Numerical simulation of fully nonlinear wave interaction with submerged structures: Fixed or subjected to constrained motion

Journal of Fluids and Structures 49 (2014) 534–553

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Numerical simulation of fully nonlinear wave interaction with submerged structures: Fixed or subjected to constrained motion W. Bai n, M.A. Hannan, K.K. Ang Department of Civil and Environmental Engineering, National University of Singapore, Kent Ridge, Singapore 117576, Singapore

a r t i c l e in f o

abstract

Article history: Received 7 May 2013 Accepted 25 May 2014 Available online 23 June 2014

The interaction between fully nonlinear water waves and fully submerged fixed or moving structures is investigated. A three-dimensional numerical wave tank model, based on potential theory is extended to include a submerged horizontal or vertical cylinder of arbitrary cross section. The vertical cylinder is allowed to have a constrained motion while attached to a rigid cable, which could represent the submerged payload of an offshore crane vessel subjected to wave actions. In this fully nonlinear time-domain approach, the higher-order boundary element method is used to solve the mixed boundary value problem based on an Eulerian description at each time step. The 4th-order Runge–Kutta scheme is adopted to update the free water surface boundary conditions expressed in a Lagrangian formulation. Interaction between waves and a submerged fixed horizontal cylinder is then computed by the model and compared with experimental and other numerical results. After that, a parametric study is performed to obtain numerical results for the vertical cylinder undergoing forced pitch motions. More simulations are carried out to investigate the hydrodynamic features of the submerged vertical cylinder in water waves attached to a cable for constrained motion and moving towards the sea bed at a constant velocity. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear wave Wave–structure interaction Submerged cylinder Constrained motion Numerical wave tank

1. Introduction Offshore installation operations are relevant to all structures situated offshore, as most of them are built onshore, then transported and installed at the operating site. For example, for an oil production platform, the topside part and the foundation part are always constructed separately, and they are only assembled at sea at the final stage. The offshore installation operations are therefore the key factor that must be taken into account in the design of offshore structures. Different types of structures require different methods of installation, and heavy lift is the most common method of installation of offshore structures. In this method, the structure is lifted off the transportation vessel by a crane vessel and lowered into position.

n

Corresponding author. E-mail address: [email protected] (W. Bai).

http://dx.doi.org/10.1016/j.jfluidstructs.2014.05.011 0889-9746/& 2014 Elsevier Ltd. All rights reserved.

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An offshore lift operation in the ocean wave environment is a very complicated process, where the interaction between response of the crane vessel and motion of the payload is coupled. Even small disturbances in the state of the system can entail the danger of collisions of the payload with the ship or other objects. Evaluating the stability of the crane vessel and the motion of the payload simultaneously is therefore a challenging task. However, we could treat the coupled system as two relatively independent sub-systems: the vessel and the payload. In the present paper, we only focus on the motion of payload which is fully submerged in waves and subject to constrained motions. The study of submerged bodies in water waves has received considerable attention for many years and several papers have been dedicated to the analytical, numerical and experimental study of the hydrodynamic response of such submerged structures. Among them, the interaction between monochromatic gravity waves and a fixed submerged horizontal circular cylinder, with its axis parallel to the crests of the incident wave, was first studied by Dean (1948), using a linearized potential theory and the conformal mapping technique. In that study Dean showed that, to the first order there is no reflection of incident waves by the circular cylinder, and that transmitted waves only undergo a phase shift when passing the obstacle. Ursell (1950) subsequently obtained the complete linear solution and reproduced Dean's conclusions. Ogilvie (1963) and Mehlum (1980) confirmed and extended the work by Dean (1948) on the diffraction problem. The work of Ursell (1950) and Ogilvie (1963) also investigated the wave radiation problem for a circular cylinder in forced oscillations and showed that it is possible to absorb all the power in a sinusoidal wave, by forcing a cylinder to move in a circular path. The first experimental study related to this phenomenon was undertaken by Chaplin (1984) in order to calculate the nonlinear forces and the nonlinear features of the reflected and transmitted waves originating from a fixed submerged horizontal cylinder. His study revealed the nonlinear components of these forces with frequencies up to 3 times the fundamental wave frequency. Since then, analysis of the hydrodynamic performance of submerged bodies has become increasingly important with the growing interest in offshore activities, especially in the ocean wave energy context which involves exploiting the wave induced motion of oscillating submerged bodies. This has led to a number of numerical studies. Vada (1987), for example, used an integral equation method based on Green's theorem to solve the second-order wave diffraction problem for a submerged cylinder of arbitrary shape. His results were in good agreement with those of Ogilvie (1963) and Chaplin (1984). Wu (1993) formulated a mathematical model to calculate the forces exerted on a submerged cylinder undergoing large-amplitude motions. He satisfied the no-flow boundary condition on the submerged body on its instantaneous position, while the free surface condition was linearized. The solution for the potential was expressed in terms of a multipole expansion. In particular, Wu (1993) obtained results for a circular cylinder undergoing prescribed motions in a wave field. Chaplin (2001) and Schønberg and Chaplin (2003) performed more detailed experimental and numerical studies of the nonlinear wave interactions with a submerged horizontal cylinder. Following that, Koo et al. (2004) developed a twodimensional fully nonlinear numerical wave flume based on the potential theory, MEL time marching scheme, using a boundary element method. This model was applied to determine wave characteristics and wave loads on submerged single and dual cylinders. Among more recent studies, Conde et al. (2009) performed another experimental study in conjunction with numerical analysis to study the fully nonlinear behaviour of a two-dimensional horizontal cylinder in waves. Guerber et al. (2010) extended a two-dimensional fully nonlinear potential wave model to include a submerged horizontal cylinder of arbitrary cross-section. Chen (2012) developed a new vortex based panel method for the numerical simulation of a 2-D potential flow around a hydrofoil submerged under a free surface with energy dissipation. An and Faltinsen (2013) studied the forced harmonic heave motions of horizontally submerged and perforated rectangular plates both experimentally and numerically at deep and shallow submergence. Some recent studies have also been undertaken to apply viscous models to fixed submerged bodies of various shapes. Bai et al. (2010), for example, studied two-dimensional submerged dikes interacting with viscous free surface waves using the Cartesian cut cell approach; Lam et al. (2012) modelled the laminar and turbulent flows around wavy rectangular cylinders; and Sarkar and Ratha (2014) carried out an experimental investigation on the flow field around submerged structures on horizontal plane beds. From the above discussion, we can notice that most of the previous numerical investigations using potential flow models mainly focused on the study of submerged horizontal circular cylinders or plates in water waves, due to the ease of implementation. The dynamic response of a fully submerged payload subject to constrained motions under the action of waves, however, is not well understood. The main difficulty associated with this process lies in the large motion that the payload might experience in waves when the payload supported by a cable is lowered down at a constant speed. In such a situation, the conventional frequency-domain analysis may not be applicable. At the same time, even in the time-domain approach, a volume-discretized method, such as the Finite Element Method (FEM), might encounter some unsolvable difficulties in the mesh generation for the highly-deformed computational domain. The authors are not aware of any published results for a fully nonlinear three-dimensional study of a submerged crane vessel payload undergoing constrained motions in waves. The present study aims at obtaining a good insight into this problem by using numerical simulation. The approach presented here uses a 3D fully nonlinear potential flow model based on the Boundary Element Method (BEM), which is an extension of the programme developed by Bai and Eatock Taylor (2006, 2007, 2009) for simulating the wave radiation, wave generation, propagation and interaction with structures. The original code has been modified to make it capable of considering fully submerged bodies either fixed or subjected to forced and/or constrained motions under wave actions. The overall approach is as follows. First the mathematical formulation is presented in Section 2, followed by its numerical

