Numerical Simulation of Vortex-induced Motion of a Deep Draft Semi-Submersible Platform

Zhenghao Liu, Weiwen Zhao, Decheng Wan* Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Key Laboratory of Ocean Engineering,

School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, China *Corresponding author

ABSTRACT The vortex-induced motion (VIM) of deep draft semi-submersibles (DDS) could have critical impact on fatigue life of mooring and riser systems. In this paper, VIM of a DDS at different current incident angles is simulated using Shear Stress Transport (SST) based delayed detached-eddy simulation (DDES) model. The numerical simulations are carried out by the in-house CFD solver naoe-FOAM-SJTU which is developed on the open source platform OpenFOAM. The nominal response of sway motion is presented and compared with experimental data. For different current incident angles, time histories, trajectories and vorticity of the DDS at different reduced velocities are reported. The result shows our CFD solver naoe-FOAM-SJTU is applicable and reliable to study VIM of semi-submersibles. KEY WORDS: SST-DDES; vortex-induced motion; naoe-FOAM-SJTU solver; deep draft semi-submersibles; mooring system INTRODUCTION Deep draft semi-submersibles (DDS) have become a trend in semi-submersible design due to its favorable heave motion characteristics compared to conventional semi-submersibles. As the draft increases, both in-line drag forces and transverse lift forces of a DDS become higher and the vortex-induced motion (VIM) of a DDS has emerged as an important issue. VIM is a complex fluid-structure interaction phenomenon, where vortex shedding occurs on a fixed or floating structure subjected in current, resulting in alternating cross-flow forces on the structure. If the vortex-shedding frequency coincides the platform’s natural frequency in calm water, the lock-in phenomenon may occur, resulting in large sway motion amplitudes of the platform. This can have significant effect on the fatigue life of mooring system and risers connected to platform. Thus, it is essential to investigate the VIM of offshore floating structures, such as spar and semi-submersible. Up to now, some researchers have studied the VIM of different types of semi-submersibles. Previous investigations were mainly carried out by model tests, which were performed in towing tanks. The models in the experiments were generally free to surge, sway and yaw motion. Waals et al. (2007) discussed the effect of mass ratio and draft on VIM of

semi-submersible platform, and found that lower mass ratio for semi-submersible may result in larger sway response. The semi-submersible with smaller column height showed much less flow induced transverse and yaw response than that with larger column height. To systematically study the VIM of a semi-submersible platform, a series of model tests were carried out at the Institute of Technological Research (IPT) by Gonçalves et al. (2012, 2013). Effects of current incidence angles, hull appendages, surface waves, external damping and draft conditions were discussed. They found that lock-in phenomenon could occur for reduced velocity (Ur) ranging from 4 to 14 at different current incidence angles. The effect of appurtenances, tow direction and wave action was investigated by Martin and Rijken (2012). They found that semi-submersibles can also exhibit vortex-induced yaw (VIY) response which may be due to shed vortices from upstream columns interacting with downstream columns. The complexity of experimental settings can lead to time consuming to study VIM of a semi-submersibles. Generally, model tests are always based on Froude scaling, which will result in much smaller Reynolds numbers at model scale than those at full scale. In addition, there are limitations in obtaining physical understanding of flow field around semi-submersibles in model tests. In recent years, numerical simulation methods based on Computational Fluid Dynamics (CFD) have been playing a more and more significant role in VIM prediction. Lee et al. (2014) studied scale effect of a deep draft semi-submersible on VIM using Reynolds averaged Navier-Stokes (RANS) method. It was found that the scale effect tends to enlarge the sway VIM motion in the full scale condition for a large reduced velocity range. Detached-eddy simulation (DES) of a DDS VIM were performed by Chen et al. (2016) and the results were compared with large eddy simulation (LES) results. The difference in VIM from the two turbulence models was not significant in their study. Kara et al. (2016) used open source toolkit OpenFOAM to study drag, decay and VIM of a DDS and the results were compared with model tests data. Three different turbulence models including URANS, DES and scale-adaptive simulation (SAS) were assessed. They proposed that DES was a powerful turbulence and recommended for CFD based VIM simulations. The objective of the present work is to investigate the VIM of a DDS

