas in the film zone are designated as uf and uG respectively. It is assumed that the slug contains dispersed bubbles which move with velocity ub , whilst the velocity of the liquid within the slug is uL . Accordingly, the liquid hold-up and the void fraction in the slug, will be denoted by Rs and ˛s respectively. In the ideal case where the liquid is not aerated Rs = 1. The relative velocities between the slug unit and the liquid and the gas are vf = ut − uf and vG = ut − uG . The shape of the film is denoted by hf and is a function of the local coordinate z (see Fig. 1). Note that z denotes in the opposite direction of x (or s). As the two phases are propagating and the slug units are formed, each film zone is picked up by the following slug. The height of the film and the velocity at picked-up are denoted by hfe and ufe and are important parameters for the heat and mass transfer in slug-flow. Nevertheless, these will not be investigated in detail in the present. Finally it is noted that the flow induces friction forces between the liquid and the wall, f , the gas and the wall, G , and on the interface between the liquid and the gas i . The primary unknowns of the problem of the steady slug-flow are the film length, lf , the film velocity, uf , and the liquid hold-up on the film, Rf . Note that it has been assumed that the liquid in the film is not aerated. The calculation of lf , uf and Rf is dictated by the derivation of the film shape hf (z). That shape is a very complex structure and in fact it is a three-dimensional turbulent flow problem with a free-surface. Here we follow the assumption made by Taitel and Barnea [14], namely to use the single dimensional approach of the channel flow theory. In order to find solutions for lf , uf (z) and Rf (z), the momentum balances in the film zone will be considered. With reference to Fig. 1 these are expressed by the following ordinary differential equations, for the liquid and the gas respectively [14]:

∂ vf ∂hf ∂P f Sf S L vf − i i + L g sin ˇ − L g cos ˇ =− + Af Af ∂z ∂z ∂z

(1)

∂hf ∂vG ∂P G SG i Si G vG + + G g sin ˇ − G g cos ˇ =− + AG AG ∂z ∂z ∂z

(2)

In Eqs. (1) and (2) P is the pressure gradient, g is the gravitational acceleration, Af and AG are the regions occupied by the liquid in the film and the gas respectively, Sf and SG are the corresponding subdivisions in the pipe’s section and final, Si is the section of the interface (see Fig. 2). Eqs. (1) and (2) are written in terms of the relative velocities vf and vG . Nevertheless, the friction forces f , G and i are given in terms of the actual velocities uf and uG according to

3

Fig. 2. Liquid film and gas regions in the pipe’s cross section.

In the friction forces given by Eqs. (3)–(5), ff , fG and fi denote the friction coefficients between the liquid and the wall, the gas and the wall and on the interface between the gas and the liquid, respectively. Note that uf and uG are considered positive along the x axis (or s) as indicated in Fig. 1. Eqs. (1) and (2) can be combined to eliminate the pressure gradient, i.e. L vf

∂vf f Sf ∂vG G SG − − G vG = AG Af ∂z ∂z



−i Si

1 1 + Af AG



+ (L − G ) g sin ␤ − (L − G ) g cos ␤

dhf dz (6)

Eq. (6) provides an expression for the shape of the film hf (z). However, before the derivation of hf (z) we need to remove the dependence of the relative velocities of the liquid film and the gas from the actual velocities of these media which are also unknowns. To this end we use the mass balance, initially for the liquid, by considering the input liquid flow rate which is denoted by WL . The mass balance principle will read

⎛ WL =

1 tu

⎜ ⎝L ARs ls +

⎞

lf

⎟

L ARf ds⎠ − X

(7)

0

where A is the cross sectional area and tu = lu /ut is the time that is needed for a slug unit to pass through a fixed point. At that time, part of the liquid moves backwards and is captured by the following slug unit. The amount of the liquid part is X and is determined by





X = (ut − uL ) L ARs = ut − uf L ARf

(8)

Accordingly





f =

1 L ff |uf |uf 2

(3)

vf = ut − uf = (ut − uL ) Rs /Rf

G =

1 G fG |uG |uG 2

(4)

A similar relation holds for the relative velocity of the gas, namely

(5)

vG = (ut − uG ) = (ut − ub ) ˛s /˛f

i =

  1 G fi |uG − uf | uG − uf 2

(9)

(10)

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I.K. Chatjigeorgiou / Applied Ocean Research 62 (2017) 1–17

Noting that Rf and ˛f are functions of hf , we substitute Eqs. (9) and (10) into Eq. (6) to obtain dhf dz

=

f Sf Af

G SG AG

−

− i Si

(L − G ) g cos ˇ − L vf

1 Af

+

1 AG



(ut −uL ) dRf dhf R2

.

It is noted that a small change in Reynolds number during the transition from laminar to turbulent flow causes the friction factor(s) in Eq. (17) to change dramatically. This problem can be tackled by using a maximum approach to smooth the friction factor. In a relevant case fk is calculated by

(11)

fk = max

+ (L − G ) g sin ˇ − G vG

f

(ut −ub )(1−Rs ) dRf 2 dhf 1−R

(

f

)

dRf dhf

=

4 D

 1−

hf

2

D

−1

(12)

The quantities Af , AG , Sf , SG , Si are literarily one-dimensional functions of hf and they are interrelated via the angle q (see Fig. 2). The structure of the flow in the cross section implies q = 2 cos

 Af =

1−



hf

(13)

D/2

2

D/2 2

 (q − sin q) , AG =

D2 4

 − Af

 − cos−1

2

hf D

(15)

 −1

+

2

hf D



−1

1−

2

hf D

2



−1

(16)

and ˛f = 1 − Rf . 2.2. The friction coefficients

fk = Ck Rekn , Rek =

Dhk uf L



, k = f, G

(17)

fk = 0.001375





1 + 2 × 10

ε  Dhk

106 + Rek

1/3 

j

G

5



−1





hf D

, jG ≥ 5m/s

(21)

Taitel and Barnea [14] suggested a simpler approach that is based on a single only value. In fact they claimed that the wavy structure of the interface cannot be accurately predicted and thus one must make some crude correlations and assumptions. In that respect and in accord with the studies of Cohen and Hanratty [25] and Shoham and Taitel [26] for inclined pipes, they suggested that the constant value fi = 0.014 can be used provided that the inner wavy flow is stratified. In fact this is the assumption taken in the present.

