G.K.Furnes ,J.Berntsen Ontheresponseofafreespanpipelinesubjectedtooceancurrents

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On the response of a free span pipeline subjected to ocean currents

G.K. Furnes

, J. Berntsen

aNorsk Hydro E&P Research Centre, Bergen, Norway

bGeophysical Institute, University of Bergen, Bergen, Norway

cDepartment of Mathematics, University of Bergen, Bergen, Norway Received 17 January 2002; accepted 20 March 2002

Abstract

A mechanistic study is performed to examine the coupling between the in-line and the cross- flow motion of a cylindrical structure subjected to current forces. The structure represents a free span pipeline but concerns marine risers as well.

A time domain model is formulated in which the in-line and cross-flow deflections are coupled through the axial tension which in turn is computed from the pipeline prolongation at any time. This formulation introduces time dependent tensions and non-linearity into the problem.

Preliminary validation of the model simulations vs. physical test data are carried out for one specific case to ensure that the sag and the in-line deflection are correctly resolved by the model. Using this as the initial condition a series of calculations are carried out to examine cross-flow induced deflections induced by an in-line prescribed deflection and vice versa.

Finally, an idealistic simulation of flow induced vibration is presented.

The model simulations demonstrate that the coupling varies with the mode shape and with which component it is initially introduced into. However, it is evident that the coupling effects may be significant and not negligible.

2003 Elsevier Science Ltd. All rights reserved.

Keywords: Vortex induced vibrations; Inline-crossflow coupling; Free span pipelines

∗ Corresponding author. Tel.:+147-5599-6237; fax:+147-5599-6928.

E-mail address: [email protected] (G.K. Furnes).

0029-8018/03/$ - see front matter2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0029-8018(02)00138-5

1. Introduction

This paper is motivated from an increasing demand in resent years to improve the understanding of the dynamics of pipeline installations in rough sea bottom ter- rains, where free spans are likely to occur. One such area is Storegga off the shelf of mid-Norway where Norsk Hydro is evaluating technical solutions for transpor- tation of gas from the Ormen Lange field.

Several aspects of free span pipelines are presented in the literature. Halse (1997) presents a dynamical analyses of a free span pipeline in uniform and in sheared flow.

For details see Halse (1997) and references therein.

A recent study by Hansen et al. (2001), examined the vibrations of a free span pipeline located in the vicinity of a trench.The objective of the work was to extend the validity of Guidelines 14 of Det Norske Veritas (June 1998) concerning free span pipelines.

The behaviour of free span pipelines subjected to ocean currents is not satisfac- torily described by today’s ‘coefficient based’ models. Firstly, the hydrodynamical forces on the pipeline are generally oversimplified by using design currents that are planar. Secondly, the models are designed to compute only planar displacements which are either in the cross-flow direction or the in-line direction. This is a major simplification which exclude any option of evaluating the coupling between in-line and cross-flow oscillations.

The model presented in this paper computes the vectorial deflections transversely to the pipeline axis assuming that the in-line and cross-flow deflections are coupled through the axial tension at any time. The tension is assumed to be spatially uniform and is computed from the prolongation induced by the displacements. In situations where the axial tension cannot be considered as uniform a method given in Furnes (2000) may be employed. The hydrodynamical forcing formulation that is used is sufficiently flexible to deal with arbitrary currents that vary continuously with space and time. Some aspects of the coupling between the in-line and cross-flow displace- ments are the focus of this paper.

2. Model formulation

Consider a pipeline that is supported at x=0 and x=L with an axis that initially is coincident with the horizontal x axis (see definition sketch in Fig. 1). The pipeline is cylindrical with uniform material properties in the axial direction and constant diameter D. At arbitrary positions along the pipline a set of point sources are intro- duced to allow for external modifications of local elastic and damping forces. One such device is depicted in Fig. 1 (at point x =a) and incorporated in the analysis, but it is trivial to extend the analysis to an arbitrary number of sources.

The equations that are used to describe the deflections r(x,t) of the pipeline due to current forces and gravity, at any time t ⬎ 0 and position 0⬍ x⬍L, are

∂

∂t冉^M∂r∂t冊^{⫹R1∂r∂t ⫹EI∂x∂4r4⫺∂x∂}冉^T∂x∂r冊^{⫽F⫹Gi⫺R2∂r∂td(x⫺a)⫺krd(x⫺a) (1)}

Fig. 1. Schematic diagram of the free span pipeline with one suspension device.

and

T⫽T0⫹EAS⫺L

L (2)

where the prolongation S⫺L of the pipeline due to the deflections, is computed from dS

dx⫽冪^1⫹冉^dxdr冊^{2. (3)}

In Eq. (1) G is the ‘reduced gravity’ directed along the z-axis

G⫽冉^prD42⫺M0冊^{g. (4)}

The terms on the right hand side of Eq. (1) are from left to right: current forces, buoyancy minus gravity force along the vertical axis z, external damping and elastic force at point a.