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implementation in Section 3. After that, Section 4 provides an analysis of the accuracy of the present numerical model based on a comparison with published numerical and experimental results. In Section 5 the numerical results are given for a fully submerged vertical cylinder subjected to forced pitch motion. Finally, the fully submerged vertical cylinder attached to a cable and subjected to a constant downward velocity in waves is discussed in Section 6, before some conclusions are drawn in Section 7. 2. Mathematical formulation To describe the three-dimensional wave–structure interaction problem, two right-handed Cartesian coordinate systems are defined. One is a space fixed coordinate system Oxyz having the Oxy plane on the mean free surface and the origin O usually at the centre of the numerical tank. In this case the z-axis is positive upwards. The other is a body fixed coordinate system O0 x0 y0 z0 with its origin O0 placed at the centre of mass of the body. When the body is in an upright position, these two sets of coordinate systems are parallel and the centre of mass is located at Xg ¼(xg, yg, zg) in the space fixed coordinate system. A sketch of the problem showing the coordinate systems along with the definition of some dimensions is shown in Fig. 1. In the figure, n is the normal unit vector pointing out of the fluid domain, and SF, SM, SW, SD and SB represent the free surface, wave maker, side wall, horizontal bottom and body surface respectively. As can be seen, the wave is generated in the tank by a piston-like wave maker and it is mounted at the left-hand end (on the negative side of the x-axis) of the tank. A damping zone is adopted at the other end of the tank using an artificial damping layer on the free surface. 2.1. Governing equations and boundary conditions The fluid is assumed to be incompressible and inviscid, and the motion irrotational. The water wave problem, therefore, can be formulated in terms of a velocity potential ϕ(x, y, z, t), which satisfies Laplace's equation within the fluid domain Ω, ∇2 ϕ ¼

∂2 ϕ ∂2 ϕ ∂2 ϕ þ þ ¼ 0; ∂x2 ∂y2 ∂z2

ð1Þ

and is also subject to various boundary conditions on all surfaces of the fluid domain. On the instantaneous free water surface, the following kinematic and dynamic conditions need to be satisfied, which are nonlinear, the source of the difficulties associated with the water wave problem: DX ¼ ∇ϕ; Dt

ð2Þ

Dϕ 1
¼  gz þ
∇ϕ2 j; Dt 2

ð3Þ

where the kinematic and dynamic conditions are expressed using the Lagrangian description. Here, D/Dt is the usual material derivative, X denotes the position of points on the free surface, and g is the acceleration due to gravity. The boundary condition on the wave maker can be given as ∂ϕ ¼ UðtÞ; ∂x

ð4Þ

Fig. 1. Sketch of definition for submerged vertical cylinder in waves.

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where U(t) is the velocity of the wave maker and the condition is imposed at its instantaneous position. The general boundary condition on other solid surfaces, including the body surface, bottom and side walls, can be expressed as ∇ϕ U n ¼ Vn ;

ð5Þ

where V n is the velocity of the solid surface in the normal direction. If small angular motions are assumed, the motions of a three-dimensional rigid body about its centre of mass can be described in terms of six components. ξ ¼ (ξx , ξ y, ξ z ) is a translational vector denoting the displacements of surge, sway and heave, and α ¼(αx , α y, αz ) is a rotational vector indicating the angles of roll, pitch and yaw respectively, measured in the anticlockwise direction. On the basis of these six components, the normal velocity becomes _  ðX  Xg Þ Un: Vn ¼ ½ξ_  α

ð6Þ

Since a time domain approach is applied to formulate the problem, initial conditions are also required. In every case considered here the body starts from rest in calm water, so the velocity potential and wave elevation at the initial time instant can be set to ϕ ¼ 0 and z ¼ 0

when t r0:

ð7Þ

Moreover, in order to avoid wave reflection from the downstream end of the numerical tank, a proper far-field condition is required: this is defined and discussed in the next section. 2.2. Higher-order boundary element formulation In this paper, the higher-order boundary element method is used to solve the mixed boundary value problem at each time step. For cases in which the body and the simulated flow are symmetric about the x–z plane and the sea bed is horizontal, a simple Rankine source and its images with respect to the symmetry plane and the sea bed can be chosen as Green's function. The integral thus only needs to be evaluated over half of the computational boundaries, and the sea bed is excluded. In this numerical approach, the surface over which the integral is performed is discretized by quadratic isoparametric elements. After assembly of the equation for each node on the whole integral surface, a set of linear algebraic equations is obtained (for details see Bai and Eatock Taylor, 2006). At any given time step, either the potential or its normal derivative on each part of the boundary is known from the corresponding boundary conditions. The resulting set of linear algebraic equations can therefore be solved to obtain the solution of the mixed boundary value problem (i.e. to find the remaining unknowns) at that time step. 2.3. Hydrodynamic forces Once the potential has been found by solving the mixed boundary value problem at each time step, the pressure on the body is expressible using the Bernoulli equation. The hydrodynamic forces F¼{f1, f2, f3} and moments M¼{f4, f5, f6} on the body can consequently be obtained by integrating the pressure over the wetted body surface:   1 f i ¼ ∬SB pni ds ¼  ρ∬SB ϕt þ ∇ϕ U ∇ϕ þ gz ni ds ði ¼ 1; 2; …; 6Þ; ð8Þ 2 where ρ is the water density. In the above equation, the main difficulty is the evaluation of the time derivative of the potential, ϕt. Estimating this quantity by a simple backward difference scheme is inaccurate and prone to instabilities, particularly in the more general case when the body is free to move. An effective method for calculating the hydrodynamic force is to solve a separate boundary value problem, as used by Wu and Eatock Taylor (2003). In this approach, some auxiliary functions are introduced, in place of computing ϕt directly. By using these auxiliary functions, the fluid–structure problem is decoupled, and can be solved easily. 2.4. Motion of a submerged cylinder attached to cable The following assumptions are used in the analysis of the submerged vertical cylinder which is attached to a cable and subjected to constrained motion: (i) The cylinder is regarded as being rigid with concentrated mass at its centre of mass, and it is allowed to have pendulum motion only without twisting. (ii) The cable is taken as a rigid rod. This assumption is valid as long as the oscillations of the load in the vertical direction are small and the cable remains in tension. (iii) The cable is attached to the centre point of the top surface of the cylinder (no rotation about this point) and the cylinder is subjected to constrained motion under wave action (motion with respect to cable origin point only).

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Based on the abovementioned assumptions, the following nonlinear equation of motion of a pendulum is implemented in order to model the cylindrical payload attached to a rigid cable: 2

ðf x cos ξ5  f z sin ξ5 ÞðLÞ ¼ mðLÞ2

d ξ5 ; dt 2

ð9Þ

where m is the mass of the cylindrical body concentrated at its centre of mass; L is the distance between the rigid cable origin and the centre of mass of the cylindrical payload; ξ5 is the angular displacement of the vertical cylinder; and fx and fz are the horizontal and vertical dynamic forces on the submerged cylinder respectively. 3. Numerical implementation The successful implementation of the mathematical formulation in a numerical simulation demands careful attention to several aspects. The relevant details can be found in Bai and Eatock Taylor (2006), and only a brief summary is given below. In the present method, structured quadrilateral meshes are distributed on the vertical wall surfaces and the side surface of the body. On the free surface, and the top and bottom of the truncated body, unstructured triangular meshes are generated by using the Delaunay triangulation method, which is well suited to fitting the complex computational domain. A double or triple node is employed on the intersections between the free surface and solid surfaces or between the side surface and the top and bottom of the truncated body. This multiple-node technique can determine the intersection lines conveniently without the need for additional treatment; we only need to update the node on the free surface and allow other multiple nodes on the solid surfaces to move with it. In order to avoid the reflection of scattered waves from the far-field computational boundaries and to simulate a sufficiently long duration in a reasonably sized domain, an artificial damping layer on the free surface is adopted to absorb the scattered wave energy on the downstream side of the numerical tank. In this numerical beach, the kinematic and dynamic boundary conditions are modified by a damping term over a finite length of the free surface (Ferrant, 1993) as follows: DX ¼ ∇ϕ  υðlÞðX  Xe Þ; Dt

ð10Þ

Dϕ 1 ¼  gz þ ∇ϕ U ∇ϕ  υðlÞϕ; Dt 2

ð11Þ

where l is the distance from the body, υ(l) is the damping coefficient and Xe ¼(xe, ye, 0) is a reference value specifying the at-rest position of the fluid particle. The free surface geometry and potential are updated by the standard 4th-order Runge–Kutta scheme, and a cosine ramp function is used over the first time period to modulate the boundary condition on the wave maker during the initial time steps. As the equation of motion only provides the body acceleration, the standard 4th-order Runge–Kutta scheme is also adopted here to compute the associated velocity and displacement of the body. The efficient generalized minimum residual (GMRES) iterative scheme with a diagonal preconditioner is used to obtain the solution of the full and asymmetric influence matrix arising from the mixed boundary value problem at each time step. Under most conditions, without regridding the numerically computed wave profile will, after a sufficiently long time, develop a saw-tooth appearance. To avoid this numerical instability, mesh regeneration on the free surface is used, leading to the need to specify the horizontal coordinates of the new nodes. Interpolation is then used to predict the vertical coordinate and the potential at each new node, based on its given horizontal coordinates. 4. Fixed horizontal cylinder beneath waves In this section, the present numerical model is implemented to simulate the experiment conducted by Chaplin (1984) who measured the nonlinear force on a horizontal fixed cylinder under waves in a wave flume. In his study, Chaplin analysed the influence of the Keulegan–Carpenter (KC) number on the harmonics of the applied force, defined as KC ¼

πH πa expðkzc Þ ¼ expðkzc Þ; 2r r

ð12Þ

where H is the wave height, a the wave amplitude, r the cylinder radius, k the wave number, and zc the submergence of the axis of the cylinder. Chaplin's results are in good agreement with Ogilvie (1963) when the wave steepness is low, but important nonlinear effects were experimentally observed for higher steepness. Grue and Granlund (1988) also carried out some experiments, focusing on the diffracted waves. For the validation of current study, case E in Chaplin's experiment is selected for the comparison of hydrodynamic forces. At the same time, the present results are also compared with another numerical study published by Guerber et al. (2010) where possible. A cylinder of radius r ¼0.051 m is placed at zc ¼  0.102 m under the undisturbed free surface, in a tank of depth d ¼0.85 m. The total length and width of the numerical tank are 25.5 m and 0.6 m respectively. At x ¼  9.375 m, i.e. the left end of the wave tank, a monochromatic wave is generated by a piston-like wave maker and the wave period is

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T ¼1.0 s. In the present study the influence of the KC number on the hydrodynamic forces is investigated by varying the wave amplitude from 0.001 m to 0.028 m. Following Chaplin (1984), the non-dimensional horizontal and vertical forces, Fx and Fz, are written as Fourier series and the amplitudes of the harmonics are computed with a FFT over the last 3 wave periods of the simulations: fx ¼ F x ð0Þ þ∑n Z 1 F x ðnÞ cos ðnωt þ ψ ðnÞ Þ; ρr 3 ω2

ð13Þ

Fig. 2. Time history of horizontal and vertical dynamic forces on the submerged horizontal cylinder for various KC numbers: (a) horizontal force and (b) vertical force.