The value of Rs and accordingly the void fraction ˛s = 1 − Rs can be obtained experimentally or through a physical model. Here the correlation of Gregory et al. [27[]] is used, i.e. Rs =

and Re denotes the Reynolds number associated with the hydraulic diameters Dhk which for the liquid and the gas are obtained by Dhf = 4Af /Sf and DhG = 4AG / (SG + Si ) [20] whilst Cf = CG . For laminar flow the friction coefficients are obtained by the Hagen-Poiseulle equation which is obtained letting Ck = 16, n = −1. For turbulent flow, Ck = 0.046 and n = −0.2 [20]. For rough pipes the friction coefficients are modified accordingly [21,22]. In this case one can invoke Moody’s equation



4

2.3. Liquid hold-up in the liquid slug zone

These are obtained through Eqs. (3)-(5). The friction coefficients ff , fG can be calculated using the Blasius correlation





for rough pipes. The critical Reynolds number in Eq. (19) is ReC ∼ = 1500. The proper determination of the friction coefficient of the interface fi is more difficult. A coarse assumption that could be made is fi = fG which however assumes that the gas velocity is small [22]. Nevertheless, the truth is that the friction coefficient on the interface depends on the particulars of the interface, i.e. whether is smooth or wavy. The most proper assumption is to consider a wavy surface which subsequently determines the value of the average friction coefficient. If the wave interface is stratified the friction factor between the gas face and the wall remains the same. Nevertheless, the friction factor on the interface depends on the wavelength of the waves generated on the interface. According to Issa et al. [23], for gas velocities higher than 5 m/s equality of the gas phase with the interface is an invalid approach. In this case it is more appropriate to employ the formula suggested by Andritsos and Hanratty [24] which reads fi = fG 1 + 15

For a stratified flow, the liquid hold-up on the film is given by [14]



fk = max

(14)

qD D , SG = D − Sf , Si = Sf = 2 − 2 cos q 2 2

1 Rf = 



16 , 0.001375 Rek

(20)

2.1. The cross section



(19)



2

Eq. (11) is an ordinary differential equation, which is literarily expressed by a single unknown, that is the z-varying shape of the film. It can be efficiently solved using well-known methods of numerical analysis and herein is processed by the Runge-Kutta method. The procedure to calculate the function hf (z) via Eq. (11) comprises several steps, whilst in addition several assumptions have to be made regarding the complicated slug-flow physical problem.

−1



16 0.046 , Rek Re0.2 k

for smooth pipes, and by

where in case of a stratified flow





ε  4

1 + 2 × 10

Dhk

106 + Rek

1/3 

where ε/Dhk denotes the relative roughness of the surface.

1+



1



1.39 us 8.66

(22)

where us is the mixture velocity that is composed by the superficial velocities of the liquid and the gas uLS and uGS respectively as us = uLS + uGS . The validity of Eq. (22) has been verified by Ferschneider [28] and was accordingly adopted by other authors as well (e.g. [29]). Here the superficial velocities are considered as known parameters and they are included in the input data. 2.4. The slug unit steady translational velocity

(18) The value for ut is obtained by the following relation [30] ut = Cus + ud

(23)

I.K. Chatjigeorgiou / Applied Ocean Research 62 (2017) 1–17

where ud is the drift velocity, namely, the velocity of the propagation of a large bubble in stagnant liquid and the factor C is related to the contribution of the mixture velocity. According to Bendiksen [31] and Zukoski [32], the drift velocity, ud , depends on the angle of

inclination ˇ and the surface tension = 2 . For small (L −G )(D/2) g  ∼ 0.001, it can be assumed that ud / gD is equal surface tension, = to 0.35 and 0.54 for a completely vertical and completely horizontal pipe, respectively. These values are very close to the potential flow theory. For intermediate inclinations one could employ the widely used formula suggested by Bendiksen [31]



ud = 0.54



gD cos ˇ + 0.35

(24)

On the other side, the factor C [31] depends on the surface tension and the Reynolds number. For nonzero surface tension ( ∼ = 0.042) Bendiksen [31] showed that for Reynolds numbers up to 105 , is C ∼ = 1.2. For very small Reynolds numbers, the value of C increases rapidly and its limit for very small surface tension and laminar flow is about C = 2. These values are still being considered as good engineering approximations [33,34]. Indeed, these are the conditions which will be considered in the present study.

2.5. The bubble velocity In a similar manner, ub is given by ub = Bus + u0

(25)

where B is the distribution parameter and u0 is the drift velocity for stagnant liquid. Wallis [35] suggested that the bubble drift velocity is equal to the bubble rise velocity u∞ which according to Harmathy [36] is given by

 u∞ = 1.54

g (L − G )

1/4



 





B ˇ = B 0o + B 90o − B 0o

sin2 ˇ

(27)

with B (0o ) = 1 and B 90o = 1.2. In the study of Mayor et al. [37] a formula similar to Eq. (23) was adopted for the bubble which, the bubble rise velocity was assumed equal to velocity, in u∞ = 0.35 gD. In the present study we will assume a constant value B = 1.2 throughout the length of the pipe, which will be considered independent from the inclination.

