URANS predictions of wave induced loads and motions on ships in regular head and oblique waves at zero forward speed

Journal of Fluids and Structures 74 (2017) 178–204

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URANS predictions of wave induced loads and motions on ships in regular head and oblique waves at zero forward speed Yuting Jin, Shuhong Chai *, Jonathan Duffy, Christopher Chin, Neil Bose National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, University of Tasmania, Launceston, TAS, 7250, Australia

highlights • • • •

URANS computations are performed on individual FLNG and LNG in waves. Wave induced loads and motions are predicted for head and oblique sea conditions. Simulation results are compared against potential flow solutions and model testing data. Comparisons are made between model and full scale computational results.

article

info

Article history: Received 23 December 2016 Received in revised form 31 May 2017 Accepted 10 July 2017

Keywords: Unsteady Reynolds-averaged Navier–Stokes Wave induced loads and motions FLNG and LNG Transfer functions Zero forward speed

a b s t r a c t The paper presents predictions on the hydrodynamics of a conceptual floating liquefied natural gas (FLNG) and a liquefied natural gas (LNG) carrier in regular head and oblique sea waves using an unsteady Reynolds-Averaged Navier–Stokes (URANS) solver. Initially, a verification and validation study is performed on estimating the numerical uncertainties in the presented URANS simulations. Using the verified numerical setup, ship hydrodynamic properties including wave induced loads as well as ship motion responses are investigated for different wave conditions. The computed time history results are decomposed by Fourier Series to estimate the wave load and ship motion transfer functions. These results show good correlation with predictions from model tests and potential flow calculations. From the computations, it is observed when increasing the wave length, wave diffraction around the ship becomes less profound and the water depth starts to influence the transfer functions. Full scale computations in head and oblique sea waves are also presented and compared with model scale predictions for investigating possible scale effects. The differences are found to be close to the numerical uncertainties, indicating minor influences of scale effects on the prediction of wave loads and ship motion responses for the tested cases. © 2017 Elsevier Ltd. All rights reserved.

Nomenclature A B

Wave amplitude (m) Beam (m)

*

Correspondence to: National Centre for Maritime Engineering and Hydrodynamics, Australian Maritime College, University of Tasmania, Locked Bag 1395, Launceston, Tasmania, 7250, Australia E-mail address: [email protected] (S. Chai). http://dx.doi.org/10.1016/j.jfluidstructs.2017.07.009 0889-9746/© 2017 Elsevier Ltd. All rights reserved.

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Bij Fifth order Stokes wave coefficients CB Block coefficient d Model test basin water depth (m) D Ship moulded depth (m) f Wave frequency (Hz) Fj Wave induced force (N) FLNG/LNG FLNG or LNG vessel alone H Wave height (m) FS Full scale k Wave number (rad/m) Kxx /Kyy /Kzz Radius of gyration (m) LCG Longitudinal centre of gravity (m) from the bow LOA Length overall (m) Lpp Length between perpendicular (m) MS Model scale Mj Wave induced moment (Nm) T Draft (m) Tp Wave period (s) TF Fj Force transfer function TF Mj Moment transfer function TF Xi Ship motion transfer function VCG Vertical centre of gravity (m) from the keel Xi1 Ship motion response ω Wave frequency (rad/s) µ Wave heading, angle between the vessel heading and wave direction (deg) λ Wave length (m) η Free surface elevation (m)

1. Introduction Prediction of ship motion responses in seaway is essential for optimising hull performance as well as evaluating the operability of on-deck facilities. Numerous computational methods were developed by previous researchers on simulating such problems, attempting to make accurate estimations on ship hydrodynamic properties and behaviour. Many efforts have focused on using potential flow (PF) theory for calculating frequency domain wave loads and ship motion responses. In the past several decades, the application of PF method has evolved from two-dimensional strip theory assuming the ship is slender and wall-sided to three-dimensional panel codes which are able to deal with more complicated hull shapes. Recent examples of using PF method for seakeeping analysis and wave added resistance calculations are available in e.g. Fang and Chen (2006), Joncquez (2009), Zhu et al. (2011), Bennett et al. (2013) and Seo et al. (2014). Despite good accuracy and rapid computational duration, PF methods have common limitations, which do not take into account of breaking waves, turbulence and viscosity (Simonsen et al., 2013). With recent outstanding developments of the computer technique, the viscous Computational Fluid Dynamics (CFD) method, typically the unsteady Reynolds-Averaged Navier–Stokes (URANS) method tends to make progress. Orihara and Miyata (2003) conducted studies on the pitch and heave motions and added resistance for a S175 container ship by solving URANS equations within an overlapping grid system. The results were close to experimental data for ship models with different bow-forms. The work demonstrates the capabilities of using CFD method to simulate ship motion responses in regular oblique waves. Irvine Jr. et al. (2008) investigated the pitch and heave motions of a surface combatant model in regular head sea waves using URANS technique. From the obtained results, an empirical formula was derived to predict the Froude number for maximum motion response as functions of ship geometry coefficients. Sadat-Hosseini et al. (2013) applied the same method to predict the motion responses and added resistance of a KVLCC2 container ship in head sea waves for two conditions: fixed in surge and free to surge. The results have been validated against experimental data and showed better agreement when compared to PF solutions. By conducting local flow analysis, it was found that the added resistance was highly dependent on the pressure variation on the upper bow of the ship and correlated with bow relative motion. Simonsen et al. (2013) studied heave and pitch motions of a model scale KCS container ship in regular head waves by means of Experimental Fluid Dynamics (EFD) and CFD approaches. Model tests with uncertainty assessment were designed and conducted to validate URANS predictions on ship motion responses and added resistance. Fair agreement was observed between the CFD results and EFD measurements. Extending the work further, Simonsen et al. (2015) carried out URANS simulations on estimating added resistance of the KCS container ship in oblique waves. The results were shown to be in good agreement with experimental data from literature.