The notation used in this paper is: t is time; x, y, z are Cartesian coordinates forming a right-handed set, in which x, y are measured in the horizontal plane and z vertically upwards; i is the imaginary unit√⫺1 directed along the z axis; g is the acceleration due to Earth’s gravity; is the density of seawater surrounding the pipeline; D is the uniform pipeline diameter; E is Young’s modulus of elasticity; I is the second moment of area; A is the wall cross section area (stress area); T is the axial tension with the value To when S=L; F is the external hydrodynamical exci- tation force; Rl and R2 are linear damping coefficients due to hydrodynamic and structure forces and the point source, respectively; k is the elastic coefficient of the point source; δis the Dirac delta function; M is the sum of structure mass MO and added mass, per unit length.

The deflection r is described by a two-dimensional vector with components ryand rz. Using a complex rather than a vector representation of r, the transverse deflection can be expressed by

r⫽ry⫹irz. (5)

The solution of Eq. (1) requires four boundary conditions and two initial con- ditions. The boundary conditions that are employed at z = 0 and z = L are expressed by

r(0,t)⫽r(L,t)⫽0 and∂²r(0,t)

∂x² ⫽∂²r(L,t)

∂x² ⫽0, (6)

which corresponds to pinned ends. Finally, the initial conditions are expressed by r(x,0)⫽r0and∂r(x,0)

∂t ⫽0 (7)

where r0is the prescribed initial position of the pipeline. For lack of a known formu- lation of how the added mass should vary with the deflection characteristics of the pipeline, e.g. from a node to an antinode, M is initially assumed to be independent of time and given by

M⫽M₀⫹p

4C_arD² (8)

where Ca is the added mass coefficient. After scaling of the axial coordinate byx

= x / L and assuming the axial tension T to be independent of x, Eq. (1) can be written as

∂²r

∂t² ⫹R1

M

∂r

∂t ⫹ EI ML⁴

∂⁴r

∂x⁴⫺ T ML²

∂²r

∂x² ⫽ F M⫹ G

Mi⫺R2

M

∂r

∂td(x⫺a˜)⫺k

Mrd(x⫺a˜) (9) where the point source position, a˜= a / L, now is defined on the interval [0 1]. The Fourier sine transform inxof r over 0 ⬍ x⬍ 1 is written as

rˆn⫽rˆ(n,t)⫽冕¹⁰r(x,t)sin(npx)dx (10)

with inverse transform

r(x,t)⫽2_n冘_⫽^⬁₁^{rˆnsin(npx). (11)}

Taking the Fourier sine transform on both sides of Eq. (9) to obtain d²rˆn

dt² ⫹R1

M drˆn

dt ⫹ EI

ML⁴冕^{01∂x∂4r4sin(npx)dx⫺MLT2}冕^{10∂x∂2r2dx⫽M1}冕⁰¹Fsin(npx)dx⫹ Gi

M冕^{10sin(npx)dx⫺RM2}冕^{10d(x⫺a˜)∂r∂tsin(npx)dx⫺Mk}冕^10d(x⫺a˜)rsin(npx)dx,

(12)

which after integration and application of boundary conditions lead to

d²rˆn

dt² ⫹R1

M drˆn

dt ⫹ EI

ML⁴n⁴p⁴rˆn⫹ T

ML²n²p²rˆn ⫽ 1

M冕¹⁰Fsin(npx)dx

⫹ Gi

Mnp[1⫺(⫺1)ⁿ]⫺R2

M冉^∂r∂t冊^{x=a˜sin(npa˜)⫺Mkrx=a˜sin(npa˜). (13)}

From Eq. (11) the inverse transform can be written as r⫽2rˆnsin(npx)⫹2_j冘_⫽^⬁₁

j⫽n

rˆjsin(jpx) (14)

which, utilized in Eq. (13), gives d²rˆn

dt² ⫹2an

drˆn

dt ⫹w²nrˆn⫽ 1

M冕¹⁰Fsin(npx)dx⫹ Gi

Mnp[1⫺(⫺1)ⁿ]

⫺2R2

Msin(npa˜)_j冘_⫽^⬁₁

j⫽n

drˆj

dtsin(jpa˜)⫺2k

Msin(npa˜)_j冘_⫽^⬁₁

j⫽n

rˆjsin(jpa˜)