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Proceedings of the Twenty-seventh (2017) International Ocean and Polar Engineering ConferenceSan Francisco, CA, USA, June 25-30, 2017Copyright © 2017 by the International Society of Offshore and Polar Engineers (ISOPE)ISBN 978-1-880653-97-5; ISSN 1098-6189

www.isope.org

and flow field around it under different reduced velocities at 0 and 45 degree incident angles. In this paper, present CFD calculations are performed by the in-house CFD solver naoe-FOAM-SJTU. The numerical results of the sway responses at 45 degree current incident angle are presented and compared with the experiments performed by Waals et al. (2007). Details of the flow field around the DDS are also given. The results show that the current approach can be an alternative tool to investigate the VIM of semi-submersible platforms. The paper is organized as follows: next section presents the numerical methods including turbulence modeling, dynamic mesh deformation and mooring system; after that we describe the details of computational model and test conditions; In the next part, the numerical results and comparison with experiment data will be given; finally, some conclusions of this work are drawn. NUMERICAL METHODS

Turbulence Modelling

Based on the open source platform OpenFOAM, the CFD solver naoe-FOAM-SJTU is designed for computing viscous flows around ocean structures (Shen et al., 2015; Wang et al., 2015; Wang and Wan, 2016; Wang et al., 2016, Zhao and Wan, 2016), and complemented with a dynamic mesh module, a 6DoF motion module, a mooring system module and turbulence models. For turbulence modelling, URANS and DES are available. In this study, a DDES (Delayed DES) based on the SST (Shear Stress Transport) RANS model (Menter et al., 1994; 2003) is employed. The incompressible flow in the SST-DDES model (Gritskevich et al., 2012) is governed by the modified Navier-Stokes equations:

0i

i

ux

∂ =∂

, (1)

[( ) ]i ji it

j j j j

u uu upt x x x x

ν ν∂∂ ∂∂ ∂+ = − + +

∂ ∂ ∂ ∂ ∂, (2)

where iu is the time-averaged velocity in RANS and the filtered velocity in LES. ν is the kinematic viscosity, tν is the eddy-viscosity. By changing the length scale in closure equations in SST model, the equations for turbulent kinetic energy k and the dissipation rate per unit of turbulent kinetic energy ω can be modified as:

( )3( )= /j

DDES k tj j j

u kk kG k lt x x x

ν α ν ∂∂ ∂ ∂+ − + +

∂ ∂ ∂ ∂ , (3)

( ) ( )2 21

( )= 1j

t kj j j

uS F CD

t x x xω ω

ωω ωγ βω ν α ν ∂∂ ∂ ∂+ − + + + −

∂ ∂ ∂ ∂ , (4)

where * 2min( ,10 ), tG G k G v Sβ ω= = , 12 , [ ]2

jiij ij ij

j i

uuS S S Sx x

∂∂= = +∂ ∂

,

Sij is the strain rate tensor. The eddy viscosity can be calculated by 1

1 1 2max( , )ta kv

a b SFω= . (5)

In Eq. 5 F1 and F2 denote the SST blending functions which can be given as follows:

41 1tanh(arg )F = , (6)

4 21 * 2 * 2

500 4arg min[max( , ), ]k

k v ky y CD y

ω

ω

αβ ω ω

= , (7)

22 2tanh(arg )F = , (8)

22 * 2

500arg max(2 , )k vy yβ ω ω

= , (9)

1 1 1 2(1 )F Fφ φ φ= + − . (10) The length scale of DDES introduced in turbulence model equations is used to switch between the RANS and LES models. The DDES length scale in Eq. 3 is given by

max(0, )DDES RANS d RANS LESl l f l l= − − , (11)

where LES DESl C= Δ , *RANSkl

β ω= and

1 1 2 1(1 )DES DES DESC C F C F= ⋅ + ⋅ − , Δ is the filter width. To prevent grid-induced separation, an empirical shielding function fd was introduced in Eq. 11. It is computed using the following relations:

211 tanh[( ) ]dC

d d df C r= − , (12)

2 2 2 20.5( )t

dry S

ν νκ

+=+ Ω

. (13)

For SST model, the values of the constants are Cd1 = 20 and Cd2 =3. The details of the constants in SST-DDES model were provided by (Gritskevich et al., 2012). The SST-DDES model (using the proposed constants) has been validated against the decaying isotropic turbulence experiment of Comte-Bellot and Corrsin (1971). In Eq. 13, Ω is the magnitude of vorticity tensor. κ is the von Karman constant and y is the distance to the nearest wall. rd equals to1 at the nearest wall and equals to 0 at the edge of the boundary layer.