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Table 1 Principal particulars of FLNG and LNG. Hull form

FLNG MarAd F series

LNG Stretched MarAd F series

Model Scale (MS)

Full Scale (FS)

Model Scale (MS)

Full Scale (FS)

LOA (m) Lpp (m) B (m) D (m) T (m) CB LCG (m) VCG (m) Kxx /B Kyy /Lpp Kzz /Lpp

4.22 4.00 0.728 0.364 0.240 0.850 2.324 0.185 0.32 0.25 0.25

422 400 72.8 36.4 24 0.850 232.4 18.5 0.32 0.25 0.25

3.24 3.00 0.451 0.282 0.100 0.875 1.814 0.070 0.39 0.23 0.23

324 300 45.1 28.2 10 0.875 181.4 7.0 0.39 0.23 0.23

To date, the majority of URANS-based seakeeping simulations were carried out in model scale in order to compare with experimental results. However, model scale flows could show significant differences against full scale flows due to scale effects. Hochkirch and Mallol (2013) highlighted that the discrepancies between model and full scale ship hydrodynamics were mainly induced by the differences in boundary layers, flow separations and wave breaking behind transom sterns. As suggested in the literature, performing full scale analysis on ship seakeeping and manoeuvring problems is of great importance. Oh and Kang (1992) and Gao and Zhou (1995) investigated the Reynolds effects on the viscous flow around ship sterns in steady current adopting a CFD method. From the obtained results, variations of ship stern flow field and corresponding hydrodynamic characteristics due to the change of Reynolds number were observed. Kim et al. (2003) investigated the scale effects in the manoeuvring forces of a bare hull DTMB 5415 at steady drift motions using URANS computation in calm water. The discrepancy of vortical flow structures on the vessel for large angle drift motions due to the differences in model scale and full scale Reynolds number was presented, demonstrating the significance of scale effects. Bhushan et al. (2009) applied URANS simulations to investigate the hydrodynamics of self-propelled full scale and model scale DTMB 5415 during zigzag manoeuvres in calm water. The study compared the trajectories of the vessel obtained from the simulations and denoted the presence of Reynolds effects on the rudder effectiveness and propeller forces. Recently, researchers have turned their attention into the full scale surface vessel seakeeping in regular waves. Tezdogan et al. (2015) performed numerical predictions on the motion responses and added resistance of full scale KCS in head seas using a commercial URANS solver. Their simulations were validated and in a closer agreement with EFD motion response data when compared with PF predictions. This paper aims to investigate the feasibility of CFD URANS method for predicting wave induced loads and motions on ships in regular waves. Computations are presented in head and oblique sea waves at zero forward speed. Two generic hull forms, representing the FLNG platform and the LNG carrier, have been studied individually to obtain baseline wave load and motion response data for future investigation of the hydrodynamic interaction between the two ships when moored alongside each other. Two scenarios are considered: FLNG alone in waves and LNG alone in waves. Initially, model scale simulations are conducted and compared with experimental measurements for verification. Applying the same numerical setup, full scale ship hydrodynamics including wave induced loads as well as ship motion responses have been studied. 2. Ship geometry and investigation conditions In the present study, the MarAd systematic series full-form ship models from Roseman (1987) have been used to represent a generic FLNG hull design and a LNG shuttle tanker. The FLNG is an innovative type of remote LNG production and storage platform, which consists of a ship-type hull equipped with LNG storage tanks and liquefaction plants (Zhao et al., 2011). The hull forms are shown in Fig. 1. The full scale FLNG has an overall length of 422 m and a displacement of approximately 600,000 tonnes at fully loaded condition. The LNG carrier is studied for an empty hull condition ready for the liquid cargo to be loaded. The geometry particulars of the FLNG and LNG are illustrated in Table 1. The stretched MarAd F series here represents a MarAd F series hull that has been lengthened by 24.0% in the x direction, but the beam and depth remain the same ratio as the original. The two ships are studied individually in head sea and oblique sea regular waves. The tested conditions and approaches are listed in Table 2. 3. CFD method In this study, the computations have been carried out using commercial URANS solver Star-CCM+. The code utilises finite volume method to resolve the integral form of incompressible RANS equation. Time dependent URANS simulations are performed to investigate the hydrodynamic behaviour of the vessel in regular waves.

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Table 2 Matrix of studied cases.

FLNG (alone)

LNG (alone)

Degrees of freedom

Method

Wave heading µ

Free to heave and pitch; Fixed in surge, sway, roll and yaw

EFD/PF/CFD

180◦ (Head sea)

Fixed in 6 DoF

EFD/PF/CFD

180◦

Free to heave, roll and pitch; Fixed in surge, sway and yaw

PF/CFD

155◦ (Oblique sea)

Fixed in 6 DoF

EFD/PF/CFD

155◦

Free to heave and pitch; Fixed in surge, sway, roll and yaw

PF/CFD

180◦

EFD/PF/CFD

180◦

PF/CFD

155◦

EFD/PF/CFD

155◦

Fixed in 6 DoF Free to heave, roll and pitch; Fixed in surge, sway and yaw Fixed in 6 DoF

Fig. 1. Generic hull forms of (a) FLNG and (b) LNG investigated in this study.

3.1. URANS equations and turbulence model The governing equations for the two phase incompressible flow combining air and water are given by the URANS equations coupled with the conservation of continuity (Rusche, 2003):

∂ρ u + ∇ · [ρ uu] = −∇ p ∗ +g · x∇ρ + ∇ · [µ∇ u + ρτ ] + σT κγ ∇γ ∂t ∇ · u = 0.

(1) (2)

Here, u = (u, v, w) is the time-averaged velocity field in Cartesian coordinates, p∗ represents the time-averaged pressure including hydrostatic, ρ = ρ (x) is the fluid density which varies with the content of air/water in the computational cells, x = (x, y, z ) are the Cartesian coordinates, g is the gravitational acceleration, µ is the dynamic molecular viscosity and τ is the Reynolds stress tensor:

τ=

2

ρ

2

µt S − kI

(3)

3

where µt is the dynamic eddy viscosity, S = 1/2 ∇ u + (∇ u)T is the fluid strain rate tensor and k is the turbulent kinetic energy per unit mass. ∇ is the gradient operator (∂/∂ x, ∂/∂ y, ∂/∂ z ). The last term in Eq. (1) represents surface tension, where σT is the surface tension coefficient which is 0.074 kg/s2 between air and water at 20 ◦ C and κγ is the surface curvature. The presence of surface tension will only have minor effects in civil engineering applications (Jacobsen et al., 2012). The above equations are solved for air and water simultaneously, where the fluids are tracked using the volume of fraction γ · γ is 0 for air and 1 for water, and any intermediate value is a mixture of the two fluids. This is commonly known as the Volume of Fluid (VOF) method. The distribution of γ is modelled by an advection equation:

(

∂γ + ∇ · [uγ ] + ∇ · [ur γ (1 − γ )] = 0. ∂t

(

))

(4)

The last term on the left-hand side is a compression term, which limits the smearing of the interface, and ur is the relative velocity vector. Using γ , the spatial variation in fluid properties, such as ρ and µ, can be derived through weighting:

ρ = γ ρwater + (1 + γ ) ρair µ = γ µwater + (1 + γ ) µair .

