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Parametric instability of a submerged floating pipeline between two floating structures under combined vortex excitations
Applied Ocean Research 59 (2016) 265–273
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Parametric instability of a submerged floating pipeline between two floating structures under combined vortex excitations Zhiqian Wang, Hezhen Yang ∗ School of Naval Architecture, Ocean and Civil Engineering, State Key Lab of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China
a r t i c l e
i n f o
Article history: Received 29 October 2015 Received in revised form 13 April 2016 Accepted 14 June 2016 Keywords: Submerged floating pipelines Parametric instability Vortex-induced vibration Dynamic response Mathieu equation
a b s t r a c t This study investigates a parametric instability prediction of a submerged floating pipeline connected between two floating structures under combined vortex excitations. The submerged floating pipeline is considered as a potential solution for fluid transportation between floaters in deepwater. However, the parametric instability combined with vortex-induced vibration is a major concern for its safety. The instability can lead to huge displacement and fatigue damage. Thus, it is essential to propose a methodology for analyzing nonlinear dynamic properties of the pipeline with combined parametric and vortex excitations. Here, coupled vibration equations of the pipeline are established, including hydrodynamic force model which contains vortex-induced load and structural model which contains parametric excitation due to motion of the two floating structures. Then Mathieu equation is derived from the homogeneous structural equation. Differences between the dynamic responses subjected to only vortex excitation and combined excitations are compared. The pipeline engineering cases and the effects of two important design factors are analyzed. The results show that the parametric instability may occur in cross-line vibration of the submerged floating pipeline between the two floaters. Even the maximum vibration amplitude of the pipeline under combined excitations is still larger than that under only vortex excitation, when the parametric instability does not take place. There is a vibration mode jump phenomenon while the parametric instability occurs. In addition, two design factors, transfer coefficient and phase angle of the floaters’ motion, can make a great impact on the parametric stability of the submerged floating pipeline. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Offshore risers and pipelines are becoming more and more important while the exploration of marine oil and gas is moving in deep and ultra-deep water. There are several new concepts for the design of offshore risers and pipelines in recent years. The submerged floating pipeline is one of these new concepts to transfer fluid and gas in ocean engineering [1]. Generally speaking, the submerged floating pipeline is floating at a certain depth of ocean with some mooring chains or tethers anchoring it to the seabed [2,3]. On the other hand, it can also be used to connect two floating structures, shown in Fig. 1, for example, the gravity actuated pipe which connects the FPSO and Spar platform at Kikeh outside Malaysia installed in 2008 [4]. Compared with the traditional pipelines that are put on or buried in the seabed, the submerged floating pipeline is considered as a more useful solution for offshore engineering with some challenging seabed
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (H. Yang). http://dx.doi.org/10.1016/j.apor.2016.06.009 0141-1187/© 2016 Elsevier Ltd. All rights reserved.
condition. First of all, there is no need to do any intervention work for the seabed because the pipeline is floating at the mid-deep water without the direct interaction with the seabed soil. Besides the submerged floating pipeline is located at a certain depth under the sea surface, so wave forces can also be ignored. Finally, high hydrostatic pressure and temperature difference in the deep and ultra-deep ocean are avoided as well. Therefore, the submerged floating pipeline could prove to be a part of the future solutions for deep water gas and fluid transportation. Although many complex environmental effects such as the wave forces and soil-pipeline interaction forces do not need to be considered for the submerged floating pipeline’s design, current forces and vortices which are also produced by the current are the major concerns. Nonlinear dynamic response for the submerged floating pipeline is a vital analysis task for its safety. And, it is necessary to study vortex-induced vibration which is thought to be the important cause of fatigue damage of the offshore risers and pipelines [5]. There are a lot of researches done by both experimental tests and numerical predictions on the vortex-induced vibration of such a circular cylinder structure. Govardhan and Williamson [6] applied several experiment tests for the vortex-induced vibration of the
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be interacted together on the pipeline. But few analysis methods consider both parametric and vortex-induced vibration excitations in the past few years [12]. The present study is meant to establish a suitable mathematic model to explore the stability of the submerged floating pipeline between two floating structures under vortex and parametric excitations. In Section 2, the governing motion equations of the pipeline subjected to combined excitations are derived, including structural model and wake oscillator hydrodynamic force model. Mathieu equation is also derived from the homogeneous structural equation. In Section 3, validation of the hydrodynamic force model is made, and the Mathieu equation is solved. The stability diagram is made for the submerged floating pipeline design. Then engineering cases and effects of two important design factors on the stability of the pipeline are discussed, and finally some suggestions to avoid the parametric instability are given. Fig 1. Submerged floating pipeline between two floating structures.
cylinder structure at low mass ratio value, and found the response could be divided into the initial, upper and lower branches. Srinil et al. [7] used a classical van der Pol wake oscillator model to simulate the fluid-structure interaction at both in-line and cross-line direction, and compared the numerical results with his experiments. Because the submerged floating pipeline can be connected with the floating structures, the motion of these floaters will also influence the dynamic response of the pipelines, which may lead to parametric instability problem of the pipeline. The parametric instability, also called parametric resonance, is another major concern for offshore structures’ design, because it can excite the very large motion [8]. In the past decade, there are several studies on analyzing the parametric instability of tethers and risers of top tensioned platforms. Park and Jung [9] implemented a finite element method to study the lateral responses of a top tensioned platform tether under combined parametric and forcing excitations. Franzini et al. [10] analyzed the parametric instability of a flexible tensioned cylinder under harmonic excitation through experiment tests. For the recent researches of the submerged floating pipeline, most of them concentrated on its static design analysis [2] and vortex-induced vibration. For example, Sergent et al. [11] applied CFD, experiments and software to analyze the fatigue life of the submerged floating pipeline because of vortex-induced vibration. Thus it is still necessary and urgent to propose a methodology to predict the parametric instability of such a pipeline between the floating structures and learn how the motion of the floaters can influence its dynamic stability. Besides, it is also important to combine the vortex-induced forces with the parametric excitation, and study the dynamic response of the pipeline under such combined excitations, because in the real sea condition, these two excitations must
2. Theory The nonlinear dynamic motion of the submerged floating pipeline is complicated, because it involves the fluid-structure interaction which means that the motion of the structure and fluid dynamic forces affects each other. Therefore to simulate this interaction, the system model could be divided into two sub-models. The first one is called the structural model reflecting the transverse and axial vibration of the submerged floating pipeline under the hydrodynamic forces, while the second one is the hydrodynamic force model showing how to calculate the hydrodynamic forces with the dynamic response of the pipeline. Before derivation of the motion equations, the following assumptions are listed to simplify the physical models: (1) The material, mechanical properties and current speed are assumed constant along the length of the pipeline and cannot change with time. (2) The buoyance and gravity of the pipeline per unit length are thought to be equal, so that the pipeline can be straight in initial condition. (3) Only the cross-line vibration of the pipeline is considered. (4) The motion of the floating structures is simplified as a harmonic function in the uniform wave. 2.1. Structural model 2.1.1. Cross-line vibration equation of the pipeline The schematic of the submerged floating pipeline system is shown in Fig. 2, while the configuration of the submerged floating pipeline can be described in Fig. 3. In this study, the pipeline is simplified as an Euler-Bernoulli beam-column with two ends hinged.
Fig. 2. Schematic of the submerged floating pipeline system.
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
267
Fig. 3. Configuration of the submerged floating pipeline.
2.1.2. Axial vibration and parametric excitation Because the submerged floating pipeline is suspended between two floating structures, the motion of the floaters such as FPSO, Spar and TLP may influence the dynamics of the pipeline. The towheads at two ends of the pipeline will have displacement which follows the motion of the floaters. If two towheads have relative displacement at the direction along the length of the pipeline, then axial vibration of the submerged floating pipeline may take place. In addition, because the floating structures do the slow-drift motion on the sea surface, the main frequency of this horizontal motion is very low, at about 0.02 rad/s, while the first natural frequency of the axial vibration of the pipeline is quite large, at above 10 rad/s. So the quasi-static method can be applied to analyzing this effect rather than using differential equations of the axial vibration. By the quasi-static method, the relationship of increasing tension force of the submerged floating pipeline and increment of its total length can be established. The function can be derived as
Fig. 4. Hydrodynamic forces on the submerged floating pipeline section.