(15)

in which 2an⫽ 1

M[R1 ⫹2R2sin²(npa˜)] (16)

and

w²n⫽ T

ML²n²p²冋^{1⫹TLEI2n2p2⫹2kLT2sinn2(npa˜)2p2} 册^{. (17)}

It is more convenient to use a matrix representation in Eq. (15). Truncating the series after m terms and defining

rˆ⫽

冢

冣

a⫽

冢

冣

w²⫽

冢

冣

A⫽

冢

冣

and

B⫽

冢

冣

Eq. (15) can be written as d²rˆ

dt² ⫹2␬drˆ

dt ⫹ ⍀²(t)rˆ⫽H(t) (23)

where

H(t)⫽ 1

M

冢

冕¹⁰Fsin(mpx)dx⯗⫹ Gi

mp[1⫺(⫺1)^m]

冣

and

␬ ⫽a⫹R₂

MAB and⍀²⫽w²⫹2k

MAB. (25)

The vector H contains the forces exerted by currents and the gravity component.

Notice that the time dependence of⍀ is governed by the tension T computed from Eqs (2) and (3) in combination with Eqs (23) and (11). In a situation where T can

be considered as constant with time, it is easy to find an explicit expression for rˆ by taking the Laplace transform in time of Eq. (23), hence

∗rˆ⫽ ∗H

(s⫹␬)²⫹ ⍀²⫺␬² (26)

where ∗ is the Laplace operator. The inverse transform of Eq. (26) can easily be evaluated using the convolution theorem, hence for damped oscillatory motions

⍀²⫺␬² ⬎ 0 rˆ⫽ 1

冑^⍀2⫺␬2冕^{t0H(u)e⫺␬(t⫺u)sin}冉^{冑⍀2⫺␬2(t⫺u)}冊^{du (27)}

which can be evaluated for prescribed values of H. As soon as rˆ is known it is introduced into the series expansion of Eq. (11) to determine r. For the applications considered in this paper it is not possible to neglect the time variation of T and the solution of Eq. (23) will be performed by a numerical time integration.

3. The hydrodynamic forcing F

Before the solution of Eq. (23) (or alternatively (27) for constant T) can be obtained it is necessary to describe the hydrodynamic forcing F to determine the H- matrix (24). Generally F is the total pressure and viscous force acting per unit length of the pipeline. A formulation based on a coupled structure and Navier–Stokes model could be used for this purpose. As the pipeline boundaries are moving in time and the flow field is three-dimensional and time varying, this will lead to a tremendous computational task that will require computational resources that are not available today. A simpler formulation will therefore be applied in this paper.

Current data may be available at discrete points along the pipeline track either from a numerical sea model, from measurements or from a combination of these.

The effects of current forces directed parallel to the pipline axis are neglected and only forces acting normal to the axes are regarded. The current vector V acting normal to the pipeline axis is thus described by a vertical component w and a horizon- tal component ν. Hence

V⫽v(x,t)⫹iw(x,t) (28)

is used to compute the current force F which will be given by F⫽1

2rCTD|V|V. (29)

Here CTis the non-dimensional coefficient

CT⫽CD⫹iCL (30)

where the real part is the drag coefficient

CD⫽C0⫹ADsin(2pfDt ⫹fD) (31) acting in the flow direction (in-line), and the complex part is the corresponding lift coefficient

C_L⫽A_Lsin(2pfLt⫹fL) (32)

which multiplied with V gives a force that is transverse to the flow. The static drag coefficient is represented by COand is fairly well established for a cylinder at rest subjected to known flow conditions (Reynolds number). Less well defined are the forcing frequencies fDand fL and the amplitudes ADand ALwhich in this paper are introduced to simulate vortex induced vibrations (VIV) or other known perturbations of the static conditions. Further, in Eqs (31) and (32)fDandfLare the phase angles between the forcing and the pipeline motion which may vary axially.

Close to the node positions of a vibrating cylinder or in a situation where its motions are small and outside the ‘lock-in’ mode, the lift frequency f_L is approxi- mately given by the value of the vortex shedding frequency fs, derived for cylinders at rest and given by

fs⫽St|V|

D (33)

where St is the Strouhal number which is also fairly well established as a function of Reynolds number for cylinders at rest.