(5) (6)

The closure of the URANS equations is achieved using the k − ω Shear-Stress Transport turbulence model on the basis of the description by Menter (1994) and Wilcox (2008). 3.2. Free surface and ship motion modelling As mentioned above, the Star-CCM+ software package utilises the VOF method to model free surface of segregated flows. It assumes immiscible fluid phases share velocity and pressure fields in a control volume and are governed by the same set

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Fig. 2. Dimensions of the computational domain for the presented deep water FLNG seakeeping simulations (a) side view; (b) front view – half domain; (c) front view – full domain.

of equations describing momentum, mass and energy transport in a single-phase flow. By solving such transport equations for the phase volume fraction, the spatial distribution of each phase at individual time steps are obtained. The motions of the ship in waves are simulated by the Dynamic Fluid Body Interaction module integrated in the 6 DoF motion solver. The translation and rotation of the model are described in the global coordinates while the hydrodynamic forces and moments are calculated with respect to the ship local coordinate system. In the present simulations, pitch, heave and roll motions are investigated as the primary concerns for ships operation in head and oblique seas. 3.3. Computational domain, boundary conditions and mesh generation The International Towing Tank Conference (ITTC) practical guidelines for ship CFD applications (ITTC, 2011) suggests for ship seakeeping analysis in waves, the inlet boundary should be at a distance of 1–2Lship away from the vessel, whereas the outlet boundary should be located 3–5Lship downstream in the domain. The computed fluid domain in this study is designed following this guideline to overcome blockage effect and avoid wave reflections. Two water depths are analysed: 2Lpp and 0.2Lpp. Fig. 2 shows the computational domain for the deep water case (2Lpp). It is worth mentioning that different domain settings are applied for head sea and oblique sea computations. For head sea condition, only half of the fluid domain and vessel’s physics are modelled with symmetric boundary condition in order to reduce computational time. For oblique sea conditions, where the ship’s hydrodynamics are asymmetric about the XZ plane, the entire fluid domain is solved. For each of the computations, the mass centre of the ship is located at an initial position of (0, 0, 0) in the earth-fixed global coordinate system. A velocity inlet boundary condition is applied at the upstream end of the fluid domain where incident regular waves are generated. At the downstream end of the fluid domain, a damping pressure outlet boundary is imposed to prevent wave reflections. The top, bottom and side walls of the domain are all selected as velocity inlets to avoid velocity gradient occurring from the boundaries as in the use of a slip-wall condition. The mesh is generated according to the ITTC (2011) recommendations of wave simulations. The generated mesh contains a minimum number of 80 cells per wavelength and 20 cells in wave height. Slow cell growth rate is selected to create smooth mesh transition between the refined hull surfaces and the outer boundaries. Hexahedral trimmer and surface remesher are adopted to generate global volume mesh as well as local refinements. To resolve the turbulent boundary layer, the prism layer mesh is utilised together with all y+ wall treatment, achieving y+ value of less than 1 along the hull for all computations. At the free surface, anisotropic trimmer refinements in both X and Z directions are applied to capture generated waves. An

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Fig. 3. Overview of computational grid (a) background region boundary conditions; (b) overset grid capable for dynamic motions; (c) overset region boundary conditions; (d) surface mesh on the FLNG ship model. Table 3 The number of elements for the presented computations. Computation conditions

Number of elements Background

Overset

Model scale Head sea Oblique sea

Total

1,868,075 3,736,150

3,118,010 6,236,020

4,986,085 9,972,170

Full scale Head sea Oblique sea

1,868,075 3,736,150

5,300,202 10,600,404

7,168,277 14,336,554

overset grid is employed in the performed computations to simulate the ship hydrodynamics. The method requires separate mesh enclosing the ship in an overlapping domain. Flow information is linearly interpolated between the background domain and the overset region. Fig. 3 illustrates the generated computational grid for the background domain, overset region and ship hull respectively. A summary of grid size for each computational case is shown in Table 3. The time-step resolution (1t = Tp /(4.8 × N )) is selected according to Star-CCM+ practice for wave capturing, where Tp is the wave period and N is the number of grid cells per wave length. The Courant number of free surface is kept less than 0.25. Details of the grid and time-step independence study are presented in a later section. 3.4. Wave generation and data processing The fifth order Stokes waves are recommended by CD-Adapco (2014) for seakeeping simulations as it resembled a real wave more closely than one generated by the method using the first order wave theory. This higher order wave theory is based on the work by Fenton (1985) and is suggested by DNV (2010) for wave generations in this study. The expression of free surface elevation η (t ) is written as: kη (t ) = (kH /2) cos ωt + (kH /2)2 B22 cos 2ωt + (kH /2)3 B31 (cos ωt − cos 3ωt ) + (kH /2)4 (B42 cos 2ωt + B44 cos 4ωt ) + (kH /2)5 (− (B53 + B55 ) cos ωt + B53 cos 3ωt + B55 cos 5ωt )

(7)

where k is the wave number defined as 2π/λ; λ is the wave length; H is the wave height ; ω is the wave frequency and Bij are coefficients in the fifth order wave solution. Existing work by Tezdogan et al. (2015) applied this theory to generate

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Fig. 4. Layout of the numerical wave probes to monitor surface elevation for wave generation validation.

regular waves in their computations. Their results show a 3.20% difference when compared with input wave height, which is considered to be acceptable for seakeeping studies. In the present study, a number of different wave conditions are investigated and presented in Table 4. The waves generated in the virtual model test basin are monitored by a set of numerical wave probes located at 1, 1.5 and 2 ship lengths from the inlet boundary as shown in Fig. 4. Wave elevations are measured at different locations from the inlet boundary without the presence of the ship model. This is to ensure consistent waves propagating through the area of interest. Fig. 5 shows comparisons between computed wave profiles and theoretical wave elevations at Wave Probe 2 for wave conditions 5 and 9. Approximately 2% difference in the magnitude of wave height is observed for both model scale and full scale waves. It is also necessary to highlight that the computed wave crests and troughs are asymmetric due to the nature of non-linear Stokes wave theory. Table 5 summarises the numerical discrepancies between CFD predicted and theoretical wave elevations for the listed wave conditions. It can be seen that the computations generate accurate waves at the locations of interest, with less than 2% difference in wave crest and trough magnitude for all the cases. The wave induced loads and motions on the ship within the defined time period are decomposed by Fourier Series. Each set of these unsteady data F (t ) can be represented as shown in Eq. (8): F (t ) = F0 +

N ∑

Fn · cos (nω0 t + γn ) ,

n = 1, 2, 3, . . .