2.6. The velocity of the liquid within the slug First, we recalculate the input liquid flow rate WL [see Eq. (7)] using the alternative method of the liquid mass balance. Hence

⎛ WL =

1 tu

⎜ ⎝uL ARs L ts +

lf uf ARf L ds

(29)

0

Combining Eqs. (8) and (29) yields lf

ut = uL Rs + ut (1 − Rs ) − lu lu

lf ˛f ds

(30)

0

A simple mode of the mass balance of the mixture, liquidgas, assumes that for constant densities, the volumetric flow rate through any cross section is constant. Applying this balance on a cross section in the liquid slug yields us = uLS + uGS = uL Rs + ub ˛s

(31)

Eq. (31) easily yields the velocity of the liquid in the slug, uL , in terms of the superficial velocities, the bubble velocity and the liquid hold-up in the slug. Finally, the relative velocities of the gas and the liquid in the film, vG and vf respectively, are immediately obtained through Eqs. (9) and (10). The numerical integration of Eq. (11) along the liquid film length lf , requires an initial value for hf , hfi for z = 0 (see point A and the equivalent point C in Fig. 1). The point C at z = 0, corresponds to the front of the slug where the liquid hold-up, Rs , can be calculated through Eq. (22). Note that in Eq. (22) the mixture velocity, us , is considered known as it is given simply by the sum of the superficial velocities uLS and uGS . Accordingly, the initial value hfi is obtained through Eq. (16) after substituting Rf by Rs . Clearly, this equation is nonlinear in terms of hfi and should be treated accordingly. 3. Structural dynamics of the curved pipeline

L2

   

ls 1 WL = uL ARs L + lu lu

(26)

Regarding the distribution parameter B, Wallis [35] pointed out that for vertical dispersed flow, lies between 1.0 and 1.5 with most probable value of about 1.2. For horizontal flows, Taitel and Barnea [14] recommended the value of B = 1, “due to the lack of evidence” as they claimed. For intermediate inclinations ˇ, van Hout et al. [35], following the study of Bendiksen [31], suggested the following correction formula for factor B

 

Again, tu , ts , tf are the times for the passage of the slug unit, the liquid slug and the film zone, respectively. Given the fact that tu = lu /ut , ts = ls /ut and tf = lf /ut , Eq. (28) is cast to

uLS gD sin ˇ

5

tf

⎞ ⎟

uf ARf L dt ⎠ 0

(28)

For the dynamics of the pipeline conveying an inner flow, we will exploit the dynamic equilibrium system formulated by Chatjigeorgiou [4]. In this study the author presented the complete three-dimensional coupled nonlinear system of dynamic equilibrium for catenary pipelines taking into account the effect of the inner flow. Briefly, the line dynamics approximation is taken for a line distorted in the three-dimensional space (Fig. 3) [38].  ˆ bˆ , A local coordinate system is defined with unit vectors tˆ, n, in tangential, normal and bi-normal directions respectively. The   ˆ bˆ runs along the complete length of the curve and system tˆ, n, its axis change orientation continuously. Assuming that at time t the origin of the local system coincides with the point p (t) on the curve, then at time t + t has moved to point p (t + t) . Let us further assume an element of the pipeline, the differential length of which in the unstretched condition is ds, where s denotes unstretched Lagragian coordinate that takes values along the pipeline (Fig. 4). Assuming a linear stress-strain relation, the differential element is distorted axially and is elongated to (1 + e) ds,   ˆ bˆ is where e is the axial strain. The local coordinate system tˆ, n, positioned on the centre of the stretched element. Under imposed external excitation and the action of the environmental loading the element experiences deformations in three directions associated with internal loading components, i.e. forces and moments. The  p of the distributed forces, external loading, denoted by the vector R includes static components, i.e. weight and buoyancy and dynamic counterparts, that is, inertia forces due to the added mass, current and wave forces and additional damping due to the drag forces which are developed because of the motions of the pipeline in its environment. In this study we neglect the effect of sea current. In

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I.K. Chatjigeorgiou / Applied Ocean Research 62 (2017) 1–17

Fig. 3. The geometry of a distorted curve in the three dimensional space.

addition we consider that the diameter of the pipeline is sufficiently small that allows neglecting the effect of the hydrodynamic diffraction and Froude-Krylov forces. Therefore the only distributed forces which are retained are the drag forces and the forces of the added inertia due to the added mass. The force and moment vectors are written as Tp = T tˆ + Sn nˆ + Sb bˆ  p = M1 tˆ + M2 nˆ + M3 b, ˆ where T is the axial force (called tenand M ˆ and Sb is sion in the sequel), Sn is the in-plane shear force (along n) ˆ The components of the vecthe out-of-plane shear force (along b). tor of moments are the torsional moment M1 and the two bending moments M2 and M3 in the out-of-plane and the in-plane direcˆ The tions respectively (due to angular rotations aroundnˆ and b). is associated with the curvatures according to vector of moments   (M1 , M2 , M3 ) = GJp 1 , EI 2 , EI 3 where G is the shear modulus, E is the Young’s modulus of elasticity, Jp is the  polar moment  and I is the second moment of the pipe. Clearly 1 , 2 , 3 denote respectively the torsional curvature and the two bending curvatures in the out-of-plane and the in-plane directions. Therefore the  = tˆ + nˆ + b. ˆ The curvatures are vector of curvatures is 1 2 3 associated with the Euler angles of rotation as [4] 1 =

∂ ∂ − sin  ∂s ∂s

(32)

2 =

∂ cos ∂s

3 =

∂ cos  cos ∂s



+

∂ cos  sin ∂s −

(33)

∂ sin ∂s

(34)



where , , in the present configuration denote respectively the angle of torsion and the two angular rotations in the out-ofplane and in-plane directions. The components of the vector of the  = ω1 tˆ + ω2 nˆ + ω3 bˆ are given by the right hand angular velocities ω sides of Eqs. (32)-(34) after replacing ∂/∂s by ∂/∂t [39]. The system of forces is completed by considering the axial component of the force due to the pressure of the fluid inside the pipe PAf . Here only the contribution of the fluid is being taken into account omitting the contribution of the gas phase as negligible. It should be noted that even the component PAf is negligible compared to the actual level of tension of steel catenary pipeline installations. Chatjigeorgiou [4] followed a Newtonian derivation approach by considering separately the dynamic equilibriums of the pipe and the fluid elements and expressed the dynamic equilibrium of the global system in terms of a system of differential equations that involves the equations of motion, the balance of moments equations and the compatibility relations. The vectorial forms of those are 3.1. Equations of motion

 m

∂V p  × V p +ω ∂t



∂V f MU +M + 1+e ∂t

  ∂Vf ∂s

 × V + f









  ∂ PAf  × Tp −  = ∂Tp +  + wω  ×b  ×n +M vω tˆ ∂s ∂s





 × tˆ + R  pw + R  +R  pa + R  −PAf fw pd (1 + e) Fig. 4. Forces and moments acting on the stretched differential element of the pipeline.