The differential equation for the cross-line vibration of the submerged floating pipeline can be derived as the following equation, according to Huarte [13]: 2
(m + ma )
4
2
∂ v ∂v ∂ v ∂ v + cs − T (t) = fh , + EI ∂t 2 ∂t ∂x4 ∂x2
(1)
where m is the mass of the pipeline per unit length; ma = (/4)Ca w D2 means the added mass for the pipeline per unit length, in which Ca is the added mass coefficient and equals to 1.0, w is density of the sea water, and D is outer diameter of the pipeline; v is the deflection of the pipeline section; cs means the structural damping coefficient per unit length; E is elastic modulus; I is moment of inertia of the pipeline section; T (t) is axial tension force; fh is the hydrodynamic force (not including the initial mass item, because it has been put in the left-hand side of the equation) on the pipeline section at the cross-line direction. The hydrodynamic forces on the submerged floating pipeline section can be described as the lift force fl and drag force fd , shown in Fig. 4. According to the figure, the total hydrodynamic force on the pipeline section at the cross-line direction is derived as: fh = fl cos − fd sin =
1 1 C w DU 2 cos − Cd w DU 2 sin , 2 l 2
(2)
where U is the current speed; Cl is the lift force coefficient; Cd is the drag force coefficient, and here is chosen to be constantly 1.2 [7]; tan = (1/U)(∂v/∂t) means the ratio of vibration speed of the pipeline section to the current speed. By the assumption that the vibration speed is much smaller than the current speed, which is based on the recommended vortex-induced vibration model by Facchinetti et al. [14], there is a relationship that cos ≈ 1 and sin ≈ tan . Therefore the hydrodynamic force at the cross-line direction can be finally described in the following form: fh =
∂v 1 1 C w DU 2 − Cd w DU . 2 l 2 ∂t
(3)
l T = ε ≈ , EA L
(4)
in which T represents the increasing tension force; A is area of the pipeline section; ε is increment of strain; l is the increment of the length of the pipeline; L is the length of the pipeline. According to Fig. 2, we assume that the horizontal motion of the floating structures at the direction along the pipeline could be considered as a harmonic function with the same frequency on the uniform waves to simplify the problem. Thus the left and right floating structures will move as the following equations:
sl = S1 cos ωf t ,
sr = S2 cos ωf t + ˇ ,
(5)
where sl and sr are the displacement of two floating structures; S1 and S2 are their amplitudes; ωf is the frequency of the motion of two floating structures; ˇ is phase angle between two floaters’ motions. Therefore the relative motion of two floating structures can be written as
s = sl − sr = S1 cos ωf t − S2 cos ωf t + ˇ .
(6)
There must be a relationship between the motion of floating structures and the displacement of the towheads, however many complex factors may affect the relationship, such as the design of the connection between the floating structure and the towhead, tensile rigidity of the pipeline, initial tension force and so on. To simplify this relationship, a transfer coefficient which means the influence of the motion of floaters on the displacement of towheads is established as a=
l . s
(7)
Substituting Eqs. (6) and (7) into Eq. (4), we can derive the increment of the tension force, which is thought as one of the parametric excitations to the submerged floating pipeline, in the following form:
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√
= (fD/2 22 ¯l) based on Tamura’s theory is the self-excitation coefficient, in which f is the coefficient of slope of the lift force and ¯l is the mean half-length of the wake oscillator; Cl0 is the lift force coefficient of stationary cylinder. The value of St, f and ¯l can be determined from experiments, and it is proved that they are dependent on the Reynolds number. Thus the real effects of the Reynolds number on VIV can also be contained in the wake oscillator model. According to the research from Tamura and Matsui [16] and Doan and Nishi [18], St equals to 0.2 and ¯l = 1.1D, and the recommended value of the lift force coefficient of stationary cylinder Cl0 is set to be 0.4. 2.3. Coupling structural model and hydrodynamic force model According to Eq. (1), Eq. (3), Eq. (9) and Eq. (10), the structural model could be combined with the hydrodynamic force model as
⎧ 2 4 ∂2 v ∂ v ∂v ∂ v ⎪ ⎪ ⎨ mt ∂t 2 + ct ∂t + EI ∂x4 − T0 + Ta cos ωf t − ϕ ∂x2 ⎪ ⎪ ⎩
Fig. 5. Schematic illustrations for the wake oscillator model.
T =
aEA S1 cos ωf t − S2 cos ωf t + ˇ = Ta cos ωf t − ϕ , L (8)
where Ta = (aEA/L)
S1 − S2 cos ˇ
2
+ S2 sin ˇ
2
, and ϕ =
arctan(S2 sinˇ/(S1 − S2 cosˇ)). Finally the tension force of the pipeline can be written in the complete form as
T (t) = T0 + T = T0 + Ta cos ωf t − ϕ ,
(9)
where T0 is the initial tension force of the submerged floating pipeline. 2.2. Hydrodynamic force model
⎧ 2 ⎪ ∂˛ 4f 2 ⎪ ⎨∂ ˛ − 2ωv (1 − 2 ˛2 ) + ωv2 ˛ = − 2 ⎪ ⎪ ⎩
Cl0
∂t
Cl = −f (˛ +
2
∂ v 1 ∂v − ωv2 2 ¯ U ∂ ∂t , t 0.5D + l
1 ∂v ) U ∂t
1
(10)
where ˛ is the angle of the oscillator; ωv = 2StU/D means vortex shedding circular frequency, in which St is the Strouhal number;
(11)
2
4f 2 1 ∂ ˛ ∂˛ ∂ v 1 ∂v + ωv2 ˛ = − , − 2ωv (1 − 2 ˛2 ) − ωv2 U ∂t Cl0 ∂t ∂t 2 0.5D + ¯l ∂t 2
where mt = m + ma means the total mass per unit length; ct = cs + (1/2) (f + Cd ) w DU is the total damping coefficient per unit length, including the structural and hydrodynamic damping coefficient. These are the equations that can represent the cross-line vibration of the submerged floating pipeline, and to solve them, the following boundary and initial conditions are needed:
⎧ ⎨
v|x=0 = v|x=L = 0, 2
⎩ ∂ v| 2 ⎧ ⎪ ⎨
∂x
(12)
2
x=0
=
∂ v | = 0. ∂x2 x=L
∂v = 0, | ∂t t=0 ∂˛ C = − l0 , = 0. | f ∂t t=0
v|t=0 =
⎪ ⎩ ˛|
The most direct method to get the hydrodynamic force data on the vibrating cylinder is to carry out experiments or direct numerical simulation (DNS). However, both of the methods are very time consuming and computationally expensive [15], especially the hydrodynamic model combined with the structural dynamic equations of the flexible structures which includes the parametric excitation. In the paper of Yang and Xiao [12], they use harmonic function to simplify the hydrodynamic lift force without considering the fluid-structure interaction. But this interaction really exists for the dynamics of the pipeline. Therefore, to completely simulate the fluid-structure interaction, an efficient numerical model which is called the wake oscillator model is chosen to compute the hydrodynamic lift force. The model supposes that vortices behind cylinder structures can be simulated as a rigid oscillator connected with the structure, which is shown in Fig. 5, thereby the hydrodynamic force is able to be calculated through this combined system using structure mechanics. The model applied in present study is based on the van der Pol equation, which is given by [16] and developed by [17], in the following form:
∂t
2
1 = − fw DU 2 ˛, 2
t=0
(13)
The direct finite difference method is applied to solving these equations in the present study. The total length of the submerged floating pipeline is divided into 100 parts, and the increment of the time is set to be /1000ωv s to ensure the stability of the finite difference method. The total simulation time period t is 1200/ωv s. 2.4. Mathieu equation Because the structural dynamic equation from Eq. (11) is the quasi-linear partial differential equation, its solution can be influenced by the solution of its homogeneous equation. As a result, to learn the features of the structure dynamic equation, we could firstly study its homogeneous form:
∂ v ∂ v ∂v ∂ v + ct − T0 + Ta cos ωf t − ϕ = 0. + EI ∂t 2 ∂t ∂x4 ∂x2 2
mt
4
2
(14)
The modal superposition method is applied to solve Eq. (14), the cross-line displacement of the submerged floating pipeline section can be written as the sum of each natural mode’s solution:
v (x, t) =
+∞
i (x) pi (t)
i = 1, 2,3. . .,
(15)
i=1
in which pi (t) is called position function which is only affected by time; i (x) = sin(ix/L) is called vibration mode function which should cater to the boundary condition of the pipeline. Substituting Eq. (15) into Eq. (14), multiplying j (x) = sin(jx/L) at both sides of the equation, and then performing the
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
40
1.4
Wake Oscillator Model Khalak's Experiment (1999)
1.2
30
A/D 0.8
25
0.6
q j 20
0.4
15
0.2
10 2
4
6
Ur
8
10
12
j/L ct dpj 2 + + ωnj + mt dt mt
2
Ta cos ωf t − ϕ
(16)
((EI j/L
4
+ T0 j/L
2
)/mt ) represents the jth
natural circular frequency of the pipeline without the consideration of parametric excitation of increasing tension force. If substituting = ((ωf t − ϕ)/2) into Eq. (16), we can finally get the equation which is named as the damped Mathieu equation:
d 2
+ 2c
dpj d
+ aj + 2qj cos (2 ) pj = 0,
(17)
2 /ω2 ); q = (2 j/L in which c = (ct /mt ωf ); aj = (4ωnj j f
2
/mt ωf2 )Ta .