3.0

KC=0.32 KC=0.65 KC=0.91

z/a

1.5 0.0 -1.5 -3.0

0

5

10

15

t/T

Fig. 3. Time history of wave elevation at x ¼0.1 on the centre line of the tank for various KC numbers.

1

Present Guerber et al 2010 Chaplin 1984

FZ

(0)

0.1

0.01

1E-3

0.1

1 KC

Fig. 4. Comparison of mean vertical force on the horizontal cylinder with other experimental and numerical data.

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fz ¼ F z ð0Þ þ∑n Z 1 F z ðnÞ sin ðnωt þ ψ ðnÞ Þ; ρr 3 ω2

ð14Þ

where ρ is the water density, the superscript (n) denotes the nth Fourier harmonic of the associated variable, ψ is the phase angle, and ω¼2π/T is the wave frequency. Fig. 2 shows the time history of horizontal and vertical dynamic forces for various KC numbers, whereas Fig. 3 represents the corresponding wave elevation history at a point with x ¼0.1 on the centre line of the tank. Because both the forces and the wave elevation are plotted relative to the incoming wave amplitude, those nondimensional properties do not change much with KC number. The change of mean vertical forces with respect to the KC number is plotted in Fig. 4 using a logarithmic scale. From this figure it is clearly seen that the present result agrees well with both the experimental result of Chaplin (1984) and the numerical analysis of Guerber et al. (2010). However, the first harmonics of the vertical and horizontal forces (depicted in Fig. 5) show considerable deviation from Chaplin's experimental results with the increase of KC number. Except for the small values of KC number, the decrease of the experimental first-order coefficient is not reproduced by the present model. It seems that the decrease occurs due to the viscous effect and the recirculation created around the body (Chaplin, 1984). This suggestion is also supported by the study of Tavassoli and Kim (2001). They observed the same kind of decrease in their simulations performed in a viscous numerical wave tank. Chaplin (1984) also noticed that the magnitudes of the horizontal 2.3

FZ

(1)

/ KC

2.1

/ KC

1.9

Chaplin 1984 Present (FX )

FX

(1)

1.7

Guerber 2010 (FX )

1.5

Present (FZ ) Guerber 2010 (FZ ) Tavassoli 2001

1.3 0.0

0.6

1.2

1.8

KC Fig. 5. First harmonic of horizontal and vertical forces on the horizontal cylinder.

Present Guerber et al 2010 Chaplin 1984 Tavassoli 2001

FZ

(2)

0.1

0.01

0.1

1

KC Fig. 6. Second harmonic of vertical force on the horizontal cylinder.

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541

and vertical forces are very close; therefore, he drew them together using the same symbol in the figure. In the present study, the similar magnitudes of the forces in the two directions are captured. Some of the results from the potential flow model of Guerber et al. (2010) are also shown in Fig. 5, and they are seen to agree quite well with those from the present analysis. Figs. 6 and 7 show the comparison of second and third harmonics of the vertical force plotted against the KC number. Compared to the first harmonic, the second and third harmonics are in good overall agreement with both the experiments and the numerical analysis, especially the third harmonic. However, in the experimental results, similar to the case of the first harmonic, slight decreases in values are also observed here at larger values of KC, which again can be attributed to the viscous effect. Overall, it can be summarized that the present study provides similar results with the experimental analysis in Chaplin (1984) up to the point where the viscous effect starts to play a significant role. For this case, it seems that viscosity of the fluid starts to have an important influence as KC number reaches the value of 0.9 in the case of the first harmonic of vertical force. At higher values of KC number, the viscous flow simulation by Tavassoli and Kim (2001) seems to agree fairly well with Chaplin's experiment. Other numerical models based on potential theory, such as Koo and Kim (2004) using the boundary element method, and Kent and Choi (2007) using the HOS method, also recorded similar kind of behaviour without any better agreement. Furthermore, the nonlinear higher harmonics can be captured quite well in the present study. The ability of the present numerical model in simulating nonlinear wave interactions with submerged structures near to the free surface is thus confirmed by this validation. It might also be worthwhile to mention here that no instabilities of the free surface or breaking of waves passing over the cylinder has been observed throughout this numerical study (within the chosen amplitude limits). In this connection, Fig. 8

Present Guerber et al 2010 Chaplin 1984 Tavassoli 2001

FZ

(3)

0.1

0.01

1E-3

0.1

1

KC Fig. 7. Third harmonic of vertical force on the horizontal cylinder.

Fig. 8. Wave past a submerged fixed horizontal cylinder at various time instants with KC ¼ 0.81: (a) t ¼12T and (b) t¼ 12.5T.