To compute forces due to time and spatial varying currents, the axial coordinate of the pipeline is divided into N intervals bounded by the coordinates x0,x1,…,xN, where xοis at x =0 and ξN is atx =1. The velocity profile will be approximated by layers of piecewise linear functions. For the interval xk⬍x⬍xk + 1the velocity components are given by

v(x,t)⫽byk(t)x⫹cyk(t) (34)

and

w(x,t) ⫽bzk(t)x⫹czk(t) (35)

such that V according to Eq. (28) is

V⫽(byk⫹ibzk)x⫹cyk⫹iczk⫽bkx⫹ck (36) where

bk⫽Vk+1⫺Vk

xk+1⫺xk

, ck⫽Vkxk+1⫺Vk+1xk

xk+1⫺xk

The depth mean current over the interval L_k=xk + 1⫺xkis then V¯_k⫽ 1

Lk冕^xxk+1k

(bkx⫹ck)dx⫽bk

2(xk+1⫹xk)⫹ck. (37)

The drag and lift coefficients and the corresponding frequencies will be approxi-

mated with piecewise constant values over the interval considered. The forcing fre- quency is then initially computed from the Strouhal relation which for interval xk

⬍ x⬍ xk + 1is given by f^(k)s ⫽S^(k)t V¯_k

Dk

where the S^(k)_t can be given a prescribed profile. Denoting C_T for interval xk ⬍ x

⬍xk + 1by C^(k)_T , the hydrodynamical forcing Eq. (29) for the interval considered may be expressed as

F^(k)⫽1

2rC^(k)T Dk冑^{v2⫹w2(v⫹iw) (38)}

which, using the component form, is converted to F^(k)y ⫽1

2rDk冑^{(bykx⫹cyk)2⫹(bzkx⫹czk)2(C(k)D(bykx⫹cyk)⫺C(k)L (bzkx (39)}

⫹czk)) and

F^(k)_z ⫽1

2rDk冑^{(bykx⫹cyk)2⫹(bzkx⫹czk)2(C(k)D(bzkx⫹czk)⫺C(k)L (bykx (40)}

⫹cyk))

where F^(k)=F^(k)y + iF^(k)z .

The forcing components in Eq. (24) can now be expressed by

冕⁰¹Fsin(npx)dx⫽_k冘_⫽^N₁冕^xxkk⫺1

F^(k)sin(npx)dx (41)

which can be integrated exactly. The model formulated above will in the subsequent sections be applied on a pipeline prototype used in a real test case. It is not the intention of present paper to emphasize the use of a ‘correct’ set of coefficients and parameters to validate the model vs. measurements. The focus is on the model’s ability to resolve the physical mechanisms, model features and robustness.

4. Decay test simulations

The model presented above will now be applied on a free span pipeline which has been subjected to extensive basin tests (Huse, 2001). The dimensions of the prototype, which is used in the model setup, are length L =194.6 m, diameter D

=0.556 m and structure mass per unit length M_o=341.8 kg / m. For further details about the structure and the test procedure see Huse (2001). The main purpose of the subsequent simulations is to obtain a better understanding of the dynamics of free

span pipelines. This is obtained through a mechanistic study where the interaction of the pipeline sag, mode shape and forcing can be isolated and studied individually.

The setup of the model simulations is based on the test case 1450 in Huse (2001) in which a uniform current ofν =0.62 m / s (w=0) was applied. In an initial series of calculations some features of the pipeline responses are studied.

4.1. Zero current speed V =0

The model is started from a state of rest by imposing an initial position of the pipeline given by r0= rz =⫺7.5sin(px) and ry=0. The initial vertical deflection ro

is introduced into the first Fourier component (Eq. (11)) such that rˆ1 = ⫺7.5 / 2 and rˆm=0 for m⬎1. The added mass coefficient and damping are arbitrarily chosen as Ca =1 and R1=51.25 kg m^⫺1 s^⫺1 respectively, and do not necessarily represent the actual test case. The coefficients R2 and k are set to zero in the simulation and as current is zero to start with, F =0.

Fig. 2a presents the initial position of the pipeline (sag at t=0 is ⫺7.5 m). The time evolution of the axial tension and displacement at x=L/2 are given in Fig. 2b and c respectively, based on seven Fourier components. A test of the convergence of the series expansion given by Eq. (11) showed that sufficient accuracy was obtained by seven terms. As the problem is symmetrical with respect to x =L / 2 it is evident that the expansion coefficients rˆn with even n, are exactly zero.

It is apparent from Fig. 2b and c that the axial tension and displacement oscillate with the same period which was found to be 3.59 s. The phase, however, is opposite (180°) such that maximum tension coincides with the deepest sag, as is expected.

The oscillations are damped with time towards a constant tension of about 600 kN when the pipeline is at rest. Fig. 2a shows the position of the pipeline at the steady state where the displacement at x = L / 2 is approximately ⫺6.5 m. These results correspond very well with those given in Huse (2001). Exactly the same simulation was repeated but with zero added mass coefficient. In this case the oscillation period was 2.73 s which indicates the importance of using a ‘correct’ added mass formu- lation.

4.2. With current speed n =0.62 m / s

Using the same setup and choice of parameters the model was again started from a state of ‘rest’ using the near stationary solutions computed in Section 4.1 as initial conditions. A uniform and steady current of ν = 0.62 m / s was used to force the pipeline from the initial position. The current speed is exactly the same as in test case 1450 in Huse (2001). Contrary to Huse’s experiment, time varying hydrodynam- ical forces are excluded from the solutions by choosing AD = AL = 0 in Eqs (31) and (32). The static drag coefficient was set to Co = 1.2.