3. Results and discussions 3.1. Validation of the wake oscillator model To take an advantage of the wake oscillator model for computing the dynamic response of the submerged floating pipeline, we should make the validation of this hydrodynamic force model at first. It is necessary to make sure how the wake oscillator model could simulate the lock-in phenomenon during the vortex-induced vibration of the cylinder structure, like the pipeline. Here the numerical simulation by the wake oscillator model is compared with the experiment data from Khalak and Williamson [19]. In their experiment, the mass ratio of the cylinder is 10.1 and structural damping ratio is 0.0013. The theoretical model of that experiment done by Khalak and Williamson can be derived as the following form for the numerical simulation:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Unstable Unstable
0 0
pj = 0j = 1, 2,3. . .,
d2 pj
Unstable
Stable
14
integration along the x direction within x ∈ (0, L), the homogeneous structural equation can be simplified as
where ωnj =
Unstable
5
mt
d2 v dv 1 + kv = − fw DU 2 ˛, + ct 2 dt dt 2
d2 ˛ 4f 2 d˛ 1 d2 v 1 dv + ωv2 ˛ = − , − 2ωv (1 − 2 ˛2 ) − ωv2 2 ¯ U dt dt dt 2 dt Cl0 0.5D + l (18)
where k = (K/L), and K refers to total spring stiffness of the system. Because in the experiment by Khalak and Williamson, the cylinder structure was considered as a rigid body, its displacement would be only dependent on time. Therefore, we could use the Runge-Kutta Method to solve these equations. Here we set f equals to 1.56, and the numerical results and the experiment data are shown in Fig. 6, in which Ur = (2U/ωn D) means reduced veloc-
5
10
15
Stable 20
25
30
aj
Fig. 6. Comparison of the numerical solution of the wake oscillator model and the experiment data.
dt 2
c=0.1 c=0.0 c=0.2 c=0.3
35
1
0 0
d2 pj
269
Fig. 7. Stability diagram of the Mathieu equation.
ity, ωn = (k/mt ) is natural circular frequency, and A is amplitude of the vibration. As is shown in Fig. 6, the numerical result by the wake oscillator model can match the lower branch data of the vortex-induced vibration from the experiment [19], which is also similar to the conclusion from the study by Ogink and Metrikine [20], though it is still cannot match the upper branch data. Thus the wake oscillator model can be applied to the simulation of the lock-in phenomenon of the vortex-induced vibration of the submerged floating pipeline, although it is still not so accurate to simulate the upper branch of vortex-induced vibration. 3.2. Stability diagram of the Mathieu equation The homogeneous structural dynamic equation of the submerged floating pipeline can be derived as the damped Mathieu equation, which has been shown in Section 2.4. It is well-known that the Mathieu equation will have two different solutions. One is the stable solution, while the other is unstable. It is effective and visual to use stability diagram to describe the stability feature of the solution of Mathieu equation. The Bubnov-Galerkin approach [21] is applied to work out such stability diagram of Eq. (17) with different damping coefficients c. The diagrams can be observed in Fig. 7. In Fig. 7, unstable zones of the solutions are above the curves, while stable zones are below. Unlike the traditional vibration resonance, the parametric instability which is also called parametric resonance will occur in several areas rather than only if the frequency of outside excitation is similar to the natural frequency of the structure. These unstable zones are continuous and always start at the points where parameter aj equals to 1, 4, 9, 16,. . .. Besides, if the parameter qj that represents the value of the parametric excitation is large enough, the solution of the damped Mathieu equation is much likely to be unstable. In addition, with the bigger damping coefficient, the stable zones will also become bigger. 3.3. Engineering cases In this section, a submerged floating pipeline is designed to connect the FPSO with a tension leg platform (TLP). The design data of the submerged floating pipeline is based on the information of one of the submerged floating pipeline in the world [4]. Also we use the low frequency surge motion as the horizontal motion of the FPSO and TLP on the waves, according to the calculation from Tahar and Kim [22] and Low [23]. The configurations of our submerged floating pipeline and the motions of the floating structures are displayed in Table 1. The coupled structural and hydrodynamic force model of the submerged floating pipeline Eq. (11) is solved by
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Table 1 Configurations of the submerged floating pipeline and the motion of floating structures. Items
Configurations
Pipeline length (m) Pipeline diameter (O.D.) (m) Pipeline wall thickness (mm) Pipeline elastic modulus (Pa) Initial tension force (kN) Current speed (m/s) Water density (kg/m3 ) Surge motion amplitude of the FPSO (m) Surge motion amplitude of the TLP (m) Surge motion circular frequency (rad/s)
1000 1.2 100 2.1 × 1011 600 0.25 1000 10.0 4.0 0.0209
0.6 0.4 0.2
Fig. 10. Time history of the angle for the center of the pipeline under only vortex excitation.
v/D 0 -0.2
0.6
-0.4
0.4
-0.6 2500
0.2
2700
2900
ωv t
3100
3300
3500
v/D 0
Fig. 8. Dynamic response for the center of the pipeline under only vortex excitation.
-0.2 -0.4
0.6 0.4
2500
2700
2900
ωv t
3100
3300
3500
0.2 Fig. 11. Dynamic response for the center of the pipeline under combined excitations (a = 0.25%).
v/D 0 -0.2
0.6
-0.4
0.4
-0.6 0
200
400
x (m)
600
800
1000
Fig. 9. Vibration mode of the pipeline under only vortex excitation (ωv t = 1000 = 3142).
the direct finite difference method as is mentioned in Section 2.3. Here the structural damping coefficient is simplified as zero, but the hydrodynamic damping coefficient is calculated.
0.2
v/D 0 -0.2 -0.4 -0.6 0
200
400
600
800
1000
x (m) 3.3.1. Comparison of the pipeline dynamic responses subjected to only vortex excitation with combined excitation Most of the researches on the vortex-induced vibration of risers and pipelines assumed that the axial tension forces were constant with time [24,25], which means only vortex excitation is forced on the cylinder structure. Fig. 8 shows the time domain dynamic response of the center of the submerged floating pipeline under only vortex excitation, and Fig. 9 displaces its vibration mode. Meanwhile the time history of angle is also shown in Fig. 10 to validate the assumption that the vibration speed is much smaller than the current speed. The maximum of the angle is 0.3767, and in this condition sin = 0.368, tan = 0.396, and cos = 0.930. This means the assumption can be accepted to establish the dynamic model for VIV simulation.
Fig. 12. Vibration mode of the pipeline under combined excitations (a = 0.25%, ωv t = 1000 = 3142).
On the other hand, in the real engineering condition, because of the horizontal motion of the floating structures, the tension force of the submerged floating pipeline will change, which is derived in Section 2.1.2. So if we consider this parametric excitation sufficiently in the calculation, the dynamic response of the pipeline would be different. Here we set the motion transfer coefficient a equals to 0.25%, and the phase angle ˇ equals to zero. Fig. 10 shows the time domain response of the center of the pipeline subjected by both vortex and parametric excitations, and Fig. 12 also displaces its vibration mode.
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
0.6
150
0.4
100
0.2
(a)
(c)
(b)
50
v/D
v/D 0
0
-0.2
-50
-0.4
-100
2500
2700
2900
ωv t
3100
3300
-150 0
3500
Fig. 13. Dynamic response for the center of the pipeline under combined excitations (a = 0.15%).
As is seen in Fig. 11, the amplitude of the vibration is regular and convergent, which means the parametric resonance does not occur, and the dynamic response is still stable. However, comparing Fig. 11 with Fig. 8, even in the stable zone, the maximum of the dynamic response of the pipeline under combined excitations will be larger than that under only vortex excitation. This result represents that if designers design the pipeline system without considering its possible parametric excitation, the calculation response of the pipeline will be smaller, but the pipeline will have more dangerous dynamic responses in the real sea condition. Therefore it is necessary to take the parametric excitation into consideration, during the design process of a submerged floating pipeline between two floating structures. 3.3.2. Effects of the motion transfer coefficient on the stability of the submerged floating pipeline In the design of the submerged floating pipeline between the floating structures, the connection of the towheads and the floaters is significant, because it directly decides the transferring motion of the towheads and the value of the parametric excitation which will influence the dynamic response of the pipeline. Therefore it is necessary to study the effects of the motion transfer coefficient on the parametric stability of the pipeline. Here, three values of the transfer coefficient a are selected to explore how the coefficient can influence the stability of the pipeline, while the phase angle is zero in all three cases. Fig. 13 shows the time domain response of the pipeline center when the transfer coefficient equals to 0.15%. The dynamic response of the center of the pipeline when the transfer coefficient equals to 0.25% has been displayed in Fig. 11 in Section 3.3.1. Finally when the transfer coefficient is 0.5%, the time domain result at the pipeline center is diagramed in Fig. 14. As is shown in Fig. 13, the vibration of the pipeline is stable when the motion transfer coefficient equals to 0.15%, which is similar to the results when the coefficient is 0.25% seen in Fig. 11. However, there is still a difference that the maximum amplitude of the vibra-
(a)
271
8
(b) 10
4
5
v/D 0
v/D 0
-4
-5
1000
2000
ωv t
3000
4000
Fig. 14. Dynamic response for the center of the pipeline under combined excitations (a = 0.5%).