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illustrates the simulated free surface along with the submerged horizontal cylinder for two different time instants. The free surface is truncated on both the left and right side; also the cylinder cross-section is magnified to provide a better representation of the image. As can be seen in this figure, the unstructured triangular mesh generated on the free surface is quite uniform and regular. In addition, we can also notice the scattered wave above the cylinder due to the presence of the cylinder. 5. Wave radiation by a fully submerged vertical cylinder undergoing pitch motion In this section, nonlinear wave radiation by a fully submerged vertical cylinder undergoing forced sinusoidal pitch motions in otherwise still water is studied. This problem concerns a moving body in the domain, which provides a solid foundation for investigating the more complicated case discussed in the next section. It is important to note that the water depth d, gravitational acceleration g and fluid density ρ are taken to be unity in all cases discussed henceforth (unless otherwise specified), and other lengths are non-dimensionalised by the water depth, yielding forces, motions and time in non-dimensional form. It should also be noted that the vertical hydrodynamic force presented in the paper is the total force obtained from Eq. (8) minus the mean hydrostatic force. Throughout the parametric study of this section, a vertical cylinder of length l¼0.2d with radius r ¼0.1d, is placed at 0.15d below the still water surface and prescribed to pitch about a point 0.3d above the still water surface. The mean position of the cylinder is at the centre of a circular cylindrical tank having a radius of R¼2.5d, which is sufficiently large as the focus here is to analyse the wave profile/elevation and hydrodynamic force on the moving cylinder. In order to prevent reflection, a damping layer is implemented around the perimeter of the cylindrical tank. Moreover, due to the symmetry of the computational domain, only half of it is considered in the computation. The cylinder starts oscillating from its rest position, and the displacement and velocity of its motion are given respectively as ξ5 ¼  a5 cos ωt;

ð15Þ

χ 5 ¼ a5 ω sin ωt;

ð16Þ

Fig. 9. (a) Circular tank outlook and (b) mesh distribution on free surface and body (half tank).

2 2 Fz /r a5 d

0.3

a5 = 60

0.2

a5 = 30 a5 = 20

0.1 0.0 -0.1

0

1

2

3

4

5

6

7

t/T

Fig. 10. Time history of vertical dynamic force for various pitch motion amplitudes at ω ¼2.0.

a5 = 60

z/a5

0.03

a5 = 30

0.01

a5 = 20

-0.01 -0.03 -2.50

-1.25

0.00 x

1.25

Fig. 11. Wave profile at t¼ 9T for various pitch motion amplitudes at ω¼ 2.0.

2.50

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543

where ω is the oscillation frequency and a5 defines the pitch amplitude of the body motion. There are 6 boundary elements in the vertical direction on the body surface and on the side wall, and the numbers of elements around the half of the cylinder and side wall are 16 and 35 respectively. Including 1199 elements on the water surface, there are about 3816 nodes distributed on the whole computational boundary. Fig. 9 provides an illustration of the computational domain. Fig. 9(a) depicts the sketch of the entire 3-D numerical wave tank with the vertical test cylinder inside the domain, whereas Fig. 9(b) shows a snapshot of part of the numerical grid used in the calculation in half of the computational domain (free surface and cylinder only) at a certain time step.

5.1. Influence of pitch motion amplitude This subsection provides an analysis of the forces on the cylinder and corresponding wave elevations in order to investigate the effect of changing amplitudes of pitch motion. Figs. 10 and 11 show the time history of the vertical dynamic force on the cylinder and the resulting wave profile with ω¼ 2.0 for three pitch angles. The vertical force in Fig. 10 is nondimensionalised by the square of pitch angle a5 multiplied by the water depth d, as the first-order term of the vertical force

(a)

(b)

0.05

0.020

0.019

0.018

z / a5

2

Fz /r a5 d

0.04

0.03

0.017

0.02 0.016

0.01 0.025

0.050

0.075

0.100

0.125

0.150

0.015

0.175

0.050

0.075

0.100

a 5 /r

0.125

0.150

0.175

a 5 /r

Fig. 12. Variation of maximum vertical wave force and wave elevation with pitching amplitude: (a) vertical dynamic force and (b) wave elevation at x ¼  0.10 on the plane of symmetry.

2

Fz /r a5 d

0.2

= = =

0.1 0.0 -0.1 -0.2

0

1

2

3

4

5

6

7

8

9

10

11

t/T

3

My /r a5 d

20

= = =

10 0 -10 -20

0

1

2

3

4

5

6

7

8

9

10

11

t/T

Fig. 13. Time history of vertical dynamic force and moment for various pitch frequencies with a5 ¼π/30: (a) force and (b) moment.

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is zero. From this figure, it is evident that the vertical dynamic force has a non-zero positive mean and no phase change is observed with increasing amplitudes. The wave profiles for various pitch amplitudes are shown in Fig. 11 (where x ¼0 corresponds to the point above the mean position of the cylinder). These profiles also seem to be similar, except for some ripples at higher amplitudes. These ripples indicate that some higher-order nonlinear components induced by the larger pitch amplitude are superimposed on the basic linear solution obtained at the smaller pitch amplitude. The overall trends of vertical force and wave elevation with respect to the pitch motion amplitude are represented in Fig. 12. The maximum vertical force and wave elevation have been obtained here by considering the last three steady periods of the entire simulation. Both the vertical force and the wave elevation show considerable nonlinear behaviour as the imposed motion amplitudes increase. The variation of vertical force with pitch amplitude is close to quadratic (as seen also in Fig. 10); the wave elevation recorded at a certain position (x ¼  0.1), however, is more complex and shows less nonlinearity with the increase of motion amplitudes. 5.2. Changing frequency of pitch motion Next, the variation of pitching frequency is considered to see how it affects the wave elevation, force and moment on the pitching cylinder. Figs. 13–15 provide the results of analysis where the cylinder oscillates with three different pitching frequencies and having an amplitude of a5 ¼π/30. As can be seen in Fig. 13 both the vertical force and moment increase quite noticeably with the increase of frequency. However, the vertical force has a frequency twice that of the moment response, which is reasonable because when the cylinder oscillates over half a cycle, the vertical force will change direction once. Also, unlike the moment, the vertical force has a non-zero mean at higher frequencies. A slight phase shift is also observed in the vertical force at higher frequencies, whereas no such changes are found to be associated with the variation of motion amplitudes in the previous subsection. Unlike the variations of the wave profiles with change of motion amplitudes, the wave profiles in Fig. 14 have fairly distinct patterns at the different frequencies; which is expected because the wave lengths are different in this case. Finally, the variation of maximum vertical force and moment with respect to kr is plotted in Fig. 15 in order to visualize the overall

= = =

0.10 z/a5

0.05 0.00 -0.05 -0.10 -2.50

-1.25

0.00 x

1.25

2.50

Fig. 14. Wave profile at t ¼9T for various pitch frequencies with a5 ¼ π/30.