Time series of the horizontal displacement (in the current direction) ry at x= L / 2 is plotted in Fig. 3a. Fig. 3b gives the vertical displacement rz at the same axial position. Although a slight distortion (vertical decay residuals) is seen in the time series of rz, it is evident that the dominating response of ry and rz to the imposed

Fig. 2. Decay test without currents: (a) Initial and final axial deflection. (b) Time evolution of axial tension at x=L / 2. (c) Time evolution of displacement at x=L / 2.

Fig. 3. Uniform current speed V=0.62 m / s: (a) Time evolution of inline displacement at x=L / 2. (b) Time evolution of crossflow displacement at x=L / 2. (c) Time evolution of the swing motion angle relative to the vertical plane.

current, is an oscillation with a period of about 11.4 s. The components oscillate in phase and indicate a swing motion of the pipeline. The angle of the swing motion (at x=L / 2), with respect to the vertical, is plotted vs. time in Fig. 3c. The maximum angle of the transient motions is close to 15° and is damped towards a stationary value of approximately 8°. A rough estimate of Huse’s data for the considered experi- ment indicates an angle of about 12°. This deviation may indicate that the static drag coefficient C0 is too small to account for the VIV drag enhancement that occurred in the experimental test case.

5. Mode shapes and periods

It is evident that as long as the material properties and the forcing are uniform in the axial direction, only symmetrical Fourier modes (rˆ1,rˆ3,…) will give contributions to the deflection vector r. The antisymmetrical modes rˆ2,rˆ4…, are then identically zero as in the previous simulations.

As the tension varies with the current speed it is in principle possible for the same mode shapes to oscillate with different frequencies, or for two different mode shapes to oscillate with the same frequency. Contrary to mathematical modelling, it is diffi- cult to achieve test conditions that are completely homogeneous in the axial direction.

The objective of the following series of calculations is to determine some oscillation characteristics of modes 2, 3 and 4 assuming the same parameters and flow conditions as in Section 4.2, except that the damping is set to zero. The idea is to demonstrate how a given modal perturbation in the crossflow plane induces inline motions and vice versa.

All the simulations started from ‘rest’ using the near stationary pipeline position computed in Section 4.2 as an initial condition, but modified with a deflection which is introduced into each of the considered Fourier modes. The amplitude of the deflec- tion was chosen as r=D m (one pipeline diameter) and is introduced into the series expansion (11) to determine the corresponding rˆn. For comparison purposes, the deflection is introduced first in horizontal component (in-line y-direction) and sub- sequently in the vertical (cross-flow z-direction). The results are presented as devi- ation from the static solutions presented above.

5.1. Imposing (rˆ2)y =D / 2

The purpose of this calculation is to examine how the static crossflow sag is affected by a second mode perturbation of the static inline deflection. Fig. 4 presents the results in which subplot A is the standard deviation (STD) of rynormalized with D based on forward integration over a time span of 100 s. Subplot C is the time variation of ry/D sampled at x = 50.0 m which is close to an antinode location (maximum STD).

It is evident from Fig. 4 panels A and C that the second mode deflection, with amplitude D, is correctly imposed into the horizontal component. The time evolution of ry/D given in panel C, indicates a pure single frequency oscillation with a period

Fig. 4. Motions due to a prescribed second mode inline deflection. (A) Axial variation of the standard deviation of ry/D. (B) Axial variation of the standard deviation of rz/D. (C) Time series of ry/D close to an antinode. (D) Time series of rz/D close to an antinode. (Bottom panels) Countours of time evolution of ry/D and rz/D.

of about 4.8 s. Panels B and D correspond to panels A and C, respectively, but display the crossflow induced deflections rz/D. The STD of rz/D is an order of magni- tude less than the STD of ry/D and indicates motions that are composed of more than one mode. Panel D presents a time series plot of rz/D at the antinode position x = 40.5 m. It shows that oscillations occur at a period of 2.4 s, which is half of the inline period, but the amplitude varies periodically with time. It is also seen from panel D that rz is slightly shifted off its equilibrium position in a direction that reduces the initial vertical sag.

The two bottom panels present contours of the ry/D and rz/D displacements with time. The time period covers the first 20 s from the initial mode shape deformation, but represents the features at any later time interval. It is interesting to observe that the pure second mode oscillation of ryinduces deflections in rzthat are a composite of the first and the third mode responses. The response appears as alternations between first mode dominance succeeded by third mode dominance. The recurrence of the alternations is roughly 21 s.