Dynamic Mesh Deformation In this study, the dynamic mesh technique is applied in all cases, which can keep the topology of the mesh unchanged during the simulation. The mesh velocity is determined by solving Laplace’s equation at each time step, and the new positions of the vertices of the mesh can therefore be obtained.

( ) 0gUγ∇ ⋅ ∇ = (14)

Where 2

1r

γ = is the quadratic inverse distance of cell center to the

nearest moving wall boundary. Ug is the grid velocity.

Mooring System Mooring system is important for the prediction of DDS VIM. In order to model the mooring system in the CFD simulation, an equivalent spring system for 6DoF motion is implemented to naoe-FOAM-SJTU solver. Thus, the motion can be modeled by means of a mass-spring-damping system. The natural frequency of this mass-spring system can be given as:

12n

kfm mπ

=+ Δ

, (15)

where k is the equivalent stiffness of the mass-spring system, m is the mass of the DDS, and mΔ is the added mass of the DDS submerged in water. For natural rotation frequency, it can be described as:

12

tnt

a

kfJ Jπ

=+

, (16)

where kt is the equivalent rotation stiffness of the mass-spring system, J is the moment of inertia of the DDS, Ja is the added moment of inertia of the DDS submerged in water. Generally, a set of horizontal springs are employed to provide the horizontal restoring force for the DDS and to match the natural frequency. Each horizontal spring line can be defined by four physical parameters: pretension, stiffness, anchor and fairlead. For different current incident angles, the four physical parameters are kept the same.

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GEOMETRY AND TEST CONDITIONS

Geometry In the present work, the deep draft semi-submersible of four square columns and four pontoons used by Waals et al. (2007) is applied for our CFD calculation. Fig. 1 illustrates the geometry of the DDS model. The column width is 14 m, the draft 35 m, and the column height above the pontoon 24.5 m in full scale. The mass of the DDS is 4.4×107 kg. The DDS length and width are estimated to be 70 m according to Chen et al. (2016). They also estimated the moment of inertia for yawing to be 4.97×1010 kg·m2. The model test and CFD simulation are performed with a scale ratio of 1:70. Main particulars of the DDS at both full scale and model scale are shown in Table 1. Table 1 Main particulars of DDS model

Parameter Notation Full-scale Model-scale Scale ratio - - 1：70

Length between columns

S (m) 56 0.8

Draft (H+P) (m) 35 0.5 Column width L (m) 14 0.2

Pontoon height P (m) 10.5 0.15

Displacement T (t) 53000 0.158 Mass M (t) 44000 0.125

Fig. 1 Geometry of DDS model Test conditions The model tests were conducted in a towing tank and for the CFD simulations, the DDS model was subjected to uniform incident current. In order to investigate the effect of incident angle, 0 and 45 degree incident angles were performed in the CFD simulation. For different incident angles, the projected column width differs which is used for the calculation of reduced velocity. The definition of the projected column width (D) is shown in Fig. 2. And the reduced velocity is

defined as nr

UTUD

= . Tn is the sway natural period The CFD

simulations were performed for the reduced velocities from 4 to 10 at both incident angles. The reduced velocities with corresponding flow velocity and the Reynolds number are shown in Table 2. The DDS model is free to 3 degree-of-freedom, i.e. surge, sway and yaw.

Fig. 2 Definition of projected column width (D) Table 2 Test conditions of the DDS model