(35)

I.K. Chatjigeorgiou / Applied Ocean Research 62 (2017) 1–17

3.1.1. Balance of moments  equations  p  ∂M c I ∂ω 1  ×M  p + tˆ × Tp (1 + e) + = 2 1 + e ∂t ∂s (1 + e) 3.1.2. Compatibility relations ∂V p   1 ∂T  × tˆ = tˆ + (1 + e) ω + × Vp EA ∂t ∂s

(36)

(37)

vector of the liquid phase in the pipeline. Further m is the mass per unit unstretched length of the pipe, and M is the mass of the liquid phase always per unit unstretched length of the pipe.  In Eq. (36) c I is a 3 × 3 diagonal square matrix with diag [c I] = c Jp , c I, c I , while c is the density of the pipe’s material. Clearly, the pipe is considered to be uniform. The distributed forces acting on the pipeline are given by the last term of the right hand side of Eq. (35). In accord with the assumptions taken in the present, the distributed forces include the weight of the pipe and the inner liquid phase





  pw + R R fw (1 + e) = − (w0 + Mg) sin cos  tˆ − (w0 + Mg) cos nˆ − (w0 + Mg) sin sin  bˆ

(38)

the added inertia due to the added mass  pa (1 + e) = −ma ∂v nˆ − ma ∂w bˆ R ∂t ∂t

(39)

and the drag forces



 (1 + e) = − 1 DO C u|u| R pd dt 2

1 + etˆ

  1 1 − DO Cdn v|v| 1 + enˆ − DO Cdn w|w| 1 + ebˆ 2 2

(40)

The drag forces have been taken in the form of Morison’s formula. In Eqs. (38)–(40) w0 is the submerged weight per unit unstrteched length of the pipe, ma is the added mass of the pipe per unit unstretched length,  is the density of the surrounding fluid, DO is the outer diameter and Cdt , Cdn are the drag coefficients in the tangential and normal (or bi-normal) directions respectively. A good engineering approximation for these constants is 0 and 1 respectively. The three-dimensional system of Eqs. (35)–(37) is simplified if we consider only plane motions, say in the in-plane or the outof-plane directions. In the present study only the former case is considered. Setting the problem in two dimensions immediately ignores torsional effects. In addition, the out-of-plane components are omitted and the system is simplified significantly. The final system that governs only the in-plane structural vibrations of the pipeline including inner flow effects will read MU ∂u ∂U ∂ m v+ +M + (M + m) 1+e ∂t ∂t ∂t



=

∂ T − PAf ∂s







∂U − 3 v ∂s





MU ∂v ∂ +m u+ 1+e ∂t ∂t

(41)

∂v + U 3 ∂s

+ T − PAf 3 − (w0 + Mg) cos ␾ −

=

∂Sn ∂s

 1 ␳DO Cdn v|v| 1 + e 2

(45)

∂ = 3 ∂s

(46)

Eqs. (41) and (42) are the equations of motion in the tangential and normal directions respectively, Eqs. (43) and (44) are the compatibility relations, while Eq. (45) is the balance of moments equation. Eq. (46) was artificially introduced to yield a system of equations with an equal number of unknowns. In Eq. (43) EAp denotes the elastic stiffness while in Eq. (45) EIp is the flexural rigidity of the pipe. Note that Ap and Ip , namely the cross sectional area and the section’s second moment, correspond to the dimensions of the pipe’s material in the cross section. In the sequel the axial stretching e will be assumed equal to zero. The dynamic parameters that influence the kinematics of the pipe and its structural behaviour are the axial and the normal structural velocities u and v respectively, the axial tension T , the shear force Sn , the angle which is formed between the tangent on the pipe and the horizontal , and the in-plane curvature 3 . In addition, U is the velocity of the inner liquid, which according to the slug-flow approximation can be either uL or uf depending on whether the node that is considered corresponds to the slug or the film zone respectively. Clearly, U, or equivalently uL and uf , are functions of the Lagrangian coordinate s (or hf ). Note that according to the steady slug-flow concept that is assumed in the present, U is independent on time and thus ∂U/∂t = 0. It must be remarked that Te = T − PAf is the actual tension along the pipe that accounts for the inner flow contribution due to the pressure gradient. According to the followed approach, T is literarily the so-called effective tension that accounts also for the hydrostatic pressure effects. Under the steady slug-flow assumption, the pressure P is independent on time and accordingly ∂Te /∂t = ∂T/∂t. Thus, the axial inner loading in the system of Eqs. (41)–(46) can be represented by the actual tension Te . For simplifying the notations, the actual tension will be denoted by T . The dynamic parameters T, Sn , 3 and denote total components, i.e. the summation of static and dynamic counterparts. These will be distinguished by the subscripts 0 and 1 respectively. Eqs. (41)–(46) can be effectively linearized in order to be expressed only in terms of the time-varying components [axial and normal motions p and q with ∂p/∂t = u and ∂q/∂t = v, dynamic tension and shear force T1 and Sn1 , dynamic curvature 31 and dynamic angle 1 ]. That is materialized by i) omitting the nonlinear terms (products of time-varying parameters) and ii) linearizing the nonlinear drag forces using Fourier series expansions and retaining only the leading (fundamental) frequency contribution. The linearization process requires assuming time-harmonic responses for the dynamic terms. Thus, letting y be the vector of the unknowns

T

(47)

it is assumed that



y = Re y˜ (s) eiωt



(44)

∂ 3 + Sn (1 + e)3 = 0 ∂s

y = T1 , Sn1 , p, q, 31 , 1

 1 − ␲␳DO Cdt u|u| 1 + e 2 (m + ma + M)