5.2. Imposing (rˆ₂)_z =D / 2

Following the same procedure as in Section 5.1 except that the second mode now is imposed into the crossflow component, the results are as given in Fig. 5. The STD of rz/D in panel B indicates that the second mode dominates the crossflow motion.

However, as the STD is not zero at the midpoint of the pipeline it is not a pure second mode motion. This is also aparent from panel D which shows the time variation of rz/D near the antinode position x=50.0 m. It is seen from this panel that the ampli- tude fluctuates slightly with time. The fluctuation leads to a periodic deviation from the prescribed initial deflection rz/ D = 1 but will not be examined in more detail in this paper.

The inline induced response (panels A and C) is more than two orders of magni- tude less than the prescribed crossflow deflection and apparently it is shifted towards a new equilibrium position that is slightly less than the static position. The period is 2.4 s which is exactly half the oscillation period of the crossflow motions. This is of particular interest in connection with vortex induced vibrations and will be treated in a subsequent section. The amplitude modulation of ry(panel C) is, as for rzin the previous section, connected to the combined first and third mode responses.

This is also demonstrated in the two bottom panels in which also the distortion of the second mode crossflow displacement is evident.

5.3. Imposing (rˆ3)y =D / 2

Fig. 6 shows the results from simulations where the third mode is imposed into the inline component. The STD distribution of ry/D in panel A indicates a shape which correspondes closely to a pure third mode motion. Notice that the ‘nodal’

points are not identically zero and hence modes different from the third mode are likely to be present too, although with minor contributions. Panel C displays the time variation of ry/D at the antinode position x=34.7 m. According to panel C it

Fig. 5. Motions due to a prescribed second mode crossflow deflection (panel details in Fig. 4).

is evident that the initial amplitude of the imposed mode is correctly prescribed. As time progresses, however, it is apparent that the amplitude of ry/D decreases from 1 to about 0.5 and subsequently increases again. By extending the time integration it was found that this amplitude modulation is periodic with a period of approxi- mately 120 s and is a consequence of the deviation from a pure single mode displace- ment.

Fig. 6. Motions due to a prescribed third mode inline deflection (panel details in Fig. 4).

The crossflow induced displacement rz/D shown in panel B is also composed of more than one mode, but evidently the third mode is dominant. Panel D presents a time series plot of the crossflow displacement at the antinode position x =34.7 m.

The displacement is subjected to significant changes with time such that increasing rz/D amplitudes are accompanied by decreasing ry/D amplitudes. It is interesting to

observe that the magnitude of the induced deflection amplitudes, rz/D, may period- ically achieve values of the same order of magnitude as the prescribed ry/D ampli- tudes. It is also noticed that the equilibrium position of rzis slightly changed towards less sag (panel D).

The two bottom panels display contour plots of ry/D and rz/D over the time span from 50-70 s covering a period when the amplitudes of ryand rz are at its minimum and maximum values, respectively. It is evident that the deflections of both compo- nents are dominated by the third mode and oscillate with a period of about 2.7 s. A closer look at the time series of rzreveals that its dominant mode shape alternates between the first and the third mode. The first mode dominates at small amplitudes of rz/D at position x =37.7 m (panel D).

5.4. Imposing (rˆ3)z =D / 2

Fig. 7 presents the results when the third mode is imposed into the crossflow component. It is apparent from panel B that the crossflow motion is dominated by the third mode, but again it is seen that additional deflection forms are present. A time series of rz/D at the antinode position x=34.7 m indicates a leading period of about 2.7 s with an amplitude variability between approximately 0.8–1.2. Contrary to the previous calculation the amplitude of the imposed component does not exhibit a corresponding long term modulation.

The response of the inline component is also third mode dominated with a leading period of 2.7 s (panels A and C). However, comparing panels A and B it is seen that the mean amplitude is typically a factor 5 less than the crossflow amplitude.

Panel C in Fig. 7 shows a time series plot of the inline displacement at the antinode position x =34.7 m. It appears from this plot that the amplitude of ry/D reaches a maximum value of close to 0.3 after about 50 s. An extended simulation indicated that this amplitude variability is periodic.

Contour plots of ry/D and rz/D are presented in the two bottom panels of Fig. 7 covering the same time span as the corresponding plots in Section 5.3. The third mode dominance in both components is apparent from these panels. For comparison purposes contour plots were also made at time intervals where the amplitude ry/D at x=34.7 m (panel C) was smallest such as at t=100 s. The conclusion from this was that ryalternates between a first mode and a third mode dominance. For instance, the third mode dominates ryaround t= 50 s and the first around t=100 s.