tion in Fig. 13 is lower than that in Fig. 11. It is easy to be understood that if the transfer coefficient is near to zero, the towheads will nearly not have the relative displacement, and the vibration will be close to the result when only considering the vortex excitation. In Fig. 14, the amplitude of the vibration reaches a significantly large value (v/D may reach 100), which implies the parametric resonance occurs and the solution of the vibration equation is unstable. Besides, it can be seen from Fig. 14 that the vibration of the pipeline is not symmetrical. In some time period, the value of the movement of the pipeline is often positive, while in other time period the value is negative. Finally, it is also interesting that the vibration mode of the pipeline changes with time when the parametric instability is happening. Fig. 15 shows the three process of the vibration mode within the total simulation time. In the first period, the mode is the third mode, shown in Fig. 15(a). Then it changes into the second mode, shown in Fig. 15(b). Finally it returns to the third mode at the last period of the calculation, shown in Fig. 15(c). It is confirmed that if the calculating time is much longer, this modes jump process will remain. This phenomenon is also witnessed in the study by Park and Jung [9]. The cause of such mode jump phenomenon can be considered as the effects of the combination of the lock-in area in vortex-induced vibration and variety of the natural frequency of the pipeline by the parametric excitation. Because the natural frequency of the vibration is changing with the various tension force, the ratio of the natural frequency to the vortex shedding frequency is also changing. Thereby the vibration may be located in the different lock-in areas. 3.3.3. Effects of the phase angle on the stability of the submerged floating pipeline The phase angle of the motion of the two floating structures would also influence the stability of the submerged floating pipeline, because it greatly decides the relative displacement of the floaters. Two cases that have the same transfer coefficient (a = 0.2%) are explored. The first case has the phase angle set to
(c)
2 1
v/D
0 -1
-8 0
200
400 600 x (m)
800
1000
-10 0
-2 200
400 600 x (m)
800
1000
-3 0
200
400 600 x (m)
800
1000
Fig. 15. Three process of the pipeline vibration mode jump (a = 0.5%): (a) ωv t = 150 = 471; (b) ωv t = 650 = 2042; (c) ωv t = 1200 = 3770.
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Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
0.6
40
(a)
0.4
(b)
20 0.2
v/D 0
v/D 0
-0.2
-20
-0.4 2500
2700
2900
ωv t
3100
3300
-40 0
3500
1000
2000
ωv t
3000
4000
Fig. 16. Dynamic response for the center of the pipeline under combined excitations (ˇ = 0◦ ).
Fig. 17. Dynamic response for the center of the pipeline under combined excitations (ˇ = 180◦ ).
be zero, while the other is 180◦ . In Fig. 16 the dynamic response of the center of the pipeline is displayed, and Fig. 17 shows the result when the phase angle is 180◦ . It is shown that the variety of the phase angle can also lead to the parametric instability of the submerged floating pipeline. At the lower phase angle, the vibration of the pipeline is more likely to be stable, because the axial tension excitation is smaller. The vibration mode jump is also observed when the parametric instability occurs in the second case (ˇ = 180◦ ). However, during the total simulation time, it just change one time, from the third mode to the second mode. The process of the vibration mode jump is diagramed in Fig. 18. It seems that how long the vibration can remain at a certain mode is dependent on the value of the parametric excitation (Fig. 18).
the submerged floating pipeline. According to the expression of the damping coefficient derived in Section 2, we can increase both the structural damping and hydrodynamic damping coefficient. There are several ways that can be used to increase the hydrodynamic damping, such as, adding bumps and helical strakes on surface of the pipelines [27,28]. Another method for avoiding the parametric instability is to decrease the value of the parametric excitation. As the stability diagram show, if qj is smaller, the stable zone will be much larger. It is derived in Section 2.4 that the increment of the axial tension force directly influences the parameter qj in the Mathieu equation. According to its expression and the discussion we have made in Sections 3.3.2 and 3.3.3, there are also several ways to reduce the value of the parametric excitation. For examples, designing a more effective mooring system to decrease the motion of the floating structures, using dynamic positioning system on the towheads to reduce the motion transfer coefficient, and choosing the proper distance between two floating structures so that the phase angle of their motion could be small enough can make great contributions to avoiding the parametric instability of the pipeline.
3.4. Safety design suggestions to avoid the parametric instability of the submerged floating pipeline It is dangerous for the submerged floating pipeline if the parametric instability happens. Because the pipeline section will have large displacement which could be observed in Figs. 14 and 17, it is much easier for the pipeline to surfer from fatigue damage. So it is quite necessary to take some measures to avoid the parametric instability for the pipeline structure’s safety. One measure is to increase the damping of the submerged floating pipeline system. The damping is considered as one of the most effective parameters to control the unstable zone and avoid the parametric resonance [26]. As is shown from the stability diagram in Section 3.2, the increasing system damping results in that the unstable zone reduces. Especially the proper damping will make the unstable zone disappeared when the value of the parametric excitation is very small. Thereby increasing the system damping coefficient can reduce the possibility of the parametric instability of
(a) 40
4. Conclusions The parametric instability of a submerged floating pipeline between two floating structures has been studied according to the derived equations which are coupled with the structural model and hydrodynamic force model. The validation of the wake oscillator model is carried out, and the stability diagram of the Mathieu equation is used to explain the stability of the solution of the motion equation. Engineering cases are applied for this work by the finite difference method, and the effects of some important design parameters are discussed. Finally some effective measures to avoid the parametric instability are provided for the guidance of
(b)
6 4
20 2
v/D 0
v/D 0 -2
-20 -4 -40 0
200
400
600
x (m)
800
1000
-6 0
200
400
x (m)
600
800
1000
Fig. 18. The process of the pipeline vibration mode jump (ˇ = 180◦ ): (a) ωv t = 240 = 754; (b) ωv t = 720 = 2262.
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
the submerged floating pipeline’s design. From the present study, the following conclusions are drawn: (1) The dynamic responses of the submerged floating pipeline under only vortex excitation and combined vortex and parametric excitations are different. Even if the parametric instability does not occur, the maximum vibration amplitude of the pipeline under combined excitations will be larger than that subjected to only vortex excitation. (2) The parametric instability can occur in the cross-line vibration of the submerged floating pipeline when the fluctuation of its axial tension force is big enough. The parametric instability can result in a very large displacement to the pipeline, which will lead to the serious structural failure and fatigue damage. (3) There is a vibration mode jump phenomenon when the parametric instability occurs, which means that the vibration mode can change periodically with the time. This is due to the lock-in phenomenon in the vortex-induced vibration and the variety of the natural frequency of the pipeline’s transverse vibration. (4) The damping coefficient and the value of the parametric excitation can make a great impact on the dynamic stability of the submerged floating pipeline. Increasing the damping coefficient including the structural and hydrodynamic damping coefficient, lowering the motion transfer coefficient and reducing the phase angle of the motion of two floating structures are effective to avoid the parametric instability. Finally, this study is only about the theoretic method to deal with the parametric instability prediction for the submerged floating pipeline between the floating structures. Experiment analysis will be carried out in the future research. Acknowledgment This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51379005 and 51009093). References [1] O. Fyrileiv, O. Aamlid, A. Venas, L. Collberg, Deepwater pipelines—status, challenges and future trends, Proc. Inst. Mech. Eng. M: J. Eng. Marit. Environ. 227 (4) (2012) 381–395, http://dx.doi.org/10.1177/1475090212450659. [2] N. Kruijt, Turkey-Cyprus Submerged Floating Freshwater Pipeline, Delft University of Technology, Netherlands, 2003. [3] G. Paulsen, T.H. Søreide, F.G. Nielsen, Submerged floating pipeline in deep water, Seattle, USA, in: Proc. of 10th Int. Offshore Polar Eng. Conf., vol. II, 2000, pp. 108–114. [4] M. Lemoël, P. Brown, P. Jean, B. Shepheard, Design of the world’s 1st gravity actuated pipe (GAP) for Murphy’s Kikeh deepwater development, east Malaysia, in: Proc. ASME 27th Int. Conf. Offshore Mech. Arct. Eng., Estoril, Portugal, 2008, http://dx.doi.org/10.1115/OMAE2008-57533 (OMAE2008-57533). [5] Y. Gao, S. Fu, T. Ren, Y. Xiong, L. Song, VIV response of a long flexible riser fitted with strakes in uniform and linearly sheared currents, Appl. Ocean Res. 52 (2015) 102–114, http://dx.doi.org/10.1016/j.apor.2015.05.006. [6] R. Govardhan, C.H.K. Williamson, Critical mass in vortex-induced vibration of a cylinder, Eur. J. Mech.: B/Fluids 23 (2004) 17–27, http://dx.doi.org/10.1016/ j.euromechflu.2003.04.001.