(8)

n=1

Fn = an =

√ 2

a2n + b2n

T

bn = −

Tp

∫

(9)

F (t ) cos (ω0 nt ) dt

(10)

0

2

Tp

∫

T

F (t ) sin (ω0 nt ) dt

(11)

0

γn = arctan

(

bn

)

an

(12)

where ω0 is the frequency of the harmonic function; Fn is the n th harmonic amplitude and γn is the corresponding phase angle. The 0th harmonic amplitude F0 in the Fourier Series is the average value of the time history of F (t ), which can also be expressed as: F0 =

1 T

Tp

∫

F (t ) dt .

(13)

0

The 1st harmonic amplitude F1 refers to the linear term from the unsteady time histories. Simonsen et al. (2013) applied the 3rd order Fourier Series to decompose the wave resistance and motion results on a ship advancing in head sea regular

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Fig. 5. CFD time history wave elevation results for wave condition 5: (a) model scale waves at numerical probes; (b) model scale waves compared to theoretical prediction at Wave Probe 2; (c) full scale waves at numerical probes; (d) full scale waves compared to theoretical prediction at Wave Probe 2. Table 4 Summary of studied wave conditions. Wave height H (m)

Wave length λ (m)

Wave frequency f (Hz)

Wave steepness H /λ

Wave/ship length λ/LFLNG (λ/LLNG )

Model scale 1 2 3 4 5 6 7 8

0.04 0.04 0.04 0.04 0.04 0.04 0.04 0.04

1.2 1.4 1.6 2.0 2.4 2.8 3.2 4.0

1.147 1.060 0.991 0.885 0.808 0.747 0.699 0.625

1/30 1/35 1/40 1/50 1/60 1/70 1/100 1/150

0.30 (0.38) 0.35 (0.44) 0.40 (0.50) 0.50 (0.63) 0.60 (0.75) 0.70 (0.88) 0.80 (1.00) 1.00 (1.25)

Full scale 9 10 11 12

4.00 4.00 4.00 4.00

240 280 320 400

0.0808 0.0747 0.0699 0.0625

1/60 1/70 1/100 1/150

0.60 (0.75) 0.70 (0.88) 0.80 (1.00) 1.00 (1.25)

Wave condition

waves. It is found that the second and third order components make a large portion (up to 50%) of the first order amplitude. This paper only focuses on the 1st harmonics of the Fourier Series, since there is no forward speed imposed on the ship and therefore no strong non-linearity is expected in the results. The 1st harmonics of the Fourier Series for heave, roll and pitch motions are represented by transfer functions TFXi (i = 3, 4, 5) respectively, TFXi =

Xi1 A

(14)

where A is the wave amplitude, Xi1 (i = 3, 4, 5) represents the 1st harmonic amplitudes for heave, roll and pitch, respectively. Applying an analogous methodology, the wave loads on the ship are expressed as transfer functions, TFFj and TFMj : TFFj = TFMj =

Fj A Mj A

(15) (16)

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Table 5 Comparison between CFD wave generation and theoretical 5th order Stokes waves. Wave condition

1 2 3 4 5 6 7 8 9 10 11 12

Wave Probe 1

Wave Probe 2

Wave Probe 3

Diff in crest Uc (%)

Diff in trough Ut (%)

Diff in crest Uc (%)

Diff in trough Ut (%)

Diff in crest Uc (%)

Diff in trough Ut (%)

−1.69% −1.41% −0.67% −1.47% −0.69% −0.87% −0.71% −0.10% −0.81% −0.68% −0.93% −0.55%

−0.68% −0.35% −0.96% −0.28% −0.55% −1.01% −0.50% −0.02% −0.56% −0.83% −0.67% −0.34%

−1.79% −2.13% −1.67% −1.12% −0.61% −0.26% −0.87%

−0.92% −0.71% −0.97% −0.39% −0.28% −0.13% −0.85%

0.19%

0.17% −0.21% −0.12% −0.33% 0.29%

−1.97% −2.13% −0.86% −0.82% −0.81% −0.65% −0.74% −0.21% −0.46% −0.32%

−0.97% −0.39% −0.63% −0.41% −0.76% −0.23% −0.78% −0.30% −0.60% −0.63% −0.34% −0.13%

−0.25% 0.03%

−0.71% 0.23%

0.03% 0.02%

Table 6 Physical model scale testing matrix. Degrees of freedom

µ

d/T

Wave condition

No. of repeats

H (m)

f (Hz)