15

0.20 0.18 0.16

10

0.12

My /r a5 d

0.10

3

2

Fz /r a5 d

0.14

0.08

5

0.06 0.04 0.02 0.00

0

0.0

0.2

0.4

0.6

kr

0.8

1.0

0.0

0.2

0.4

0.6

kr

Fig. 15. Variation of vertical force and moment with pitching frequency: (a) force and (b) moment.

0.8

1.0

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Table 1 Test cases for fully submerged cylinder attached to rigid cable. Number of cases

cbl/d

a

Influence of different cable length 3 0.4, 0.6, 0.8 0.015 Variation of motion amplitude of wave maker 5 0.5 0.005–0.025 5 0.5 0.005–0.025 Effect of wave frequency 5 0.5 0.01 5 0.5 0.015 Cylindrical payload subjected to constant downward velocity 4 0.7 0.005 4 0.7 0.01 4 0.7 0.015 4 0.7 0.02

ω

D/d

mod/d

2.0

0.2

N.A.

2.0 2.25

0.2 0.2

N.A. N.A.

1.50–2.50 1.50–2.50

0.2 0.2

N.A. N.A.

2.0 2.0 2.0 2.0

0.15 0.15 0.15 0.15

0.005–0.02 0.005–0.02 0.005–0.02 0.005–0.02

Trajectory

0.020

x XZ projection

nt Vertical displaceme

0.015

0.010 0.005 0.000 -1

Hor izon 0 tal d ispla cem ent

1

0

4

8

12

16

20

t/T

Fig. 16. 3D cylinder positioning with respect to time with cable length of 0.4d (initial cylinder position at centre line).

effect of frequency change. Both the force and moment tend to increase with increase of frequency. However, the rate of increase of the force seems to be higher than that of the moment. It might also be worthwhile to note that a close look at Figs. 11 and 14 confirms that the damping layer applied around the periphery of the cylindrical wave tank is able to prevent wave reflections from the boundary of the domain. As can be seen, at both the ends of these wave profiles the wave elevations are almost ‘zero’, which means the damping layer implemented in those areas (see Fig. 9(a) for tank sketch) has effectively damped out the radiated waves. Therefore, it can be said that the results presented in this paper are not affected by wave reflections, and the current configuration of the damping layer appears to be acceptable.

6. Fully submerged cylinder attached to rigid cable and subjected to constrained motion In this section a comprehensive investigation is presented to understand the fully nonlinear wave interaction with a completely submerged vertical cylindrical payload which is attached to a rigid cable and is allowed to have pendulum motion only with respect to the cable origin point. The same rectangular numerical wave tank depicted in Fig. 1 is employed throughout the study of this section. The water depth d, gravitational acceleration g and the fluid density ρ are taken to be unity as before. This time, however, the density of the cylindrical payload has to be considered as well and is taken as 1.2ρ: this is found to be enough to maintain a constant tension in the cable, thus justifying the rigid cable simplification considered during the formulation of the problem. The payload is taken to be a vertical cylinder of length l¼ 0.2d with a radius r ¼0.1d. The total dimensionless tank length is taken as 7.5 in most of the cases. During the simulation, only half of the wave tank is considered due to symmetry, and the half-width is taken as 0.5, which is equivalent to five times the cylinder radius. The wave maker is placed at x¼  4.375 (at the left end side of the tank) and a monochromatic wave is generated by this pistonlike wave maker undergoing the following motion: ( S0 ðtÞ ¼ a cos ðωtÞ; ð17Þ U 0 ðtÞ ¼ aω sin ðωtÞ;

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where S0(t) and U0(t) are the displacement and velocity of the wave maker respectively, and in this section a and ω are the corresponding motion amplitude and frequency of the wave maker. The damping zone is situated near the right hand end of the wave tank and is formulated based on the methodology described in Section 3. The damping layer starts at a distance of one incident wave length measured from the far end of the tank. In generating meshes on the half numerical tank including the cylindrical body, a total of 2206 elements is used. In the horizontal plane, 60 elements are placed in the x direction and 5 elements in the y direction along the boundary line of the tank. 6 elements are placed in the vertical direction on each solid boundary surface. The resulting vertical solid tank surface along the x direction contains 360 elements whereas the vertical solid tank surface along the y direction has only 30 elements. Another 204 elements are used to simulate half of the cylindrical payload (96 elements on the side surface and the rest on the top and bottom surfaces). The remaining 1582 elements are generated on the free surface. A number of test cases are designed to perform a comprehensive study of the scenarios described in the following subsections. Table 1 represents a complete list of those cases analysed in each subsection. Here, D represents the vertical distance between the undisturbed free surface level and the cylinder top surface. The definition of the variables cbl and mod will be given in Sections 6.1 and 6.4 respectively.

6.1. Influence of different cable length The influence of cable length variation on cylinder response and resultant moment is considered, and the findings are illustrated in Figs. 16–18. It is important to note that the cable length would change when the origin point of the cable cbl =0.4d cbl =0.6d cbl =0.8d

3

My / r a

40 20 0 -20 -40

0

5

10

15

20

t/T

Fig. 17. Moments about the origin of the cable for different cable lengths at a¼ 0.015 and ω¼ 2.0.

cbl (sin

5

)/a

1.5 0.9 0.3 -0.3 -0.9 -1.5

0

cbl =0.4d cbl =0.6d cbl =0.8d

5

10

15

20

t/T

Fig. 18. Change in horizontal displacements due to the variation of cable length at a¼ 0.015 and ω¼ 2.0.

3.0

27 = 2.00 = 2.25

= 2.00 = 2.25

2.8

25

2

T/r a

3

My / r a

2.6 23

21

19 0.00

2.4

2.2

0.05

0.10 ka

0.15

2.0 0.00

0.05

0.10

0.15

ka

Fig. 19. Comparison of maximum moment about the cable origin and maximum dynamic cable tension for different motion amplitudes of wave maker at two wave frequencies: (a) moment and (b) tension.