5.5. Imposing (r4)y =D / 2

Referring to Fig. 8 it is seen that the specified inline component oscillates with a shape that is close to a pure fourth mode, with a period of approximately 1.7 s (panels A and C). The amplitude, as represented by the deflection at the antinode position x = 27 m, is stable at a value of ry/ D⬇1. The response observed in the crossflow component is represented in panels B and D and indicates a significant contribution from the first mode. The time series of rz/D near the midpoint of the pipeline (panel D) exhibits amplitude variations of about 0.35 at a leading period of

Fig. 7. Motions due to a prescribed third mode crossflow deflection (panel details in Fig. 4).

3.7 s. An offset of the equilibrium position of rzwith approximately 0.35D is also apparent in panel D.

The contour plots for ry/D and rz/D support that the inline motion is nearly a pure fourth mode displacement while the crossflow is dominated by a slightly distorted first mode motion as suggested from panels B and D. Notice the different time resol- ution in the contour plots.

Fig. 8. Motions due to a prescribed fourth mode inline deflection (panel details in Fig. 4).

5.6. Imposing (rˆ4)z =D / 2

The fourth mode dominates also the crossflow displacements rz, but as seen from Fig. 9 panel B, it is affected by a significant contribution from other mode shapes as well. This is apparent in panel B where the time series at an antinode indicates

Fig. 9. Motions due to a prescribed fourth mode crossflow deflection (panel details in Fig. 4).

amplitude variations that deviates slightly from the prescribed. The period of the most prominent signal is about 1.7 s.

The STD profile of the inline motion in panel A has a shape that indicates that the first mode dominates at a period of 3.7 s (panel C). The inline response amplitude is typically a factor 25 less than the crossflow amplitude and its equilibrium position

is shifted by approximately⫺0.05·D. Contour plots of the time variation of ry/D and rz/D presented in the two bottom panels, show a distorted first mode motion for the inline component and a fourth mode displacements (distorted) for the crossflow component. The coupling between the inline and the crossflow deflections are investi- gated above for some idealistic cases. It is clear from these calculations that the effects vary with the mode number and with whether the initial imposition is in the inline component or the crossflow component. However, the coupling between the components is so strong that it cannot be neglected in general as it is in many models today.

The above simulations were carried out for one specific current speed (pipeline plane approximately 8° from the vertical) and the effect of different current speeds remains to be examined.

6. Vortex induced vibrations

Under conditions in which vortex induced vibrations (VIV) are important the cor- responding forces are simulated through the periodic drag and lift coefficients given in Eqs (31) and (32). In addition to the static drag Co, which will be given the same value as above, there are four parameters that need to be prescribed; the drag and lift amplitude factors ADand AL, the corresponding phase anglesfD andfLand the oscillation frequencies fD and fL. These four parameters are not universal but vary with the oscillation charateristics.

As indicated above (Section 3) it is possible to estimate the lift frequency fL, due to vortex shedding, from Eq. (33) given that the pipeline is in a state of rest or in a situation where the motions are small. As the vibration amplitudes increase the vortex shedding is modified towards the natural frequency of the structure. However, as the added mass changes with the vibration characteristics, it is likely that the actual vibration frequency will be different from the natural frequency of the structure as well as the Strouhal frequency. A model simulation will therefore require an iterative adjustment of the three parameters, namely the added mass and the drag and lift coefficients AD and AL. This procedure will need experimental data as for example provided by Gopalkrishnan (1993) and Vikestad (1998). The present model is developed with flexibility to account for this. It is also able to take account of spatial varying characteristics of the drag and lift coefficients as well as the phase angles between the forcing and the pipeline motions.

As the purpose of the present paper is to improve the understanding of the oscil- lation mechanisms of a free span pipeline subjected to time varying axial tension, it is justifiable to fix the value of AD and ALinitially, without applying an iterative adjustment towards empirical values. The values that were chosen are uniform in the axial direction and given by AD =0.8, AL= 0.4 and added mass as prescribed earlier. Using a Strouhal number St = 0.2 which for this specific case is chosen uniform in the x-direction, the frequency computed from Eq. (33) is 0.22 s^⫺1 which corresponds to a period of 4.48 s. This period is very close to the period of the

second mode oscillation of rz (4.8 s, Section 5.2) which has a response in the (ry)- component at a period that is exactly half of this. The factor two between the vertical and horizontal oscillation periods coincides with the corresponding periods of the cross-flow and in-line VIV.

In simulations of VIV it is natural to relate the in-line frequency fDto the cross- flow frequency fL by fD =2 fL where in this case fL initially is estimated from the Strouhal frequency fs=0.22 s^⫺1. To close the problem it remains to establish values forfDandfL. It is evident from the discussion above, that it is impossible to generate asymmetrical modes (e.g. mode 2) from a uniform hydrodynamical forcing. In spite of that, the considered experiment in Huse (2001) gave significant contributions from mode 2 at a period of 4.8 s. It is therefore justifiable to introduce a non-uniform axial distribution of fDand fL.