FLNG alone

Free to heave and pitch Fixed in 6 DoF Fixed in 6 DoF

180◦ 180◦ 155◦

3.3 3.3 3.3

0.04 0.04 0.04

0.625–1.147 0.699–1.060 0.625–0.808

3 repeats for f = 0.885 Hz 3 repeats for f = 0.808 Hz 3 repeats for f = 0.747 Hz

LNG alone

Fixed in 6 DoF Fixed in 6 DoF

180◦ 155◦

8.0 8.0

0.04 0.04

0.625–1.147 0.625–1.147

3 repeats for f = 0.885 Hz 3 repeats for f = 0.885 Hz

where Fj and Mj (j = x, y, z) are the 1st harmonic amplitudes for wave induced force and moment on the ship hull. It must be noted that in the performed computations, the wave induced loads are evaluated referring to its body fixed coordinate system. 4. Physical model scale tests A series of model tests were conducted to obtain validation data for the presented CFD simulations. Uncertainty analysis was carried out for the tested conditions. Comparisons between the experimental measurements and numerical predictions are made and discussed in the results section. 4.1. Model test setup The experimental study was performed in the Model Test Basin at the Australian Maritime College, University of Tasmania. The basin is 35 m long, 12 m wide, and the water depth set for the tests in this study was 0.8 m, which corresponds to the water depth to draft ratio of d/TFLNG = 3.3 or d/TLNG = 8.0. Regular waves were generated by a multi-element wavemaker with 16 paddles. The model tests were carried out at the scale of 1:100 for both FLNG and LNG models. The objective of these tests was to validate the setup for the CFD computations. The tested cases are listed in Table 6. Fig. 6 gives an overview of the model test setup in the Model Test Basin. Wave energy dissipation devices were located at the end of the basin to reduce the influence of wave reflection. During the tests, the FLNG/LNG model was mounted to a carriage by a starboard forward post, a port forward post and an aft post. Force transducers were placed at the base of the posts for the measurement of wave loads. Ball joints were used to connect the base of the load cells to the ship model. The aft ball joint incorporated a slide oriented parallel to the longitudinal centreline of the model. The forward starboard ball joint incorporated a slide oriented perpendicular to the aft slide. The port ball joint did not incorporate a slide. The forward and aft post connections were located at the ship model’s VCG. For the free heave and pitch tests, the posts were free to move vertically so the ship was able to heave and pitch in waves. Linear variable differential transducers (LVDTs) were attached to the posts in order to measure the vertical displacement of the vessel. For the constrained cases, the posts were fixed in the vertical plane, constraining the model in 6 DoF motions. Challenges were encountered to fix the FLNG hull firmly due its large displacement. 4.2. Experimental results and uncertainties Ship motion responses and wave loads on the model hull are derived from data recordings of the LVDTs and load transducers. Examples of force, moment and motion measurements are given in Fig. 7. Experimental uncertainties are investigated in order to provide data for the CFD verification and validation study. Uncertainty analysis is carried out using a similar method as suggested by Duffy (2008). A summary of the experimental uncertainties for each of the tested cases are

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Fig. 6. Schematic illustration of experimental setup at model test basin.

Table 7 Experimental uncertainties of the measurements for FLNG. f (Hz)

1.147 1.060 0.991 0.885 0.808 0.747 0.699 0.625

FLNG free to heave and pitch in head sea

FLNG 6 DoF motion fixed in head sea

FLNG 6 DoF motion fixed in oblique sea

UD X31 (%)

UD X51 (%)

UD Fx (%)

UD Fz (%)

UD My (%)

UD Fx (%)

UD Fy (%)

UD Fz (%)

UD My (%)

UD Mz (%)

14.68 10.25 5.15 5.92 3.38 2.30 2.48 3.76

7.15 3.53 4.76 2.17 3.72 4.29 3.62 2.10

2.35 2.43 2.45 2.36 2.44 2.30 2.24 2.51

7.20 6.89 3.54 8.72 5.60 2.44 2.59 6.68

2.72 2.01 2.59 1.67 1.68 2.59 2.00 1.63

– – – 1.88 1.81 1.89 1.93 2.29

– – – 2.11 1.35 3.00 10.5 6.10

– – – 6.71 4.11 3.89 7.57 12.46

– – – 2.14 4.34 3.56 4.21 2.00

– – – 1.64 2.09 1.25 1.21 1.06

presented in Tables 7 and 8. For estimation purposes it is assumed that the uncertainties on the motion and force/moment measurements arise from four sources: LVDT calibration factor; LVDT voltage measurement; load transducer calibration factor; load transducer voltage measurement. Appropriate confidence interval estimates from manufacturer specifications for each of these sources are used to find the uncertainty in each of the LVDT and load transducer measurements. In turn, these estimates are used to find the uncertainty on the wave induced motion, force and moment predictions. Uncertainty sources that are smaller than 1/4th or 1/5th of the largest sources are considered negligible (Longo and Stern, 2005). 5. CFD verification and validation study Before proceeding to the systematic computations, it is necessary to validate the CFD simulations against experimental data. Therefore, the model testing condition, FLNG free to heave and pitch in 0.808 Hz head sea waves is replicated in CFD computation and based on the obtained results, a formal verification and validation study is performed to assess the

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Table 8 Experimental uncertainties of the measurements for LNG. f (Hz)

1.147 1.060 0.991 0.885 0.808 0.747 0.699 0.625

LNG 6 DoF motion fixed in head Sea

LNG 6 DoF motion fixed in oblique sea

UD Fx (%)

UD Fz (%)

UD My (%)

UD Fx (%)

UD Fy (%)

UD Fz (%)

UD My (%)

UD Mz (%)

3.36 3.40 3.20 3.57 3.78 4.57 4.16 2.78

2.91 4.31 3.68 2.39 2.95 5.98 8.24 3.84

3.48 2.25 2.49 3.30 2.46 2.18 1.63 1.86

3.79 3.17 3.06 3.54 4.43 3.96 3.42 2.72

9.01 2.91 2.57 4.46 8.58 5.74 3.53 2.49

4.16 4.61 2.64 2.27 5.54 7.18 4.50 3.35

2.37 2.25 2.94 2.56 1.69 1.66 1.85 2.16

4.10 5.23 15.02 6.83 3.30 3.16 3.23 3.93

Table 9 Number of elements for in the grid convergence study. Mesh configuration

Fine (1) Medium (2) Coarse (3)

Number of elements Background

Overset

Total

1,868,075 1,868,075 1,868,075

9,285,487 3,118,010 1,095,193

11,153,562 4,986,085 2,963,268

numerical uncertainties following the procedures provided by Wilson et al. (2001) and Stern et al. (2001). It is assumed that the numerical uncertainty USN comprises of iterative convergence uncertainty UI , grid uncertainty UG and time-step uncertainty UT , giving the following expression: 2 USN = UI2 + UG2 + UT2 .

(17)

Tezdogan et al. (2015) estimated the iterative convergence uncertainty for container ship motion response simulations in Star-CCM+ URANS solver. Their analysis indicates UI of less than 0.20% in simulation results, which is considered to be negligible. As similar computational settings are applied, grid and time-step uncertainties are of primary interest in the present study. The grid and time-step convergence study are both conducted with triple solutions systematically. The grid uncertainty analysis is performed with the smallest time-step, whilst the time-step independence study is carried out with the finest mesh. In the grid convergence study, refinement is applied only to the overset region while the mesh size in the background domain remains unaltered. By scaling the mesh √ base size in the overset region, the coarse, medium and fine grid conditions are created by means of a refinement ratio, rG = 2. Based on this ratio, the number of elements for each mesh configuration is listed in Table 9. Referring to Simonsen et al. (2012) and Jin et al. (2016a, b), the grid uncertainty UG of computations with unstructured trimmer mesh can be estimated based on Richardson extrapolation. Variations of simulation results between the coarse (S3 ), medium (S2 ) and fine (S1 ) mesh conditions are identified as,

εG32 = S3 − S2 εG21 = S2 − S1 .

(18) (19)

The numerical convergence ratio can be calculated from Eq. (20), RG = εG21 /εG32 .