EIp

(43)

∂ ∂v = + u 3 ∂t ∂s



− Sn 3 − (w0 + Mg) sin ␾



∂u 1 ∂T = − v 3 EAp ∂t ∂s (1 + e)

In Eqs. (35)–(37) V p = utˆ + vnˆ + wbˆ is the vector of the structural velocities of the pipeline and V f = U tˆ + vnˆ + wbˆ denotes the

7



(48)

where Re denotes the real part of the complex quantity in the brackets, ω is the frequency of the excitation and



(42)

˜ 31 , ˜1 y˜ (s) = T˜1 , S˜ n1 , p˜ , q˜ ,

T

(49)

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expresses the complex spatial variation of the dynamic terms that respond harmonically with frequency ω. The tildes will be dropped in the sequel. With the above remarks the governing in-plane, twodimensional, nonlinear set of Eqs. (41)–(46) is reduced to the following linearized equivalent:

∂T1 ∂U = −ω2 mp + MU − iωMU 30 q + Sn1 30 ∂s ∂s +Sn0 31 + (w0 + Mg) cos 0 1

(50)

∂Sn = −ω2 (m + ma + M) q + MU 2 31 + iωMU( 1 − p 30 ) ∂s −T1 30 − T0 31 − (w0 + Mg) sin 0 1 +i

4 DO Cdn ω2 |q|q 3

(51)

∂p T1 + q 30 = EAp ∂s

(52)

∂q = 1 − p 30 ∂s

(53)

∂ 31 Sn1 =− EIp ∂s

(54)

∂ 1 = 31 ∂s

(55)

The system of Eqs. (50)–(55) can be treated as a two-point boundary value problem. Here it is assumed that the bottom end of the curved pipeline is attached to the bottom and the pipeline is completely suspended (without having a bottom lying part). The required six boundary conditions are specified by assuming pinned ends (zero curvatures and accordingly moments), zero motions at the bottom and specified harmonic motions at the top with circular

frequency ω and amplitudes pa and qa in the axial and normal directions respectively. Analogous two-point boundary value problems have been considered (and solved) by Chatjigeorgiou [4,40–42] using finite difference formulations. Here the solution of the concerned boundary value problem was realized using Matlab R2014b and its dedicated routines along the interval s ∈ [0, L]. 4. Combining the pipeline dynamical model with the steady slug-flow model The better explain how the two systems are combined, i.e. the steady slug-flow model with the linear dynamical model of the curved pipeline, aiming to investigate the effect of the two-phase flow on the dynamics of the pipeline, which is simultaneously subjected to forced motions imposed at its top terminal point, the discussion that follows will refer to a specific case under certain assumptions. Clearly, the method is generic and can be applied to any structural configuration with different particulars of the inner flow. The pipeline that is considered is a fully suspended catenary riser. The installation properties are properly chosen such that the angle that is formed at the bottom end is nearly zero. The length of the pipeline is L = 2025.1 m and is installed in a deep water field with depth 1000 m. The pipeline’s material is steel with density 7800 kg/m3 . The Young’s modulus of elasticity is E = 207 GPa, while the inner and outer diameters D and DO are equal to 0.385 m and 0.429 m, respectively. According to the aforementioned figures the mass m, the added mass ma and the submerged weight w0 , all defined per unit unstretched length, will be 219.41 kg/m, 148.16 kg/m and 699 N/m respectively. Further, the elastic stiffness EAp and the flexural rigidity EIp are equal to 5.83 × 109 N and 1.209 × 108 Nm2 . At the static position it is assumed that the pipeline is empty, while the zero angle at the bottom is achieved by applying a top tension force equal to 1860 kN. Under the above conditions the angle at the top will be equal to 51.3◦ . The catenary configuration and the variation of the static components of the pipeline are shown in Fig. 5.

Fig. 5. Details of the pipeline at static equilibrium: the in-plane geometry of the catenary (Xc , Zc ) and the functions of tension T (s), curvature 3 (s), and angle (s). All values are given normalized by the associated maximum: Xc,max = 1688 m, Zc,max = 1000 m, smax = L = 2025.1 m, Tmax = 1860 kN, 3max = 5.99 × 10−4 1/m, max = 51.3◦ .

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Fig. 6. Film and slug zones at the lower portion of the catenary pipeline conveying a two-phase flow.

In the initial static position, at t = 0, which is shown by the solid line of Fig. 5, the pipeline is still. Then the two phases of the inner stream, the gas and the liquid, start to flow within the pipe and the pipe can be subjected to forced motions or not. After the passing of the transient interval, the flow covers the complete length of the pipe, the slugs are formed and the phenomenon achieves the steady-state condition. The formulation of the slug-flow inside a pipe involves several difficulties and uncertainties. Issues of concern involve the type of the flow, whether it is turbulent or laminar with small surface tension that resembles potential flow, the computation of the friction

forces, between the gas and the wall, the liquid and the wall, and especially on the interface, the liquid hold-up in the slug zone, the slug unit translational velocity and the bubble velocity. A major uncertainty in the slug-flow formulations is the length of the slug zone, ls . The reported studies suggest a large dispersion for both horizontal and vertical pipes [43–47]. The situation is deteriorated when the pipeline has a curved shape, with a continuous change in the angle of inclination for which there is not sufficient information. Given the fact that the film length lf , is one of the problem’s unknowns which must be calculated such that Eq. (30) is satisfied, it is more suitable to assume a given value for the slug unit length

Fig. 7. Film and slug zones at the top part of the catenary pipeline conveying a two-phase flow.