Current heading

Reduced velocity

Flow velocity

(m/s) Re

0°

4 0.0332 0.664×104

6 0.0498 0.996×104 8 0.0664 1.328×104

10 0.0829 1.658×104

45°

4 0.0469 1.322×104 6 0.0704 1.992×104 8 0.0938 2.656×104

10 0.1172 3.316×104 Mesh Generation An automatic mesh generation tool snappyHexMesh provided by OpenFOAM is applied to generate the meshes for our computation. The tool generates mesh on an original Cartesian background mesh, splitting hexahedral cells into split-hex cells and then snap to the STL geometry. For different current incident angles, the model was rotated to aimed angle. Fig. 3 shows the arrangement of the computational domain. The boundary conditions are as follows: At the inlet, an inflow condition is adopted, in which the velocity is specified with Dirichlet condition and pressure is extrapolated from the interior solution (zeroGradient). At the outlet, zeroGradient condition is applied for velocity and Dirichlet condition for pressure. As the tests were conducted at low Froude number conditions, the effect of free surface on the VIM of the DDS is ignored. Thus, at the top boundary, the symmetry condition is enforced. Other boundaries (i.e. bottom boundary, left and right boundary) are also applied as symmetry condition. The computational domain extends to -3(L+S) < x < 6(L+S)、−3(L+S) < y < 3(L+S) and −3(L+S) < z <0. To capture the wake, the mesh regions were refined locally which is shown in Fig. 5. Fig. 4 shows the DDS model surface mesh. The total grid number is 3.58 M. The time step is 0.01 s in each case. The value of first layer y plus is less than 5.

Fig. 3 Arrangement of the computational domain

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Fig. 4 Surface mesh of the DDS model

Fig. 5 Local refined computational mesh around the DDS model RESULTS AND DISCUSSION

Free Decay Analysis For the CFD computation, free decay test has been done to verify the natural period of the DDS model. The same mooring setup was used for in-line, transverse and yaw motion with the 0 and 45 degree incident angles. Although no detailed mooring information was provided by Waals et al. (2007), the natural periods at full scale for surge, sway and yaw motion were given as 132 s, 205 s and 49 s for both incident angles. The corresponding natural periods were 15.7 s, 24.4 s and 5.9 s at model scale. Table 3 shows the natural periods for the CFD calculations in comparison with the model tests. Fig. 6 and Fig. 7 show the time histories of free decay motion at 0 and 45 degree incident angles. The natural period for surge, sway and yaw motion is 15.4 s, 24.1 s and 6.2 s, respectively. The natural periods in 45 degree incident angle is almost equal to those in 0 degree incident angle. This is because the difference in added mass can be neglected. The reduced velocity Ur for both incident angles is based on the corresponding sway natural period in calm water. Table 3 Natural periods from the free decay tests Current heading

Motion direction EFD period (s)

CFD period (s)

0° Surge (in-line) 15.7 15.4

Sway (transverse) 24.4 24.1 Yaw 5.9 6.2

45° Surge (in-line) 15.7 15.4

Sway (transverse) 24.4 24.1 Yaw 5.9 6.2

(a)

(b)

Fig. 6 Time history of free decay test for 0 degree incident angle: (a) surge and sway motion (b) yaw motion

(a)

(b)

Fig. 7 Time history of free decay test for 45 degree incident angle: (a) surge and sway motion (b) yaw motion

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Two methods namely nominal response and maximum response were used to evaluate the semi-submersible motion by Waals et al. (2007). Nominal response is based on the averaged motion amplitude and maximum response is calculated by maximum and minimum motion amplitude. Considering the DDS VIM is strongly modulated, the nominal response is adopted to analyze the DDS motion in this study. It is given by

Nominal2 [ ( )]( / ) y tA D

Dσ= (17)

where σ is the standard deviation of y(t). Since no nominal response for 0 degree incident angle was provided by Waals et al. (2007), the nominal response for 45 degree is presented and compared with the experimental data. As shown in Fig. 8, our CFD results correspond well with experimental results at Ur = 4 and 10. However, the error of nominal response between CFD results and experimental results is significant at Ur = 6 and 8. This may be due to the overprediction of VIM in model tests, which was proposed by Ma et al. (2013).

Fig. 8 VIM response for 45 degree incident angle

Time History of the Surge, Sway and Yaw Motion Fig. 9 and Fig. 10 show the time histories of surge, sway and yaw motion for 0 and 45 degree incident angles at Ur = 6 and Ur = 10, respectively. The in-line and transverse motions are normalized by the projected column width D. As shown in Fig. 9, the amplitudes of the surge, sway and yaw motion at Ur = 10 are obviously larger than amplitudes at Ur = 6. With the increase of reduced velocity, the in-line offset distance becomes larger. Both reduced velocities correspond to a region before lock-in for the sway motion, as illustrated by Waals et al. (2007). For both cases, no regular motion can be found. As shown in Fig. 10, the surge and yaw motion for 45 degree incident angle present similar trend for 0 degree incident angle. However, different trends can be found for sway motion. At Ur = 6, the sway motion becomes regular and the amplitude is larger than that at Ur = 10. This indicates that the sway motion is under lock-in condition at Ur = 6, while it is under post-lock-in condition at Ur = 10.