(20)

Depending on the value of the convergence ratio, four conditions can be predicted, (i) (ii) (iii) (iv)

0 < RG < 1, monotonic convergence RG < 0; |RG | < 1 oscillatory convergence 1 < RG , monotonic divergence RG < 0; |RG | > 1, oscillatory divergence.

It is noted that for grid divergence (iii) and (iv), the numerical uncertainty cannot be estimated. For the oscillatory convergence (ii), the numerical uncertainty can be estimated by bounding the error based on oscillation with upper limit SU and lower limit SL using Eq. (21),

⏐ ⏐ ⏐1 ⏐ ⏐ UG = ⏐ (SU − SL )⏐⏐ . 2

(21)

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Fig. 7. Time history measurement of wave induced motions and forces/moments from experiment: (a) heave motion of FLNG in 0.699 Hz head sea waves; (b) pitch motion of FLNG in 0.699 Hz head sea waves; (c) surge force of FLNG fixed in 6 DoF in 0.808 Hz head sea waves; (d) pitch moment of FLNG fixed in 6 DoF in 0.808 Hz head sea waves; (e–f) surge and sway force of LNG fixed in 6 DoF in 0.699 Hz oblique sea waves; (g–h) pitch and yaw moment of LNG fixed in 6 DoF in 0.699 Hz oblique sea waves.

∗ For grid convergence (i), the generalised Richardson extrapolation can be employed to estimate the numerical error δRE G1 and order of accuracy pG as,

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Table 10 Grid convergence study. rG

Solutions

√ √2 √2 √2 √2

TF x3 TF x5 TF Fx TF Fz TF My

2

S1

S2

S3

0.245 0.231 673.1 3664.1 3478.9

0.243 0.229 671.1 3675.7 3469.3

0.239 0.226 668.6 3621.9 3458.1

RG

Convergence

∗ δRE (%S1 ) G1

UG (%S1 )

EFD

0.65 0.65 0.80 −0.22 0.86

MC MC MC OC MC

−1.65 −1.58 −1.19 −1.66

2.69 2.61 2.13 0.73 3.63

0.229 0.212 – – –

RT

Convergence

∗ δRE (%S1 ) T1

UT (%S1 )

EFD

−0.91 −0.64

OC OC MC MC OC

– –

1.04 2.00 1.44 2.00 1.26

0.229 0.212 – – –

–

Table 11 Time step convergence study.

TF x3 TF x5 TF Fx TF Fz TF My

rT

Solutions T1

T2

T3

2 2 2 2 2

0.245 0.231 673.1 3664.1 3478.9

0.240 0.237 669.6 3651.1 3534.9

0.245 0.228 663.9 3633.0 3446.9

εG21

∗ δRE = G1

ln εG32 /εG21 ln (rG )

−0.83 −0.90 –

(22)

p

rG G − 1

(

pG =

0.61 0.72 −0.64

) .

(23)

There are two ways to estimate the grid uncertainty depending on whether the solutions are close to the asymptotic range. This is determined by the correction factor CG defined as, p

CG =

rG G − 1 p rGGest

(24)

−1

where pGest is an estimate for the limiting order of accuracy. If CG → 1, the solutions are close to the asymptotic range. In ∗ this case, the numerical error δSN , benchmark result SC and uncertainty UGC can be estimated from, ∗ ∗ δSN = CG × δRE G1

(25)

∗ SC = S − δSN

{( UGC =

(26)

} )⏐ ∗ ⏐ ⏐ 2.4(1 − CG ) + 0.1 ⏐δRE G1 |1 − CG | < 0.125 ⏐ ⏐ . |1 − CG | ≥ 0.125 |1 − CG | ⏐δ ∗ ⏐ 2

(27)

RE G1

When CG is sufficiently greater than 1, which means the solutions are far away from the asymptotic range, the numerical uncertainty UG can be calculated from Eq. (28),

{(

}

∗ ⏐| 9.6(1 − CG )2 + 1.1 ⏐δRE | ⏐ ∗ ⏐G1 1 − CG < 0.125 . ⏐ |1 − CG | ≥ 0.125 (2 |1 − CG | + 1) ⏐δ

UG =

)⏐

⏐

(28)

RE G1

The same methodology is applied to analyse the time-step convergence uncertainty UT in the present computations. The time-step refinement ratio is set to be rT = 2. The numerical uncertainty assessment is conducted for the case when the FLNG is free to pitch and heave in head sea regular waves (wave condition 5). The grid and time-step uncertainty results are listed in Tables 10 and 11. Reasonably small numerical uncertainties are estimated for the predicted forces, moment and motion transfer functions. This implies that the current numerical configuration is suitable for further systematic computations. To validate the simulation results against the EFD data, it is necessary to check if the absolute comparison error E is smaller than the validation uncertainty UV calculated using, UV =

√

2 USN + UD2 .

(29)

Here, UD is the experimental uncertainty. As a result from the convergence study, the numerical uncertainties USN of the predicted forces, moments and transfer functions are given in Table 12. Due to the model test program performed, experimental results are only available for the heave and pitch transfer functions, TF x3 and TF x5 . Confidence is achieved on the accuracy of the computation since the validation uncertainty UV of TF x3 and TF x5 are greater than the comparison error E.

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Table 12 Validation of performed numerical predictions.

TF x3 TF x5 TF Fx TF Fz TF My

USN (%CFD)

UD (%CFD)

Uv (%CFD)

E (%CFD)

3.89 4.72 3.63 2.78 4.86

3.38 3.72 – – –

5.15 6.01 – – –

2.74 5.87 – – –

Table 13 Computational matrix for CFD and PF. Model

Degrees of freedom

Depth (m)

d/T FLNG (d/TLNG )

µ

Wave condition

Model scale (1:100)

FLNG/LNG

Fixed in 6 DoF Free to pitch and heave Free to pitch and heave Fixed in 6 DoF Free to pitch, heave and roll

d d d d d

= 0.8 = 0.8 = 8.0 = 0.8 = 8.0

3.3 (8.0) 3.3 (8.0) 33.3 (80.0) 3.3 (8.0) 33.3 (80.0)

180◦ 180◦ 180◦ 155◦ 155◦

1–8 1–8 1–8 1–4 1–4

Full scale

FLNG/LNG

Free to pitch and heave Free to pitch, heave and roll

d = 800.0 d = 800.0

33.3 (80.0) 33.3 (80.0)