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Fig. 8. Velocities in the liquid film and the slug zone at the lower portion of the catenary pipeline conveying a two-phase flow.

lu and determine lf (which univocally define ls as well), until Eq. (30) is satisfied. Clearly, the whole process should be put into an iterative procedure in which Eq. (11) is solved at every step for a given (assumed) film length lf ∈ [0, lu ]. The correct value for lf (and subsequently ls = lu − lf ) will be that, which satisfies Eq. (30). For the purposes of the present paper a constant value of lu = 20D was assumed and that was considered independent from the angle of inclination (or ˇ, see Fig. 1). To combine the slug-flow model with the pipe’s dynamical model, the static equilibrium problem of the pipe is solved using a

uniform discretization of N nodes. Accordingly, the uniform spacing L s is set equal to lu , namely s = N−1 = lu . The solution of the static equilibrium problem will provide, along with the static tension, the shear force and the curvature, the static angle k , k = 1, . . ., N − 1, as a scalar function along the pipe as well. Hence, the slug-flow model can be employed for each of the N − 1 slug unit lengths lu letting the angle ˇ equal to k . The iteration process involves first the calculation of the film length lf assuming values lf ∈ [0, lu ] until Eq. (30) is satisfied. After fixing the value for lf , Eq. (11) is re-employed for the last time for determining all of the details of the flow in

Fig. 9. Velocities in the liquid film and the slug zone at the top part of the catenary pipeline conveying a two-phase flow.

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Fig. 10. Gas velocity in the lower portion of the catenary pipeline conveying a two-phase flow.

the slug-unit, including the shape of the film length hf , the holdup in the film zone Rf , the void fraction ˛f , the liquid area in the film zone Af , the mass per unit length in the same zone mf = L Af and the velocity of the liquid in the film uf . In the corresponding slug zone in the same slug unit, the area covered by the liquid and the velocity of the latter are manually set equal to Af = D2 /4 and uL respectively. All these parameters are obtained as scalar functions of z and they are accordingly transformed into functions of the Lagrangian coordinate s.

5. Numerical results 5.1. Slug-flow particulars inside the pipe Examples of the particulars of a steady slug-flow inside the catenary pipeline of Fig. 5 are shown in Figs. 6–11. The depicted results have been obtained under the following assumptions: the liquid phase is an API 48 crude oil with density L = 790 kg/m3 and kinematic viscosity L = 3.8 × 10−6 cSt, while the gas phase

Fig. 11. Gas velocity at the top part of the catenary pipeline conveying a two-phase flow.

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Fig. 12. Slug-flow effect on the internal loading components of the fixed  riser;  the results are given normalized by the maximum values of the associated static components:

Tension: T1 /max (T0 ), Shear force: Sn1 /max (Sn0 ), Curvature: 31 /max 30 ; only the real parts are shown as the imaginary parts are zero.

is methane with density G = 0.675 kg/m3 and kinematic viscosity G = 1.615 × 10−5 cSt; those values correspond to 15 o C and pressure 1.013 bars [48]; the superficial velocities for the liquid and the gas are uLS = 0.4 m/s and uGS = 2 m/s respectively; the flow is considered laminar with nearly zero surface tension ∼ = 0.001; accordingly, for the computation of friction coefficients, it was assumed that Cf = CG = 16 and n = −1 [see Eq. (17)]; for the calculation of the slug unit translational  velocity it was assumed that the drift velocity is obtained by ud / gD = 0.4 and the coefficient in Eq. (23) is C = 2; the coefficient in Eq. (25) is B = 1.2; the friction coefficient on the interface was taken equal to fi = 0.014. It is recalled that the specific value has been suggested for horizontal and inclined pipelines. Hence, this value is in accord with the present formulation according to which each slug unit is inclined with respect to the horizontal (clearly with variable inclinations) whereas the sequence of the slug units coincides with the catenary configuration of the pipeline. It should also be mentioned, that according to the numerical computations, random changes in the friction coefficient on the interface around the suggested value fi = 0.014 [25,26] have negligible impacts on the global structural behaviour of the pipeline. The pairs of Figs. 6–11 show the shape of the film, the velocities of the liquid in the film and in the slug zones and the velocity of the gas in the elongated bubbles between the slugs, respectively. Figs. 6, 8 and 10 correspond to the lower portion of the pipeline, while Figs. 7, 9 and 11 correspond to the top part of the pipeline. The liquid mass distribution varies the way depicted in Figs. 6 and 7. These figures show that the length of the slug zones increase towards the upper part of the pipeline and as a consequence the length of the film zones is decreased. The corresponding velocities uf and uL can be traced in Figs. 8 and 9, again for the lower and the upper parts of the pipeline. Here a very interesting characteristic can be observed, namely the velocity of the liquid in each film zone, for a significant part of it, is negative. The low gas pressure has a role on it. That implies that a part of the liquid follows the opposite direction and accordingly is captured by the following slug. In addition, as the angle of inclination is increased towards the upper part of the pipeline, the velocity of the liquid volume that moves

towards the opposite direction is increased. Clearly, that is a factor that complicates the pipeline’s dynamical problem given the fact that parts of the liquid inside it, move in different directions. 5.2. The dynamics of the pipeline subjected to internal slug-flow and forced excitations The effect of the slug-flow to a still pipeline is discussed with the aid of Fig. 12. Here the internal loading components are shown, given normalized by the associated maximum values of the static components. Note that the maximum static values of tension and the shear force occur at the top and the bottom of the pipeline, respectively, while the maximum of the static curvature is detected just after the fixed node of the bottom end (Fig. 5). To obtain the depicted results and investigate solely the effect of the slug-flow to an immovable pipeline with a catenary configuration, the problem was considered dynamically and solved by employing the governing dynamic system of Eqs. (50)–(55) assuming pa = qa = ω = 0. The solution reveals that the imaginary parts of the complex parameters are zero and accordingly only the real parts are shown in Fig. 12. According to the depicted results, the global contribution of the slug-flow to the internal loading components is less than 10% of the corresponding maximum static values. Given the fact that no external motions were assumed, it can be said that the slugflow effect is relatively significant. Here, it is very interesting to observe that the flow induced tension and curvature, which should be considered as quasi-static effects, change direction along the pipeline and the transition points nearly coincide. Following the tension trend, the pipeline is stretched at its lower part and assumes a “compressive loading” effect [1] at its upper portion. Similarly, the slug-flow reduces the large static curvature (and accordingly the bending moment) at the lower part. This effect could be considered as beneficial for the pipeline as the large bending stresses which are developed at the lower part due to the geometry of the catenary, are somehow blunted. The slug-flow effect on the catenary pipeline, when excited by top imposed harmonic motions is investigated through Figs. 13–16. It is anticipated that the pursuit of the effect of slug-flow on the