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547

moves. A series of analysis has therefore been undertaken for different positions of the origin point, corresponding to different cable lengths. The initial underwater position of the cylinder is assumed to remain unchanged during this process, which is 0.2d below the undisturbed free surface. Moreover, the motion amplitude of the wave maker a/r and the wave frequency ω are also kept constant at 0.15 and 2.0 respectively for all the analysis of this subsection. Hence, the wave length λ is also constant at 1.57d. Fig. 16 shows a 3D trajectory of the cylinder in time around the central y-axis for the case with cbl ¼0.4d, where cbl denotes the distance between the cable origin (assumed to be fixed) and the top surface of the cylinder. The curve at the very right of Fig. 16 provides the “XY projection” of the entire trajectory to reflect the overall pattern of the path followed by the cylinder and the maximum displacement attained by it during the full simulation period. As can be seen, the displacement of the cylinder is not symmetric about the centre line and its magnitude is not in a regular pattern, due to the interaction with fully nonlinear waves. The time series of horizontal displacement for this case is also presented later in Fig. 18 with two other cases. In Fig. 17 the time history of the moment with different cable lengths is plotted. A significant increase in moment due to the increase of cable length is visible there. This variation in moment seems to have a proportional relationship with the cable length. However, no such clear conclusion is evident in case of the nondimensionalised horizontal displacement of the cylindrical payload depicted in Fig. 18. Some differences can be identified,

2.5 = 2.00 = 2.25

2.2

5

(d/a)

1.9

1.6

1.3

1.0 0.00

0.05

0.10

0.15

ka Fig. 20. Change in maximum pitch angle due to the variation of motion amplitude of wave maker at two wave frequencies.

3.50

= 1.5 = 2.0 = 2.5

0.00

2

T/r a

1.75

-1.75 -3.50

0

5

10

15

20

t/T

Fig. 21. Cable tension for different wave frequencies at a¼0.01.

4

= 1.5 = 2.0 = 2.5

0

5

(d/a)

2

-2 -4

0

5

10

15

t/T

Fig. 22. Angle of pitch subjected to various wave frequencies at a ¼0.01.

20

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but the overall amplitudes for various cable lengths seem to be similar. This leads to the following conclusion: variation of cable length does not have a noticeable impact on the horizontal displacement of the cylinder, when other parameters like motion amplitude of the wave maker and frequency remain the same.

6.2. Variation of motion amplitude of wave maker In this subsection, the moment, cable tension and angular displacement of the cylinder about the cable origin are calculated for several amplitudes of the wave maker oscillating at two different wave frequencies: ω¼2.0 and 2.25. For each of these frequencies five different motion amplitudes of the wave maker are considered ranging from 0.005 to 0.025. The cylinder is placed 0.2d below the undisturbed free surface and a constant cable length, cbl ¼0.5d, is considered throughout this and rest of the subsections, unless otherwise specified. Fig. 19 shows the overall variation of the maximum moment and cable tension, whereas Fig. 20 represents the change in maximum angular displacement, due to the variation of motion amplitudes of the wave maker. If the problem considered were linear, in principle the results should be invariant with the change in motion amplitudes of the wave maker (a straight horizontal line in the figures). However, from these figures, it is evident that the effect of nonlinearity is visible in all cases. In addition, higher frequencies produce lower responses.

30

3.0 a =0.010 a =0.015

a =0.010 a =0.015

25

2

3

T/r a

My / r a

2.5

2.0

20

15

0.2

0.4

1.5

0.6

0.2

0.3

0.4

0.5

0.6

0.7

kr

kr

5.0 a =0.010 a =0.015

4.1

5

(d/a)

3.2

2.3

1.4

0.5

0.2

0.3

0.4

0.5

0.6

0.7

kr Fig. 23. Variation in maximum values of key parameters with respect to wave frequency and motion amplitude of wave maker: (a) moment, (b) cable tension and (c) pitch angle.

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6.3. Effect of wave frequency The wave interaction with the submerged cylindrical payload attached to a cable is next simulated under the action of waves of several frequencies. At first a single motion amplitude of the wave maker a ¼0.01 is considered. As can be seen in Figs. 21 and 22, the time history of the cable tension in this case is more regular compared to the angular displacement of the cylinder. Both the tension and pitch angle tend to undergo a phase shift at higher wave frequencies and the angle of pitch also becomes lower at higher wave frequencies. A similar analysis has been conducted for another motion amplitude of the wave maker, a¼0.015. The overall response patterns of the maximum moment, cable tension and angular displacement for both of these two amplitudes are then plotted in Fig. 23 against the dimensionless wave number. All the three parameters seem to decrease at higher frequencies, though the moment and cable tension show slight increase at lower frequencies. Beside these, higher wave amplitude generally seems to produce higher response, especially in the angular displacement. However, the moment and cable tension are not very sensitive to the incident wave amplitude. Nevertheless, as a whole it may be concluded at higher frequencies the responses are reduced, and when combined with lower wave amplitudes this would lead to even smaller angular displacements. 6.4. Cylindrical payload attached to cable and subjected to constant downward velocity The final subsection of this study investigates the behaviour of a cylindrical payload under the coupled influence of wave action and downward motion of a rigid cable. This simulation can be correlated with the practical phenomenon of lowering the payload from an offshore crane vessel through waves. In order to understand the scenario, the moment, cable tension and angular displacement of the cylinder are studied and compared from various point of views. A comparatively longer cable length (cbl ¼0.7d) is chosen to study the present situation, and the cylinder in this case is initially placed at 0.15d below the undisturbed free surface. At the beginning of the simulation period, several wave cycles are initially generated before the cylinder is set into downward motion, in order to allow the waves to reach the location of the cylinder. It was found that a time of t¼5.5T is quite sufficient for this initial time before the cylinder is given a constant downward motion. A new variable “mod” is introduced here to represent the constant velocity of the downward movement, defined here as distance per wave period, instead of the usual distance per second.

2

a = 0.005 a = 0.01 a = 0.02

(d/a)

1

5

0 -1 -2 0

5

10

15

20

t/T

Fig. 24. Angle of pitch of a cylindrical payload attached to a cable in waves of various motion amplitudes of wave maker with mod¼ 0.01.