180◦ 155◦

9–12 9–12

Scale

6. Computational results The results gathered from CFD computations are compared with model testing data and those obtained from PF solver AQWA, which utilises a three-dimensional panel method solving inviscid, irrotational and incompressible fluid around floating structures. For more theoretical explanation, reference can be made to the manual of the software (ANSYS, 2013). The presented CFD simulations are firstly conducted at the model test basin water depth of d = 0.8 m. Based on the same setup, the hydrodynamics of the ships are computed in the water depth of d = 8.0 m and its corresponding full scale conditions (d = 800 m). The computation matrix is given in Table 13. Discussions on the model scale and full scale ship hydrodynamics in different wave environments are presented with respect to wave induced loads and motions. 6.1. Wave induced loads in head seas (ship fixed in 6 DoF) First, wave induced forces and moments on individual FLNG and LNG vessel are computed at model scale when the ship is fixed in 6 DoF. Simulations are carried out at both model test basin water depth (d/TFLNG = 3.3 or d/TLNG = 8.0) and deeper water (d/TFLNG = 33.3 or d/TLNG = 80.0) conditions for different wave frequencies. Time history predictions of wave induced surge and heave force as well as pitch moment on the FLNG and LNG individually in head seas over eight encountered wave periods are given in Fig. 8. As explained previously, Fourier Series expansion is applied on these time history recordings for different wave conditions to obtain frequency domain results. Illustrations of wave force/moment transfer functions for the studied wave frequencies are given in Fig. 9. The CFD computational results are compared with that from PF simulations and EFD. Discrepancies between numerical and experimental results are presented in Tables 14 and 15. For the conditions when the FLNG and LNG are fixed in 6 DoF in head sea waves, difference between model scale experiments and PF predictions on wave induced loads varies between 0.76% and 41.85% in the frequency domain. It is observed that the CFD simulations show slightly better performance with less comparison error (0.60%–34.93%) on the force and moment predictions. Relatively large discrepancies are observed on TF Fz of the FLNG between the numerical prediction and experimental data. This may due to slack connections between the vessel and the testing rig. As mentioned in the experiment setup, challenges were encountered to fix the FLNG hull firmly due to its large displacement (close to 590 kg). However, it is necessary to highlight that CFD predictions of head sea wave forces and moments generally agree well with PF solutions. This is expected, because force and moment results from the studied conditions are inertia dominated whilst viscous effects are less prominent. 6.2. Motion responses in head seas (ship free to heave and pitch) The previous section demonstrated the capability of presented CFD computations on predicting wave induced loads on large floating structures fixed in 6 DoF. A satisfactory level of accuracy was achieved when comparing the CFD results with model testing data and PF solutions. In this section, based on the existing CFD model, a more sophisticated condition of floating structures operating in head sea waves is modelled by releasing heave and pitch motions on the ship in the numerical wave basin. Initially, model scale simulations are carried out at the model test water depth (d/TFLNG = 3.3 or d/TLNG = 8.0) for the FLNG vessel in order to make comparisons against EFD measurements. After a satisfactory level of correlation is

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Table 14 Comparison of numerical simulation results against EFD for FLNG fixed in 6 DoF in head sea waves. f (Hz)

1.147 1.060 0.991 0.885 0.808 0.747 0.699 0.625

E(%) TF Fx

E(%) TF My

E(%) TF Fz

CFD

PF

CFD

PF

CFD

PF

–

–

−6.20 −6.52

– 25.82 −14.46 16.25 7.11 −0.03 −17.92 –

–

−9.20 −10.65 −1.17

– 22.50 −15.76 16.44 8.95 −5.00 −24.44 –

– 41.85 40.12 13.18 32.23 25.84 28.69 –

2.65 5.27 12.79 23.89 –

4.73 11.13 19.05 –

−3.75 14.71 −22.38 29.38 24.13 20.48 –

Table 15 Comparison of numerical simulation results against EFD for LNG fixed in 6 DoF in head sea waves. f (Hz)

1.147 1.060 0.991 0.885 0.808 0.747 0.699 0.625

E(%) TF Fx

E(%) TF My

E(%) TF Fz

CFD

PF

CFD

PF

CFD

PF

−0.60 −6.92

−1.03 −3.54

6.07 19.47 0.85 −6.22 4.40 −2.36

6.31 17.52 0.97 1.21 8.01 −1.44

−33.44 −17.76 −5.50 −24.16 −5.33

−23.93 −15.61 −2.42 −17.46

−13.03 −30.64 −34.93 −0.16 −6.81 −14.83 −6.40 −14.50

−13.77 −25.29 −31.95

4.45 9.99 −2.45 2.78

7.22 −4.00 0.94

0.76

−6.10 −13.88 2.57

−7.54

Table 16 Comparison of numerical simulation results against EFD for FLNG free to heave and pitch in regular head sea waves at d/TFLNG = 3.3. f (Hz)

1.147 1.060 0.991 0.885 0.808 0.747 0.699 0.625

E(%) TF x3

E(%) TF x5

CFD

PF

CFD

PF

7.99 18.59 22.03 1.50 2.81 18.13 11.43 11.33

31.45 26.86 34.31 1.82 5.01 14.40 7.51 9.24

−5.50 −1.16 −14.80

−10.26 −6.18 −12.66

5.00 6.24 −12.50 1.25 5.47

1.91 −2.56 −3.91 0.04 5.97

established, the CFD computations are extended to predict deep water (d/TFLNG = 33.3 or d/TLNG = 80.0) wave loads and ship motions with the depth of the numerical tank being enlarged ten times. Furthermore, simulations are conducted at full scale for both FLNG and LNG vessel in head sea regular waves (d/TFLNG = 33.3 or d/TLNG = 80.0) to investigate possible scale effects. Wave induced loads and motions for the studied FLNG and LNG hulls at different water depth are given in the frequency domain in Fig. 10. Comparisons are made between the results obtained from CFD, PF (AQWA) and EFD studies. Due to the model test program performed, experimental data are only available for the FLNG heave and pitch motions. The numerical predictions show good correlation with the EFD measurements for the FLNG. The comparison error varies from 1.16% to 22.05% for the CFD simulations and 0.04%–34.31% for the PF results respectively, as given in Table 16. It is seen that the obtained numerical results demonstrate discrepancies in low frequency waves for the simulated model test water depth (d/TFLNG = 3.3 or d/TLNG = 8.0) and deeper water (d/TFLNG = 33.3 or d/TLNG = 80.0) conditions. This can be explained by the difference in the behaviour of deep water and intermediate water waves. For a given wave frequency, when the water depth is less than half of the deep water wave length, the wave celerity reduces and results in shortened wave length comparing to that in deep waters. Full scale predictions of head sea wave induced loads and motions on the FLNG/LNG in deep water are presented in Fig. 10. It is observed that for both the FLNG and LNG hulls, full scale ship motions and wave forces agree very well with model scale results. The difference varies from 0.10% to 2.78% and is found to be less than the estimated numerical uncertainties. This demonstrates viscous effects are not significant in the presented cases and consequently, scale effects have minor impact on the predictions. This finding is also implied by the similarities between CFD and PF results. In addition, time history predictions of full scale ship motions and wave loads are compared with model scale recordings as shown in Fig. 11, once again demonstrating scale effects are negligible in the current study.