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13

Fig. 13. Slug-flow effect on the internal loading components of the catenary riser subjected to top imposed excitations with pa = 1m, qa = 0m, ω = 1rad/s;  the results are

given normalized by the maximum values of the associated static components: Tension: |T1 |/max (T0 ), Shear force: |Sn1 |/max (Sn0 ), Curvature: | 31 |/max 30 .

pipeline dynamics, by means of comparisons of the linear transfer functions of the various dynamic components, would not be very informative. Hence, it was chosen to seek the details of the associated effect for given excitation properties. Here, two excitation cases are considered, i.e. separately axial and normal motions, with amplitude 1 m and equal frequency 1 rad/s.

Fig. 13 shows the internal loading components, again normalized by the maximum values of the associated static counterparts, while Fig. 14 shows the motions in axial and normal directions. The excitation properties assumed for obtaining the depicted calculations correspond to a severe “compression” loading condition as it derives an axial loading velocity of 1 m/s. It should be noted herein that the axial velocity has been identified as the primarily respon-

Fig. 14. Slug-flow effect on the motions of the catenary riser subjected to top imposed excitations with pa = 1m, qa = 0m, ω = 1rad/s.

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Fig. 15. Slug-flow effect on the internal loading components of the catenary riser subjected to top imposed excitations with pa = 0m, qa = 1m, ω = 1rad/s;  theresults are

given normalized by the maximum values of the associated static components: Tension: |T1 |/max (T0 ), Shear force: |Sn1 |/max (Sn0 ), Curvature: | 31 |/ max 30 .

sible factor for the cause of “compression” and the development of large bending stresses, typically at the lower part of catenary pipelines [49]. Figs. 13 and 14 compare the no-flow case, when the pipeline is empty and the slug-flow case. In both cases the pipeline is excited with the same excitation properties. According to the depicted data, the maximum values of the dynamic components exceed, or are

very close, to the static maxima. Clearly, this has to do with the severity of the excitation. It is also clear that the existence of the slug-flow amplifies obviously all dynamic parameters. The increase in the dynamic tension (Fig. 13) is caused by the slight increase in axial stretching (Fig. 14). Also, the amplification of the curvature (and the bending moment) (Fig. 13) is tightly correlated with the normal motions (Fig. 14). For all components considered, it is

Fig. 16. Slug-flow effect on the motions of the catenary riser subjected to top imposed excitations with pa = 0m, qa = 1m, ω = 1rad/s.

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Fig. 17. Comparative results for the dynamic tension amplification for two slug unit lengths; excitation properties pa = 1m, qa = 0m, ω = 1rad/s; the results are given normalized by the maximum values of the associated static components: Tension: |T1 |/max (T0 ).

evident that the existence of the slug-flow does not change the variation patterns along the pipeline. The evident impact of the slug-flow pattern should be contrasted against the conclusions on the effect of other types of flows in pipelines such as the “plug-flow”. In relevant studies (e.g. [4]) it has been reported that uniform flows with constant velocity profiles have negligible impacts on the in-plane dynamics of pipelines subjected to external excitations while it appears that only the out-

of-plane vibrations are affected and especially due to the Coriollis effects which introduce an additional damping component into the system. The case of the normal excitation is investigated with the aid of Figs. 15 and 16. Fig. 15 contrasts the variations of the dynamic tension and the dynamic curvature for both the no flow and the slug-flow cases as functions of the Lagrangian coordinate s. The dynamic amplification of shear force has been omitted only for dis-

Fig. 18. Comparative results for the dynamic curvature amplification for two slug unit lengths; properties pa = 1m, qa = 0m, ω = 1rad/s; the results are given  excitation 

normalized by the maximum values of the associated static component: Curvature: | 31 |/max 30 .

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Fig. 19. Comparative results for the axial and normal motions for two slug unit lengths; excitation properties pa = 1m, qa = 0m, ω = 1rad/s.

play purposes. Although the dynamic tension amplification due to the normal excitation is evidently very small, one can easily observe the significant impact of the slug-flow along the greatest part of the pipeline. Further, the existence of the internal slug-flow significantly modifies the magnitudes of the dynamic curvature along the pipeline. It is also evident that the slug-flow changes the wavy pattern of the function of curvature. Note that the depicted strong variations are due to the high excitation frequency. For lower frequencies these variations would be less intense. Finally, as in the axial excitation case, the functions of the dynamic tension and the dynamic curvature are closely correlated with the functions of axial and normal motions, respectively (see Fig. 16). 5.3. The effect of the slugging frequency Clearly, the study of slug-flow inside pipes involves several important uncertainties associated with the plethora of issues that affect the formation of slug units. The situation becomes much more complicated for continuously oscillating pipelines such as marine risers. It is evident therefore that the detailed and comprehensive investigation of the contribution of each parameter is impossible in the context of a single paper. Nevertheless, it is important to analyse to a rational extent the impact of a factor which is expected to have a non-negligible effect to the dynamics of the pipeline. That is the slugging frequency which in the case of the present steady slug flow model is described by slug units of different lengths. Different slug unit lengths yield a different number of total slugs in the pipeline and apparently different slugging impacts. The slugging frequency therefore, represented by the total number of slugs in the pipeline, becomes an uncertainty that should be considered. This task is implemented herein using a different riser configuration compared to that investigated in §5.1 and §5.2. In particular, the outer diameter of the particular pipeline was increased to 0.434 m, that results in mass m = 245.4 kg/m, added mass ma = 151.6 kg/m, submerged weight w0 = 922 N/m, elastic stiffness EAp = 6.524 × 109 N and flexural rigidity EAp = 1.373 × 108 Nm2 . The length of the pipeline and the pretension were maintained the same while the installation depth was increased to