(a)

(b)

Fig. 9 Time histories of surge, sway and yaw motion for 0 degree incident angle: (a) Ur = 6 (b) Ur = 10

(a)

(b)

Fig. 10 Time histories of surge, sway and yaw motion for 45 degree incident angle: (a) Ur = 6 (b) Ur = 10

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VIM Trajectories for Different Incident Angles To investigate the spatial displacement of the DDS VIM, the trajectories in XY plane for 0 and 45 degree incident angles are presented in Fig. 11 and Fig. 12. The motion amplitude is normalized by the projected column width, thus the results for different incident angles can be compared. As seen, for both incident angles, the in-line offset distance increases with the reduced velocity from 4 to 10. There are no eight-shaped trajectories which are typically presented in spar VIM. The sway motion amplitude increases with the reduced velocity for 0 degree incident angle, while the sway motion first increases and then decreases for 45 degree incident angle. This may be due to the difference of lock-in regime for different incident angles. For 0 degree incident angle, although the VIM is dominated by sway motion, the surge motion covers larger displacement than that of 45 degree incident angle. For 45 degree incident angle, the VIM is dominated by sway motion at Ur = 4, 6, 8, while at Ur = 10, the trajectory demonstrates larger in-line motion displacement and smaller sway motion displacement compared to Ur = 6, 8.

Fig. 11 Trajectories in XY plane for 0 degree incident angle

Fig. 12 Trajectories in XY plane for 45 degree incident angle Vorticity Analysis Fig. 13 and Fig. 14 illustrate the z-vorticity contours for 0 and 45 degree incident angles at Ur = 6 and Ur = 10, respectively. The contours are captured at z/L = -0.5. For 0 degree incident angle, it can be seen that strong vortexes are generated behind the upstream columns, which changes the upstream region vortex distribution of downstream columns. This affects the vortex shedding from downstream columns. The downstream columns therefore show a slightly weaker vortex strength. As shown in Fig. 14, for 45 degree incident angle, downstream column also presents weaker vortex strength due to the effect of vortex shedding from upstream columns. However, different

vortex interaction between upstream columns and downstream columns can be found for 45 degree incident angle compared to 0 degree incident angle. The three upstream columns show little vortex interaction with one another. The distance between upstream column and downstream becomes larger, thus the front region of downstream column is less affected by vortex shedding from upstream columns, especially for case at low flow velocity.

(a)

(b)

Fig. 13 Z-vorticity contours for 0 degree incident angle: (a) Ur = 6 (b) Ur = 10

(a)

(b)

Fig. 14 Z-vorticity contours for 45 degree incident angle: (a) Ur = 6 (b) Ur = 10

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CONCLUSIONS This paper presents numerical simulation of the DDS VIM using SST-DDES model. Simulations at four different reduced velocities and two current incident angles are performed by the in-house CFD solver naoe-FOAM-SJTU. Predicted nominal sway motion at 45 degree incident angle is compared with the experimental results and our CFD results correspond well with experimental results at Ur = 4 and 10. For simulations at different current incident angles, the time histories of surge, sway, yaw motion, the trajectories in XY plane, the Z-vorticity at z/L = -0.5 are presented. The surge and yaw motion for 45 degree incident angle present similar trend for 0 degree incident angle. Different trend for sway motion indicates the current incident angle has great impact on the lock-in regime of a semi-submersible. Different vortex interaction between upstream columns and downstream columns can be found for 45 degree incident angle compared to 0 degree incident angle. The distance between upstream column and downstream becomes larger, thus the front region of downstream column is less affected by vortex shedding from upstream columns. This study shows the naoe-FOAM-SJTU solver implemented with SST-DDES model is capable for the simulation of semi-submersible platforms. ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (51379125, 51490675, 11432009, 51579145), Chang Jiang Scholars Program (T2014099), Shanghai Excellent Academic Leaders Program (17XD1402300), Shanghai Key Laboratory of Marine Engineering (K2015-11), Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (2013022), Innovative Special Project of Numerical Tank of Ministry of Industry and Information Technology of China(2016-23/09) and Lloyd's Register Foundation for doctoral student, to which the authors are most grateful. REFERENCES Chen, CR, and Chen, HC (2016). “Simulation of Vortex-induced

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