The most favorable conditions for transfer of energy to an oscillator is when the rate of work done on the oscillator is at a maximum. This situation is frequently refered to as energy resonance. The total rate of work along the span of the pipeline can be expressed by

Ek⫽冕^{01F·∂r∂tdx. (42)}

It is assumed that the phase anglesfDandfL correspond to an axial distribution that results in a maximum value for Ek. A series of calculations were therefore perfor- med in whichfD,fLand fLwere varied to determine the combination of these three parameters that gave the maximum mean value of Ek. The forcing frequency was limited to a band that excited only the second mode. It was found that the axial profile offLhad a significant impact on Ek whilefD had a minor effect and could be taken as uniform during the subsequent simulations. The maximum mean value of Ek was achieved when fL= ⫺p/ 2 for x⬍ L / 2 and fL= p/ 2 for x⬎ L / 2.

Using the above given set of parameter values the model was started from a state of rest using the static defections as initial condition and integrated forward over a time span of 300 s. The same damping was used as in Section 4. After subtraction of the static deflection the results of the calculations are as plotted in Fig. 10. Panels A and B show the time varying axial deflection of ry/D and rz/D, respectively, cover- ing a time span from 260-300 s. The crossflow component in panel B shows a nearly pure second mode deflection with antinode displacements of more than two pipeline diameters. This is a substantial displacement which exceed the observations in Huse (2001). However, as indicated introductorily, the purpose of this paper is to examine mechanisms and the application of correct drag and damping coefficients have been secondary.

Notice a minor displacement in the nodal point indicating a contribution from other modes as well. The inline displacements (panel A) are much less and have a shape that indicates a deflection which is composed of the first and the third mode.

The maximum deflection is less than one pipeline diameter.

Panels C and D present time series of inline and crossflow displacements, respect- ively, at x = 48 m. This position is close to the position of an antinode. The time

Fig. 10. Time variations of the deviation from the static deflection when VIV is introduced. (A). Axial variation of inline ry/D. (B) Axial variation of crossflow rz/D. (C) Time series of ry/D close to an antinode.

(D) Time series of r_z/D close to an antinode. (E) Trajectory of the displacement near an antinode. (F) Trajectory of the displacement near the node.

series covers a period of 40 s and indicates fairly consistent amplitudes and a crossflow period that is twice the inline. The trajectory of the pipeline displacements is plotted in panels E and F covering the time from 260-300 s. Panel E is recorded at x =48 m and panel F at x =96 m. The last position is close to the antinode. It

appeared from the calculations that, except from the near nodal point, the trajectories exhibit the figure eight shape which is typical for many VIV excited motions.

7. Concluding remarks

A model has been formulated to examine some dynamical features of free span- ning pipelines subjected to current forces. The main focus has been on the coupling between the cross-flow and the in-line motions assuming that the time varying part of the axial tension can be computed from the momentary prolongation caused by current induced deflections.

The simulations demonstrated that in-line displacements can induce a significant cross-flow displacement and vice versa. The effect of this coupling varies with the mode shape and with which component it is initially introduced into. It is evident that models dealing with free spanning pipeline problems should account for this coupling. The second mode cross-flow motion is of particular interest as it induces vibrations in the in-line component with a period that is exactly half the period of the cross-flow motion. Under favourable flow conditions the vortex shedding fre- quency may thus lead to a resonant interaction between the in-line and cross-flow components.

Acknowledgements

The first author is very much indebted to Dr. H. Berends for a number of dis- cussions during the progress of this work.

References

Furnes, G.K., 2000. Deflection of flexible risers subjected to ocean currents. In: Proceedings of the Tenth ISOPE Conference, Seattle,

Gopalkrishnan, R., 1993. Vortex-induced forces on oscillating bluff cylinders. D.Sc. thesis, Department of Ocean Engineering, MIT, Boston

Halse, K.H., 1997. On vortex shedding and prediction of vortex-induced vibrations of circular cylinders.

Dr. Ing. Thesis. Department of Marine Structure, Fakully of Marine Technology, NTNUI, Trondheim.

Hansen, E.A., Bryndum, M., Mork, K., Verley, L., Sortland, L., Nes, H., 2001. Vibrations of a free spanning pipeline located in the vicinity of a trench. In: Proceedings of OMAE’01, Rio de Janeiro, Huse, E., 2001. Private communication.

Vikestad, K., 1998. Multi-frequency response of a cylinder subjected to vortex shedding and support motions. Dr. Ing. thesis, Department of Marine Structures, NTNU, Trondheim.