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Parametric instability of a submerged floating pipeline between two floating structures under combined vortex excitations Zhiqian Wang, Hezhen Yang ∗ School of Naval Architecture, Ocean and Civil Engineering, State Key Lab of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, China
a r t i c l e
i n f o
Article history: Received 29 October 2015 Received in revised form 13 April 2016 Accepted 14 June 2016 Keywords: Submerged floating pipelines Parametric instability Vortex-induced vibration Dynamic response Mathieu equation
a b s t r a c t This study investigates a parametric instability prediction of a submerged floating pipeline connected between two floating structures under combined vortex excitations. The submerged floating pipeline is considered as a potential solution for fluid transportation between floaters in deepwater. However, the parametric instability combined with vortex-induced vibration is a major concern for its safety. The instability can lead to huge displacement and fatigue damage. Thus, it is essential to propose a methodology for analyzing nonlinear dynamic properties of the pipeline with combined parametric and vortex excitations. Here, coupled vibration equations of the pipeline are established, including hydrodynamic force model which contains vortex-induced load and structural model which contains parametric excitation due to motion of the two floating structures. Then Mathieu equation is derived from the homogeneous structural equation. Differences between the dynamic responses subjected to only vortex excitation and combined excitations are compared. The pipeline engineering cases and the effects of two important design factors are analyzed. The results show that the parametric instability may occur in cross-line vibration of the submerged floating pipeline between the two floaters. Even the maximum vibration amplitude of the pipeline under combined excitations is still larger than that under only vortex excitation, when the parametric instability does not take place. There is a vibration mode jump phenomenon while the parametric instability occurs. In addition, two design factors, transfer coefficient and phase angle of the floaters’ motion, can make a great impact on the parametric stability of the submerged floating pipeline. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Offshore risers and pipelines are becoming more and more important while the exploration of marine oil and gas is moving in deep and ultra-deep water. There are several new concepts for the design of offshore risers and pipelines in recent years. The submerged floating pipeline is one of these new concepts to transfer fluid and gas in ocean engineering [1]. Generally speaking, the submerged floating pipeline is floating at a certain depth of ocean with some mooring chains or tethers anchoring it to the seabed [2,3]. On the other hand, it can also be used to connect two floating structures, shown in Fig. 1, for example, the gravity actuated pipe which connects the FPSO and Spar platform at Kikeh outside Malaysia installed in 2008 [4]. Compared with the traditional pipelines that are put on or buried in the seabed, the submerged floating pipeline is considered as a more useful solution for offshore engineering with some challenging seabed
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (H. Yang). http://dx.doi.org/10.1016/j.apor.2016.06.009 0141-1187/© 2016 Elsevier Ltd. All rights reserved.
condition. First of all, there is no need to do any intervention work for the seabed because the pipeline is floating at the mid-deep water without the direct interaction with the seabed soil. Besides the submerged floating pipeline is located at a certain depth under the sea surface, so wave forces can also be ignored. Finally, high hydrostatic pressure and temperature difference in the deep and ultra-deep ocean are avoided as well. Therefore, the submerged floating pipeline could prove to be a part of the future solutions for deep water gas and fluid transportation. Although many complex environmental effects such as the wave forces and soil-pipeline interaction forces do not need to be considered for the submerged floating pipeline’s design, current forces and vortices which are also produced by the current are the major concerns. Nonlinear dynamic response for the submerged floating pipeline is a vital analysis task for its safety. And, it is necessary to study vortex-induced vibration which is thought to be the important cause of fatigue damage of the offshore risers and pipelines [5]. There are a lot of researches done by both experimental tests and numerical predictions on the vortex-induced vibration of such a circular cylinder structure. Govardhan and Williamson [6] applied several experiment tests for the vortex-induced vibration of the
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be interacted together on the pipeline. But few analysis methods consider both parametric and vortex-induced vibration excitations in the past few years [12]. The present study is meant to establish a suitable mathematic model to explore the stability of the submerged floating pipeline between two floating structures under vortex and parametric excitations. In Section 2, the governing motion equations of the pipeline subjected to combined excitations are derived, including structural model and wake oscillator hydrodynamic force model. Mathieu equation is also derived from the homogeneous structural equation. In Section 3, validation of the hydrodynamic force model is made, and the Mathieu equation is solved. The stability diagram is made for the submerged floating pipeline design. Then engineering cases and effects of two important design factors on the stability of the pipeline are discussed, and finally some suggestions to avoid the parametric instability are given. Fig 1. Submerged floating pipeline between two floating structures.
cylinder structure at low mass ratio value, and found the response could be divided into the initial, upper and lower branches. Srinil et al. [7] used a classical van der Pol wake oscillator model to simulate the fluid-structure interaction at both in-line and cross-line direction, and compared the numerical results with his experiments. Because the submerged floating pipeline can be connected with the floating structures, the motion of these floaters will also influence the dynamic response of the pipelines, which may lead to parametric instability problem of the pipeline. The parametric instability, also called parametric resonance, is another major concern for offshore structures’ design, because it can excite the very large motion [8]. In the past decade, there are several studies on analyzing the parametric instability of tethers and risers of top tensioned platforms. Park and Jung [9] implemented a finite element method to study the lateral responses of a top tensioned platform tether under combined parametric and forcing excitations. Franzini et al. [10] analyzed the parametric instability of a flexible tensioned cylinder under harmonic excitation through experiment tests. For the recent researches of the submerged floating pipeline, most of them concentrated on its static design analysis [2] and vortex-induced vibration. For example, Sergent et al. [11] applied CFD, experiments and software to analyze the fatigue life of the submerged floating pipeline because of vortex-induced vibration. Thus it is still necessary and urgent to propose a methodology to predict the parametric instability of such a pipeline between the floating structures and learn how the motion of the floaters can influence its dynamic stability. Besides, it is also important to combine the vortex-induced forces with the parametric excitation, and study the dynamic response of the pipeline under such combined excitations, because in the real sea condition, these two excitations must
2. Theory The nonlinear dynamic motion of the submerged floating pipeline is complicated, because it involves the fluid-structure interaction which means that the motion of the structure and fluid dynamic forces affects each other. Therefore to simulate this interaction, the system model could be divided into two sub-models. The first one is called the structural model reflecting the transverse and axial vibration of the submerged floating pipeline under the hydrodynamic forces, while the second one is the hydrodynamic force model showing how to calculate the hydrodynamic forces with the dynamic response of the pipeline. Before derivation of the motion equations, the following assumptions are listed to simplify the physical models: (1) The material, mechanical properties and current speed are assumed constant along the length of the pipeline and cannot change with time. (2) The buoyance and gravity of the pipeline per unit length are thought to be equal, so that the pipeline can be straight in initial condition. (3) Only the cross-line vibration of the pipeline is considered. (4) The motion of the floating structures is simplified as a harmonic function in the uniform wave. 2.1. Structural model 2.1.1. Cross-line vibration equation of the pipeline The schematic of the submerged floating pipeline system is shown in Fig. 2, while the configuration of the submerged floating pipeline can be described in Fig. 3. In this study, the pipeline is simplified as an Euler-Bernoulli beam-column with two ends hinged.
Fig. 2. Schematic of the submerged floating pipeline system.
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
267
Fig. 3. Configuration of the submerged floating pipeline.
2.1.2. Axial vibration and parametric excitation Because the submerged floating pipeline is suspended between two floating structures, the motion of the floaters such as FPSO, Spar and TLP may influence the dynamics of the pipeline. The towheads at two ends of the pipeline will have displacement which follows the motion of the floaters. If two towheads have relative displacement at the direction along the length of the pipeline, then axial vibration of the submerged floating pipeline may take place. In addition, because the floating structures do the slow-drift motion on the sea surface, the main frequency of this horizontal motion is very low, at about 0.02 rad/s, while the first natural frequency of the axial vibration of the pipeline is quite large, at above 10 rad/s. So the quasi-static method can be applied to analyzing this effect rather than using differential equations of the axial vibration. By the quasi-static method, the relationship of increasing tension force of the submerged floating pipeline and increment of its total length can be established. The function can be derived as
Fig. 4. Hydrodynamic forces on the submerged floating pipeline section.