4

a = 0.005 a= 0.01 a = 0.02

2

T/r a

2 0 -2 -4

0

5

10

15

20

15

20

t/T

40

a = 0.005 a= 0.01 a = 0.02

3

My / r a

20 0 -20 -40

0

5

10 t/T

Fig. 25. Cable tension and moment on a cylindrical payload attached to a cable in waves of various motion amplitudes of wave maker with mod ¼ 0.01: (a) tension and (b) moment.

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mod = 0.005 mod = 0.01 mod = 0.02

45

-0.10

30

mod = 0.02

-0.15

3

My / r a

15 0 -15

-0.20

-30

4

8

12

16

t /T

mod = 0.005 mod = 0.01 mod = 0.02

4

-0.25

20

draft

-45

-0.30 -0.35

2

T/r a

2

-0.40

0

-0.45

-2

-4

4

8

12 t /T

16

20

-0.50 -1.0

-0.5

0.0 x /a

0.5

1.0

Fig. 26. Peak analysis (both positive and negative) for moment and cable tension due to different downward velocities: (a) moment, (b) tension and (c) real time cylinder positioning.

The interaction of waves with a moving cylinder is first studied for various motion amplitudes of the wave maker. A constant wave frequency ω¼2.0 and a constant downward velocity mod¼0.01d are considered. Figs. 24 and 25 present the time history of the angular displacement, cable tension and moment respectively. As can be seen, all the parameters decrease in time as the cylinder moves towards the sea bed, because the wave energy is concentrated near the free water surface. However, the decreasing trends in the moment and cable tension are more regular and somewhat more pronounced, especially in case of the cable tension. On contrary, the decrease in the angular displacement is fairly irregular and very hard to visualize, particularly before 15 wave periods. This indicates that the cylinder attached to a cable can experience quite substantial response under the wave action even at deeper positions. Also, it seems that when a cylinder attached to a rigid cable is moving downward, changing of incident wave amplitudes generally does not make a significant difference to the response. After the above analysis, several cases were simulated with different downward velocities of the cylinder, keeping other inputs the same. The motion amplitude of the wave maker and the wave frequency here are fixed at 0.015 and 2.0 respectively. The findings are presented in Fig. 26. Fig. 26(a) and (b) show the time history of the moment and cable tension, highlighting the peaks (both negative and positive) only. The decreases in values with time for both of the parameters are clearly visible here; especially with higher mod, the decrease is more pronounced. The behaviour shown is as expected, because at the same t/T the cylinder with the higher mod moves to a deeper position. In Fig. 26(c) it seems that the cylinder is showing considerable movement when it is considerably below the free surface even with higher downward speed. Figs. 27–29 show the overall trends in the variation of maximum moment, angular displacement and cable tension for several motion amplitudes of the wave maker, in relation to the change of downward velocity of the cylindrical payload. As can be seen, all of these parameters decrease with the increase of downward velocity. The reduction in the moment, however, is more pronounced compared to that of the pitch angle and cable tension. Apart from these, overall the moments seem to stay similar with the increase of incident wave amplitude whereas the other two parameters show the opposite behaviour; the angular displacement rises with the increase of incident wave amplitude and at the same time the cable tension decreases.

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38

a =0.005 a =0.01 a =0.015 a =0.02

3

My / r a

36

34

32 0.000

0.005

0.010

0.015

0.020

0.025

mod Fig. 27. Changes in moment with respect to motion amplitude of wave maker for various moving downward speeds.

1.8

a =0.005 a =0.01 a =0.015 a =0.02

1.4

5

(d/a)

1.6

1.2

1.0 0.000

0.005

0.010

0.015

0.020

0.025

mod Fig. 28. Changes in pitch angle with respect to motion amplitude of wave maker for various moving downward speeds.

Finally, some snapshots of the numerical wave tank simulating the fully submerged vertical cylinder attached with a cable and subjected to wave action are presented in Fig. 30 for illustration purpose. 7. Conclusions A three-dimensional fully nonlinear numerical wave model is presented to simulate the wave interaction with fully submerged structures: fixed or subjected to constrained motions. For the purpose of validation, a fixed horizontal cylinder subjected to wave actions is simulated by using the present numerical model, and comparisons are made with other numerical and experimental results. In this case, the viscosity of the fluid plays a significant role. Apart from this effect, the present study provides similar results to the experimental analysis. In addition, the higher harmonics can be captured quite well in the present study, which demonstrates the capability of the present model for solving highly nonlinear problems. Wave radiation by a fully submerged vertical truncated cylinder undergoing forced pitching motions is also analysed. This case is then extended to simulate the scenario of a cylindrical payload hanging from a rigid cable and subjected to wave actions. Variation of cable length does not have a noticeable effect on the horizontal displacement of the cylinder. A significant increase in moment due to the increase of cable length is however evident in that case. The moment, cable

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3.4

a =0.005 a =0.01 a =0.015 a =0.02

3.2

2

T/r a

3.0

2.8

2.6

2.4 0.000

0.005

0.010

0.015

0.020

0.025

mod Fig. 29. Changes in cable tension with respect to motion amplitude of wave maker for various moving downward speeds.

Fig. 30. Snapshots of underwater cylinder attached to a rigid cable without downwards motion at 4 different time instants, a¼ 0.025, ω¼ 2.0, cbl ¼0.5: (a) t ¼15T, (b) t ¼15.25T, (c) t¼ 15.5T and (d) t ¼15.75T.

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tension and angular displacement all seem to decrease at higher frequencies. When the cylinder is allowed to move towards sea bed at a constant velocity while still under wave action, all the hydrodynamic parameters decrease in time as the cylinder moves towards the sea bed. It is found that the variation of incident wave amplitudes does not affect the moment significantly, whereas the increase of downward moving velocity can reduce the moment, cable tension and angular displacement. Acknowledgement The authors would like to thank Professor Rodney Eatock Taylor for his helpful discussions and comments on this piece of work. References An, S., Faltinsen, O.M., 2013. An experimental and numerical study of heave added mass and damping of horizontally submerged and perforated rectangular plates. Journal of Fluids and Structures 39, 87–101. Bai, W., Eatock Taylor, R., 2006. Higher-order boundary element simulation of fully nonlinear wave radiation by oscillating vertical cylinders. 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