1800 m. All the other input data were kept the same. Under these conditions the two-dimensional configuration of the catenary of the pipeline resembles more a marine riser which is typically nearly vertical along its complete suspended length. The angle which is formed between the tangent of the specific pipeline and the perpendicular at its top terminal point is equal to 6.2o . The structural model under consideration was investigated assuming two slugging frequencies represented by slug units equal to 20D and 30D. A single excitation condition was considered that was assumed to be axial with amplitude pa = 1 m and circular frequency ω = 1 rad/s. The associated computations have been plotted in Figs. 17–19 for the dynamic tension, the dynamic curvature and the axial and normal motions, respectively. The trends of the variation patterns for the investigated components are similar. However, it is immediately apparent that the slugging frequency, indeed affects the dynamics of the riser. Although the differences in magnitude appear to be small, it can be rigorously claimed that the slugging frequency plays an important role to the global dynamics of marine risers. In this particular case, the reduction of the slagging frequency leads to stronger internal loading and larger structural displacements. The concerned finding necessitates conducting further analyses on the subject with hopefully unsteady formulations of the slug-flow which are expected to yield non-uniform and non-periodic slugging.

6. Conclusions The present study investigated the effect of slug-flow on risertype pipelines for marine applications. The model that was adopted for internal slug-flow was assumed to be steady. Several assumptions and empirical correlations were implemented to describe the details of the slug-flow, both in the liquid film and in the slugs. The procedure accounted for the variation of the angle of inclination along the, assumed, catenary pipeline. The dynamic problem of the structure was set into the two-dimensional space omitting the out-of-plane vibrations. The original nonlinear governing set was properly linearized, retaining however the internal flow terms, which were used to combine the two models, i.e. the slug-flow

I.K. Chatjigeorgiou / Applied Ocean Research 62 (2017) 1–17

model and the pipeline’s dynamical model. The solution of the problem was achieved in the frequency domain using appropriate solution techniques. It was found that the effect of slug-flow is very important on the dynamics of risers in contrast to the marginal effect of other models of internal flow such as the “plug-flow”. In general, the existence of slug-flow significantly amplifies the dynamic components, without being able however to change their variation patterns. It is remarked that further studies are required to understand better the details of the slug-flow inside pipes, especially when these are considered continuously moving. The present study was based on a two-stage procedure, firstly solving the slug-flow problem inside the pipe and secondly employing the derived results in terms of the liquid phase mass distribution, velocities and pressures to the structural model of the pipe. That was allowed by the assumption of the steady nature of the slug-flow. A more efficient model should combine together the slug-flow and the dynamical structural model of the pipe solving both simultaneously in the time domain. That would allow taking into account an unsteady slug-flow pattern. Finally it should be remarked that most of the details that govern the slug-flow pattern inside a pipe are based on empirical correlations which in turn in most of the cases rely on observations and analyses of experimental measurements. Hence the involvement of vague issues is inevitable. Even under these conditions however, the various recommended approximations are generally accepted as being reliable and constitute the bases of contemporary studies for both the steady and the unsteady slug-flow modes. Nevertheless, as far as marine risers and the effect of slug-flow are concerned, contemporary studies are scarce. Future studies in the specific concept should account for unsteady models parametrizing simultaneously the contribution of the various factors which originate from empirical formulae. References [1] M.P. Païdoussis, Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 1, London Academic Press, London, UK, 1998. [2] M.P. Païdoussis, Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 2, Elsevier Academic Press, Amsterdam, The Netherlands, 2001. [3] I.K. Chatjigeorgiou, Three dimensional nonlinear dynamics of submerged extensible catenary pipes conveying fluid and subjected to end-imposed excitations, Int. J. Non Linear Mech. 45 (2010) 667–680. [4] I.K. Chatjigeorgiou, On the effect of internal flow on vibrating catenary risers in three dimensions, Eng. Struct. 32 (201) (2016) 3313–3329. [5] K.N. Karagiozis, M.P. Païdoussis, M. Amabili, A.K. Misra, Nonlinear stability of cylindrical shells subjected to axial flow: theory and experiments, J. Sound Vib. 309 (2008) 637–676. [6] K.N. Karagiozis, M.P. Païdoussis, M. Amabili, Effect of geometry on the stability of cylindrical clamped shells subjected to internal fluid flow, Comput. Struct. 85 (2007) 645–659. [7] K.N. Karagiozis, M. Amabili, M.P. Païdoussis, A.K. Misra, Nonlinear vibrations of fluid-filled clamped circular cylindrical shells, J. Fluids Struct. 21 (2005) 579–595. [8] M. Amabili, K.N. Karagiozis, M.P. Païdoussis, Effect of geometric imperfections on non-linear stability of circular cylindrical shells conveying fluid, Int. J. Non Linear Mech. 44 (2009) 276–289. [9] S.A. Katifeoglou, I.K. Chatjigeorgiou, S.A. Mavrakos, Effects of fully developed turbulent internal flow on marine risers’ dynamics, 17–22 June 2012, Rhodes, Greece, in: Proceedings of the 22nd International Ocean and Polar Engineering Conference (ISOPE 2012), 2, International Society of Offshore and Polar Engineers, 2016, pp. 410–417. [10] https://www.orcina.com/SoftwareProducts/OrcaFlex/index.php. [11] A. Ortega, A. Rivera, O.J. Nydal, C.M. Larsen, On the dynamic response of flexible risers caused by internal slug flow, in: Proceedings of the 31 st International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2012), 1–6 July 2012, Rio de Janeiro, Brasil, American Society of Mechanical Engineers, 2016, No. 83316. [12] A. Ortega, A. Rivera, C.M. Larsen, Flexible riser response induced by combined slug flow and wave loads, in: Proceedings of the 32nd International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2013), 9–14 June 2013, Nantes, France, American Society of Mechanical Engineers, 2016, No. 10891. [13] M.H. Patel, F.B. Seyed, Internal flow-induced behaviour of flexible risers, Eng. Struct. 11 (1989) 266–280.

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