The differential equation for the cross-line vibration of the submerged floating pipeline can be derived as the following equation, according to Huarte [13]: 2
(m + ma )
4
2
∂ v ∂v ∂ v ∂ v + cs − T (t) = fh , + EI ∂t 2 ∂t ∂x4 ∂x2
(1)
where m is the mass of the pipeline per unit length; ma = (/4)Ca w D2 means the added mass for the pipeline per unit length, in which Ca is the added mass coefficient and equals to 1.0, w is density of the sea water, and D is outer diameter of the pipeline; v is the deflection of the pipeline section; cs means the structural damping coefficient per unit length; E is elastic modulus; I is moment of inertia of the pipeline section; T (t) is axial tension force; fh is the hydrodynamic force (not including the initial mass item, because it has been put in the left-hand side of the equation) on the pipeline section at the cross-line direction. The hydrodynamic forces on the submerged floating pipeline section can be described as the lift force fl and drag force fd , shown in Fig. 4. According to the figure, the total hydrodynamic force on the pipeline section at the cross-line direction is derived as: fh = fl cos − fd sin =
1 1 C w DU 2 cos − Cd w DU 2 sin , 2 l 2
(2)
where U is the current speed; Cl is the lift force coefficient; Cd is the drag force coefficient, and here is chosen to be constantly 1.2 [7]; tan = (1/U)(∂v/∂t) means the ratio of vibration speed of the pipeline section to the current speed. By the assumption that the vibration speed is much smaller than the current speed, which is based on the recommended vortex-induced vibration model by Facchinetti et al. [14], there is a relationship that cos ≈ 1 and sin ≈ tan . Therefore the hydrodynamic force at the cross-line direction can be finally described in the following form: fh =
∂v 1 1 C w DU 2 − Cd w DU . 2 l 2 ∂t
(3)
l T = ε ≈ , EA L
(4)
in which T represents the increasing tension force; A is area of the pipeline section; ε is increment of strain; l is the increment of the length of the pipeline; L is the length of the pipeline. According to Fig. 2, we assume that the horizontal motion of the floating structures at the direction along the pipeline could be considered as a harmonic function with the same frequency on the uniform waves to simplify the problem. Thus the left and right floating structures will move as the following equations:
sl = S1 cos ωf t ,
sr = S2 cos ωf t + ˇ ,
(5)
where sl and sr are the displacement of two floating structures; S1 and S2 are their amplitudes; ωf is the frequency of the motion of two floating structures; ˇ is phase angle between two floaters’ motions. Therefore the relative motion of two floating structures can be written as
s = sl − sr = S1 cos ωf t − S2 cos ωf t + ˇ .
(6)
There must be a relationship between the motion of floating structures and the displacement of the towheads, however many complex factors may affect the relationship, such as the design of the connection between the floating structure and the towhead, tensile rigidity of the pipeline, initial tension force and so on. To simplify this relationship, a transfer coefficient which means the influence of the motion of floaters on the displacement of towheads is established as a=
l . s
(7)
Substituting Eqs. (6) and (7) into Eq. (4), we can derive the increment of the tension force, which is thought as one of the parametric excitations to the submerged floating pipeline, in the following form:
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√
= (fD/2 22 ¯l) based on Tamura’s theory is the self-excitation coefficient, in which f is the coefficient of slope of the lift force and ¯l is the mean half-length of the wake oscillator; Cl0 is the lift force coefficient of stationary cylinder. The value of St, f and ¯l can be determined from experiments, and it is proved that they are dependent on the Reynolds number. Thus the real effects of the Reynolds number on VIV can also be contained in the wake oscillator model. According to the research from Tamura and Matsui [16] and Doan and Nishi [18], St equals to 0.2 and ¯l = 1.1D, and the recommended value of the lift force coefficient of stationary cylinder Cl0 is set to be 0.4. 2.3. Coupling structural model and hydrodynamic force model According to Eq. (1), Eq. (3), Eq. (9) and Eq. (10), the structural model could be combined with the hydrodynamic force model as
⎧ 2 4 ∂2 v ∂ v ∂v ∂ v ⎪ ⎪ ⎨ mt ∂t 2 + ct ∂t + EI ∂x4 − T0 + Ta cos ωf t − ϕ ∂x2 ⎪ ⎪ ⎩
Fig. 5. Schematic illustrations for the wake oscillator model.
T =
aEA S1 cos ωf t − S2 cos ωf t + ˇ = Ta cos ωf t − ϕ , L (8)
where Ta = (aEA/L)
S1 − S2 cos ˇ
2
+ S2 sin ˇ
2
, and ϕ =
arctan(S2 sinˇ/(S1 − S2 cosˇ)). Finally the tension force of the pipeline can be written in the complete form as
T (t) = T0 + T = T0 + Ta cos ωf t − ϕ ,
(9)
where T0 is the initial tension force of the submerged floating pipeline. 2.2. Hydrodynamic force model
⎧ 2 ⎪ ∂˛ 4f 2 ⎪ ⎨∂ ˛ − 2ωv (1 − 2 ˛2 ) + ωv2 ˛ = − 2 ⎪ ⎪ ⎩
Cl0
∂t
Cl = −f (˛ +
2
∂ v 1 ∂v − ωv2 2 ¯ U ∂ ∂t , t 0.5D + l
1 ∂v ) U ∂t
1
(10)
where ˛ is the angle of the oscillator; ωv = 2StU/D means vortex shedding circular frequency, in which St is the Strouhal number;
(11)
2
4f 2 1 ∂ ˛ ∂˛ ∂ v 1 ∂v + ωv2 ˛ = − , − 2ωv (1 − 2 ˛2 ) − ωv2 U ∂t Cl0 ∂t ∂t 2 0.5D + ¯l ∂t 2
where mt = m + ma means the total mass per unit length; ct = cs + (1/2) (f + Cd ) w DU is the total damping coefficient per unit length, including the structural and hydrodynamic damping coefficient. These are the equations that can represent the cross-line vibration of the submerged floating pipeline, and to solve them, the following boundary and initial conditions are needed:
⎧ ⎨
v|x=0 = v|x=L = 0, 2
⎩ ∂ v| 2 ⎧ ⎪ ⎨
∂x
(12)
2
x=0
=
∂ v | = 0. ∂x2 x=L
∂v = 0, | ∂t t=0 ∂˛ C = − l0 , = 0. | f ∂t t=0
v|t=0 =
⎪ ⎩ ˛|
The most direct method to get the hydrodynamic force data on the vibrating cylinder is to carry out experiments or direct numerical simulation (DNS). However, both of the methods are very time consuming and computationally expensive [15], especially the hydrodynamic model combined with the structural dynamic equations of the flexible structures which includes the parametric excitation. In the paper of Yang and Xiao [12], they use harmonic function to simplify the hydrodynamic lift force without considering the fluid-structure interaction. But this interaction really exists for the dynamics of the pipeline. Therefore, to completely simulate the fluid-structure interaction, an efficient numerical model which is called the wake oscillator model is chosen to compute the hydrodynamic lift force. The model supposes that vortices behind cylinder structures can be simulated as a rigid oscillator connected with the structure, which is shown in Fig. 5, thereby the hydrodynamic force is able to be calculated through this combined system using structure mechanics. The model applied in present study is based on the van der Pol equation, which is given by [16] and developed by [17], in the following form:
∂t
2
1 = − fw DU 2 ˛, 2
t=0
(13)
The direct finite difference method is applied to solving these equations in the present study. The total length of the submerged floating pipeline is divided into 100 parts, and the increment of the time is set to be /1000ωv s to ensure the stability of the finite difference method. The total simulation time period t is 1200/ωv s. 2.4. Mathieu equation Because the structural dynamic equation from Eq. (11) is the quasi-linear partial differential equation, its solution can be influenced by the solution of its homogeneous equation. As a result, to learn the features of the structure dynamic equation, we could firstly study its homogeneous form:
∂ v ∂ v ∂v ∂ v + ct − T0 + Ta cos ωf t − ϕ = 0. + EI ∂t 2 ∂t ∂x4 ∂x2 2
mt
4
2
(14)
The modal superposition method is applied to solve Eq. (14), the cross-line displacement of the submerged floating pipeline section can be written as the sum of each natural mode’s solution:
v (x, t) =
+∞
i (x) pi (t)
i = 1, 2,3. . .,
(15)
i=1
in which pi (t) is called position function which is only affected by time; i (x) = sin(ix/L) is called vibration mode function which should cater to the boundary condition of the pipeline. Substituting Eq. (15) into Eq. (14), multiplying j (x) = sin(jx/L) at both sides of the equation, and then performing the
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
40
1.4
Wake Oscillator Model Khalak's Experiment (1999)
1.2
30
A/D 0.8
25
0.6
q j 20
0.4
15
0.2
10 2
4
6
Ur
8
10
12
j/L ct dpj 2 + + ωnj + mt dt mt
2
Ta cos ωf t − ϕ
(16)
((EI j/L
4
+ T0 j/L
2
)/mt ) represents the jth
natural circular frequency of the pipeline without the consideration of parametric excitation of increasing tension force. If substituting = ((ωf t − ϕ)/2) into Eq. (16), we can finally get the equation which is named as the damped Mathieu equation:
d 2
+ 2c
dpj d
+ aj + 2qj cos (2 ) pj = 0,
(17)
2 /ω2 ); q = (2 j/L in which c = (ct /mt ωf ); aj = (4ωnj j f
2
/mt ωf2 )Ta .
3. Results and discussions 3.1. Validation of the wake oscillator model To take an advantage of the wake oscillator model for computing the dynamic response of the submerged floating pipeline, we should make the validation of this hydrodynamic force model at first. It is necessary to make sure how the wake oscillator model could simulate the lock-in phenomenon during the vortex-induced vibration of the cylinder structure, like the pipeline. Here the numerical simulation by the wake oscillator model is compared with the experiment data from Khalak and Williamson [19]. In their experiment, the mass ratio of the cylinder is 10.1 and structural damping ratio is 0.0013. The theoretical model of that experiment done by Khalak and Williamson can be derived as the following form for the numerical simulation:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
Unstable Unstable
0 0
pj = 0j = 1, 2,3. . .,
d2 pj
Unstable
Stable
14
integration along the x direction within x ∈ (0, L), the homogeneous structural equation can be simplified as
where ωnj =
Unstable
5
mt
d2 v dv 1 + kv = − fw DU 2 ˛, + ct 2 dt dt 2
d2 ˛ 4f 2 d˛ 1 d2 v 1 dv + ωv2 ˛ = − , − 2ωv (1 − 2 ˛2 ) − ωv2 2 ¯ U dt dt dt 2 dt Cl0 0.5D + l (18)
where k = (K/L), and K refers to total spring stiffness of the system. Because in the experiment by Khalak and Williamson, the cylinder structure was considered as a rigid body, its displacement would be only dependent on time. Therefore, we could use the Runge-Kutta Method to solve these equations. Here we set f equals to 1.56, and the numerical results and the experiment data are shown in Fig. 6, in which Ur = (2U/ωn D) means reduced veloc-
5
10
15
Stable 20
25
30
aj
Fig. 6. Comparison of the numerical solution of the wake oscillator model and the experiment data.
dt 2
c=0.1 c=0.0 c=0.2 c=0.3
35
1
0 0
d2 pj
269
Fig. 7. Stability diagram of the Mathieu equation.
ity, ωn = (k/mt ) is natural circular frequency, and A is amplitude of the vibration. As is shown in Fig. 6, the numerical result by the wake oscillator model can match the lower branch data of the vortex-induced vibration from the experiment [19], which is also similar to the conclusion from the study by Ogink and Metrikine [20], though it is still cannot match the upper branch data. Thus the wake oscillator model can be applied to the simulation of the lock-in phenomenon of the vortex-induced vibration of the submerged floating pipeline, although it is still not so accurate to simulate the upper branch of vortex-induced vibration. 3.2. Stability diagram of the Mathieu equation The homogeneous structural dynamic equation of the submerged floating pipeline can be derived as the damped Mathieu equation, which has been shown in Section 2.4. It is well-known that the Mathieu equation will have two different solutions. One is the stable solution, while the other is unstable. It is effective and visual to use stability diagram to describe the stability feature of the solution of Mathieu equation. The Bubnov-Galerkin approach [21] is applied to work out such stability diagram of Eq. (17) with different damping coefficients c. The diagrams can be observed in Fig. 7. In Fig. 7, unstable zones of the solutions are above the curves, while stable zones are below. Unlike the traditional vibration resonance, the parametric instability which is also called parametric resonance will occur in several areas rather than only if the frequency of outside excitation is similar to the natural frequency of the structure. These unstable zones are continuous and always start at the points where parameter aj equals to 1, 4, 9, 16,. . .. Besides, if the parameter qj that represents the value of the parametric excitation is large enough, the solution of the damped Mathieu equation is much likely to be unstable. In addition, with the bigger damping coefficient, the stable zones will also become bigger. 3.3. Engineering cases In this section, a submerged floating pipeline is designed to connect the FPSO with a tension leg platform (TLP). The design data of the submerged floating pipeline is based on the information of one of the submerged floating pipeline in the world [4]. Also we use the low frequency surge motion as the horizontal motion of the FPSO and TLP on the waves, according to the calculation from Tahar and Kim [22] and Low [23]. The configurations of our submerged floating pipeline and the motions of the floating structures are displayed in Table 1. The coupled structural and hydrodynamic force model of the submerged floating pipeline Eq. (11) is solved by
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Table 1 Configurations of the submerged floating pipeline and the motion of floating structures. Items
Configurations
Pipeline length (m) Pipeline diameter (O.D.) (m) Pipeline wall thickness (mm) Pipeline elastic modulus (Pa) Initial tension force (kN) Current speed (m/s) Water density (kg/m3 ) Surge motion amplitude of the FPSO (m) Surge motion amplitude of the TLP (m) Surge motion circular frequency (rad/s)
1000 1.2 100 2.1 × 1011 600 0.25 1000 10.0 4.0 0.0209
0.6 0.4 0.2
Fig. 10. Time history of the angle for the center of the pipeline under only vortex excitation.
v/D 0 -0.2
0.6
-0.4
0.4
-0.6 2500
0.2
2700
2900
ωv t
3100
3300
3500
v/D 0
Fig. 8. Dynamic response for the center of the pipeline under only vortex excitation.
-0.2 -0.4
0.6 0.4
2500
2700
2900
ωv t
3100
3300
3500
0.2 Fig. 11. Dynamic response for the center of the pipeline under combined excitations (a = 0.25%).
v/D 0 -0.2
0.6
-0.4
0.4
-0.6 0
200
400
x (m)
600
800
1000
Fig. 9. Vibration mode of the pipeline under only vortex excitation (ωv t = 1000 = 3142).
the direct finite difference method as is mentioned in Section 2.3. Here the structural damping coefficient is simplified as zero, but the hydrodynamic damping coefficient is calculated.
0.2
v/D 0 -0.2 -0.4 -0.6 0
200
400
600
800
1000
x (m) 3.3.1. Comparison of the pipeline dynamic responses subjected to only vortex excitation with combined excitation Most of the researches on the vortex-induced vibration of risers and pipelines assumed that the axial tension forces were constant with time [24,25], which means only vortex excitation is forced on the cylinder structure. Fig. 8 shows the time domain dynamic response of the center of the submerged floating pipeline under only vortex excitation, and Fig. 9 displaces its vibration mode. Meanwhile the time history of angle is also shown in Fig. 10 to validate the assumption that the vibration speed is much smaller than the current speed. The maximum of the angle is 0.3767, and in this condition sin = 0.368, tan = 0.396, and cos = 0.930. This means the assumption can be accepted to establish the dynamic model for VIV simulation.
Fig. 12. Vibration mode of the pipeline under combined excitations (a = 0.25%, ωv t = 1000 = 3142).
On the other hand, in the real engineering condition, because of the horizontal motion of the floating structures, the tension force of the submerged floating pipeline will change, which is derived in Section 2.1.2. So if we consider this parametric excitation sufficiently in the calculation, the dynamic response of the pipeline would be different. Here we set the motion transfer coefficient a equals to 0.25%, and the phase angle ˇ equals to zero. Fig. 10 shows the time domain response of the center of the pipeline subjected by both vortex and parametric excitations, and Fig. 12 also displaces its vibration mode.
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
0.6
150
0.4
100
0.2
(a)
(c)
(b)
50
v/D
v/D 0
0
-0.2
-50
-0.4
-100
2500
2700
2900
ωv t
3100
3300
-150 0
3500
Fig. 13. Dynamic response for the center of the pipeline under combined excitations (a = 0.15%).
As is seen in Fig. 11, the amplitude of the vibration is regular and convergent, which means the parametric resonance does not occur, and the dynamic response is still stable. However, comparing Fig. 11 with Fig. 8, even in the stable zone, the maximum of the dynamic response of the pipeline under combined excitations will be larger than that under only vortex excitation. This result represents that if designers design the pipeline system without considering its possible parametric excitation, the calculation response of the pipeline will be smaller, but the pipeline will have more dangerous dynamic responses in the real sea condition. Therefore it is necessary to take the parametric excitation into consideration, during the design process of a submerged floating pipeline between two floating structures. 3.3.2. Effects of the motion transfer coefficient on the stability of the submerged floating pipeline In the design of the submerged floating pipeline between the floating structures, the connection of the towheads and the floaters is significant, because it directly decides the transferring motion of the towheads and the value of the parametric excitation which will influence the dynamic response of the pipeline. Therefore it is necessary to study the effects of the motion transfer coefficient on the parametric stability of the pipeline. Here, three values of the transfer coefficient a are selected to explore how the coefficient can influence the stability of the pipeline, while the phase angle is zero in all three cases. Fig. 13 shows the time domain response of the pipeline center when the transfer coefficient equals to 0.15%. The dynamic response of the center of the pipeline when the transfer coefficient equals to 0.25% has been displayed in Fig. 11 in Section 3.3.1. Finally when the transfer coefficient is 0.5%, the time domain result at the pipeline center is diagramed in Fig. 14. As is shown in Fig. 13, the vibration of the pipeline is stable when the motion transfer coefficient equals to 0.15%, which is similar to the results when the coefficient is 0.25% seen in Fig. 11. However, there is still a difference that the maximum amplitude of the vibra-
(a)
271
8
(b) 10
4
5
v/D 0
v/D 0
-4
-5
1000
2000
ωv t
3000
4000
Fig. 14. Dynamic response for the center of the pipeline under combined excitations (a = 0.5%).
tion in Fig. 13 is lower than that in Fig. 11. It is easy to be understood that if the transfer coefficient is near to zero, the towheads will nearly not have the relative displacement, and the vibration will be close to the result when only considering the vortex excitation. In Fig. 14, the amplitude of the vibration reaches a significantly large value (v/D may reach 100), which implies the parametric resonance occurs and the solution of the vibration equation is unstable. Besides, it can be seen from Fig. 14 that the vibration of the pipeline is not symmetrical. In some time period, the value of the movement of the pipeline is often positive, while in other time period the value is negative. Finally, it is also interesting that the vibration mode of the pipeline changes with time when the parametric instability is happening. Fig. 15 shows the three process of the vibration mode within the total simulation time. In the first period, the mode is the third mode, shown in Fig. 15(a). Then it changes into the second mode, shown in Fig. 15(b). Finally it returns to the third mode at the last period of the calculation, shown in Fig. 15(c). It is confirmed that if the calculating time is much longer, this modes jump process will remain. This phenomenon is also witnessed in the study by Park and Jung [9]. The cause of such mode jump phenomenon can be considered as the effects of the combination of the lock-in area in vortex-induced vibration and variety of the natural frequency of the pipeline by the parametric excitation. Because the natural frequency of the vibration is changing with the various tension force, the ratio of the natural frequency to the vortex shedding frequency is also changing. Thereby the vibration may be located in the different lock-in areas. 3.3.3. Effects of the phase angle on the stability of the submerged floating pipeline The phase angle of the motion of the two floating structures would also influence the stability of the submerged floating pipeline, because it greatly decides the relative displacement of the floaters. Two cases that have the same transfer coefficient (a = 0.2%) are explored. The first case has the phase angle set to
(c)
2 1
v/D
0 -1
-8 0
200
400 600 x (m)
800
1000
-10 0
-2 200
400 600 x (m)
800
1000
-3 0
200
400 600 x (m)
800
1000
Fig. 15. Three process of the pipeline vibration mode jump (a = 0.5%): (a) ωv t = 150 = 471; (b) ωv t = 650 = 2042; (c) ωv t = 1200 = 3770.
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Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
0.6
40
(a)
0.4
(b)
20 0.2
v/D 0
v/D 0
-0.2
-20
-0.4 2500
2700
2900
ωv t
3100
3300
-40 0
3500
1000
2000
ωv t
3000
4000
Fig. 16. Dynamic response for the center of the pipeline under combined excitations (ˇ = 0◦ ).
Fig. 17. Dynamic response for the center of the pipeline under combined excitations (ˇ = 180◦ ).
be zero, while the other is 180◦ . In Fig. 16 the dynamic response of the center of the pipeline is displayed, and Fig. 17 shows the result when the phase angle is 180◦ . It is shown that the variety of the phase angle can also lead to the parametric instability of the submerged floating pipeline. At the lower phase angle, the vibration of the pipeline is more likely to be stable, because the axial tension excitation is smaller. The vibration mode jump is also observed when the parametric instability occurs in the second case (ˇ = 180◦ ). However, during the total simulation time, it just change one time, from the third mode to the second mode. The process of the vibration mode jump is diagramed in Fig. 18. It seems that how long the vibration can remain at a certain mode is dependent on the value of the parametric excitation (Fig. 18).
the submerged floating pipeline. According to the expression of the damping coefficient derived in Section 2, we can increase both the structural damping and hydrodynamic damping coefficient. There are several ways that can be used to increase the hydrodynamic damping, such as, adding bumps and helical strakes on surface of the pipelines [27,28]. Another method for avoiding the parametric instability is to decrease the value of the parametric excitation. As the stability diagram show, if qj is smaller, the stable zone will be much larger. It is derived in Section 2.4 that the increment of the axial tension force directly influences the parameter qj in the Mathieu equation. According to its expression and the discussion we have made in Sections 3.3.2 and 3.3.3, there are also several ways to reduce the value of the parametric excitation. For examples, designing a more effective mooring system to decrease the motion of the floating structures, using dynamic positioning system on the towheads to reduce the motion transfer coefficient, and choosing the proper distance between two floating structures so that the phase angle of their motion could be small enough can make great contributions to avoiding the parametric instability of the pipeline.
3.4. Safety design suggestions to avoid the parametric instability of the submerged floating pipeline It is dangerous for the submerged floating pipeline if the parametric instability happens. Because the pipeline section will have large displacement which could be observed in Figs. 14 and 17, it is much easier for the pipeline to surfer from fatigue damage. So it is quite necessary to take some measures to avoid the parametric instability for the pipeline structure’s safety. One measure is to increase the damping of the submerged floating pipeline system. The damping is considered as one of the most effective parameters to control the unstable zone and avoid the parametric resonance [26]. As is shown from the stability diagram in Section 3.2, the increasing system damping results in that the unstable zone reduces. Especially the proper damping will make the unstable zone disappeared when the value of the parametric excitation is very small. Thereby increasing the system damping coefficient can reduce the possibility of the parametric instability of
(a) 40
4. Conclusions The parametric instability of a submerged floating pipeline between two floating structures has been studied according to the derived equations which are coupled with the structural model and hydrodynamic force model. The validation of the wake oscillator model is carried out, and the stability diagram of the Mathieu equation is used to explain the stability of the solution of the motion equation. Engineering cases are applied for this work by the finite difference method, and the effects of some important design parameters are discussed. Finally some effective measures to avoid the parametric instability are provided for the guidance of
(b)
6 4
20 2
v/D 0
v/D 0 -2
-20 -4 -40 0
200
400
600
x (m)
800
1000
-6 0
200
400
x (m)
600
800
1000
Fig. 18. The process of the pipeline vibration mode jump (ˇ = 180◦ ): (a) ωv t = 240 = 754; (b) ωv t = 720 = 2262.
Z. Wang, H. Yang / Applied Ocean Research 59 (2016) 265–273
the submerged floating pipeline’s design. From the present study, the following conclusions are drawn: (1) The dynamic responses of the submerged floating pipeline under only vortex excitation and combined vortex and parametric excitations are different. Even if the parametric instability does not occur, the maximum vibration amplitude of the pipeline under combined excitations will be larger than that subjected to only vortex excitation. (2) The parametric instability can occur in the cross-line vibration of the submerged floating pipeline when the fluctuation of its axial tension force is big enough. The parametric instability can result in a very large displacement to the pipeline, which will lead to the serious structural failure and fatigue damage. (3) There is a vibration mode jump phenomenon when the parametric instability occurs, which means that the vibration mode can change periodically with the time. This is due to the lock-in phenomenon in the vortex-induced vibration and the variety of the natural frequency of the pipeline’s transverse vibration. (4) The damping coefficient and the value of the parametric excitation can make a great impact on the dynamic stability of the submerged floating pipeline. Increasing the damping coefficient including the structural and hydrodynamic damping coefficient, lowering the motion transfer coefficient and reducing the phase angle of the motion of two floating structures are effective to avoid the parametric instability. Finally, this study is only about the theoretic method to deal with the parametric instability prediction for the submerged floating pipeline between the floating structures. Experiment analysis will be carried out in the future research. Acknowledgment This work was financially supported by the National Natural Science Foundation of China (Grant Nos. 51379005 and 51009093). References [1] O. Fyrileiv, O. Aamlid, A. Venas, L. Collberg, Deepwater pipelines—status, challenges and future trends, Proc. Inst. Mech. Eng. M: J. Eng. Marit. Environ. 227 (4) (2012) 381–395, http://dx.doi.org/10.1177/1475090212450659. [2] N. Kruijt, Turkey-Cyprus Submerged Floating Freshwater Pipeline, Delft University of Technology, Netherlands, 2003. [3] G. Paulsen, T.H. Søreide, F.G. Nielsen, Submerged floating pipeline in deep water, Seattle, USA, in: Proc. of 10th Int. Offshore Polar Eng. Conf., vol. II, 2000, pp. 108–114. [4] M. Lemoël, P. Brown, P. Jean, B. Shepheard, Design of the world’s 1st gravity actuated pipe (GAP) for Murphy’s Kikeh deepwater development, east Malaysia, in: Proc. ASME 27th Int. Conf. Offshore Mech. Arct. Eng., Estoril, Portugal, 2008, http://dx.doi.org/10.1115/OMAE2008-57533 (OMAE2008-57533). [5] Y. Gao, S. Fu, T. Ren, Y. Xiong, L. Song, VIV response of a long flexible riser fitted with strakes in uniform and linearly sheared currents, Appl. Ocean Res. 52 (2015) 102–114, http://dx.doi.org/10.1016/j.apor.2015.05.006. [6] R. Govardhan, C.H.K. Williamson, Critical mass in vortex-induced vibration of a cylinder, Eur. J. Mech.: B/Fluids 23 (2004) 17–27, http://dx.doi.org/10.1016/ j.euromechflu.2003.04.001.
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