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Fig. 8. Time history predictions of wave induced forces/moments on FLNG/LNG fixed in 6 DoF from CFD: (a) surge force in 0.808 Hz head sea waves; (b) surge force in 0.699 Hz head sea waves; (c) pitch moment in 0.808 Hz head sea waves; (d) pitch moment in 0.699 Hz head sea waves; (e) heave force in 0.808 Hz head sea waves; (f) heave force in 0.699 Hz head sea waves.

To gain an understanding of the wave pattern around the ships, full scale contours of wave diffraction around ship hulls after eight encountered wave periods are obtained and illustrated in Fig. 12. It can be seen that as the wave length increases, wave diffraction becomes less noticeable. 6.3. Wave induced loads in oblique seas (ship fixed in 6 DoF) Adopting a similar approach as for modelling the head sea condition, CFD simulations are performed to predict wave loads and ship motions in the regular oblique sea waves at model test water depth d = 0.8 m (d/TFLNG = 3.3 or d/TLNG = 8.0). Firstly, wave induced loads are computed when the FLNG/LNG is fixed in 6 DoF in the oblique sea waves. Illustrations of time history results over eight encountered wave periods are shown in Fig. 13. Applying Fourier Series expansion, the computed time histories are converted to transfer functions in the frequency domain. Comparisons of these results are made between CFD, PF and EFD as shown in Fig. 14. Discrepancies between numerical and experimental data are presented in Tables 17 and 18.

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Fig. 9. EFD, PF and CFD predictions of wave induced force/moment transfer functions for FLNG/LNG fixed in 6 DoF in head sea waves: (a) surge force transfer function of FLNG; (b) surge force transfer function of LNG; (c) heave force transfer function of FLNG; (d) heave force transfer function of LNG; (e) pitch moment transfer function of FLNG; (f) pitch moment transfer function of LNG.

For the condition when FLNG/LNG is fixed in 6 DoF individually in oblique sea waves, the CFD simulations show similar level of accuracy as PF predictions when comparing against EFD data. The comparison error varies from 0.06% to 38.14% for the studied wave frequencies. Again, it is demonstrated that CFD predictions of oblique sea wave forces and moments are very close to the PF solutions.

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Fig. 10. EFD, PF and CFD predictions of wave induced force/moment transfer functions for FLNG/LNG free to heave and pitch in head sea waves: (a) surge force transfer function of FLNG; (b) surge force transfer function of LNG; (c) heave motion transfer function of FLNG; (d) heave motion transfer function of LNG; (e) pitch motion transfer function of FLNG; (f) pitch motion transfer function of LNG.

6.4. Motion responses in oblique seas (ship free to heave, roll and pitch) In the previous section, model scale CFD simulations of FLNG/LNG fixed in 6 DoF in oblique sea waves were performed individually at model test water depth (d/TFLNG = 3.3 or d/TLNG = 8.0). The computed wave forces and moments show good

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Fig. 11. Model scale and full scale time history predictions of wave induced loads and motions in 0.699 Hz deep water head sea waves by CFD: (a) model scale surge force; (b) full scale surge force; (c) model scale heave motion; (d) full scale heave motion; (e) model scale pitch motion; (f) full scale pitch motion.

Table 17 Comparison of numerical simulation results against EFD for FLNG fixed in 6 DoF in oblique sea waves at d/TFLNG = 3.3. f (Hz)

0.808 0.747 0.699 0.625

E(%) TF Fx

E(%) TF Fy

E(%) TF Fz

E(%) TF My

E(%) TF Mz

CFD

PF

CFD

PF

CFD

PF

CFD

PF

CFD

PF

−21.25 −26.36 −16.30 −10.23

−17.56 −22.32 −13.67 −5.05

−17.07 −30.25 −30.22

−13.54 −27.04 −30.67

−9.95 −25.09

8.42

17.98 −2.31 −3.95 −23.81

−12.79 −25.00

4.92

13.75 −2.50 −3.75 −16.06

−27.20 −26.36 −18.19 −13.59

−22.16 −23.73 −15.32 −10.85

2.67

10.47

−13.26

−9.60

correlation with EFD results and PF solutions. After releasing heave, roll and pitch motions of the vessel in the CFD model, motion responses in the oblique sea waves are investigated for a deeper water condition (d/TFLNG = 33.3 or d/TLNG = 80.0). Furthermore, full scale CFD computations of the FLNG/LNG free to heave, roll and pitch in oblique seas are performed and compared with model scale results. An illustration of time history predictions on oblique sea wave induced motions is presented in Fig. 15. It can be seen that the model scale results agree well with full scale predictions for heave, roll and pitch motions. Comparisons of the frequency

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Fig. 12. Full scale CFD contour representations of wave diffractions around the FLNG and LNG for different wave frequencies (a) FLNG in 0.808 Hz head sea waves; (b) FLNG in 0.747 Hz head sea waves; (c) FLNG in 0.699 Hz head sea waves; (d) FLNG in 0.625 Hz head sea waves.

Table 18 Comparison of numerical simulation results against EFD for LNG fixed in 6 DoF in oblique sea waves at d/TLNG = 8.0. f (Hz)

0.808 0.747 0.699 0.625

E(%) TF Fx

E(%) TF Fy

E(%) TF Fz

E(%) TF My

E(%) TF Mz

CFD

PF

CFD

PF

CFD

PF

CFD

PF

CFD

PF

−5.00 −0.06

−1.80

−38.14 −27.00 −15.32 −7.95

−34.79 −18.33 −10.60 −3.77

−2.80 −15.14 −16.90 −3.41

−3.27

−10.00

−3.53

1.65 −8.93 2.32

2.11 4.57 7.58

8.22 8.84 10.33

9.29 0.57 −0.89 15.38

14.64 3.91 1.85 17.73

13.58 7.50

7.07 17.69 10.40

domain results are made against PF solutions as shown in Fig. 16. It is worth mentioning here that the roll motion transfer functions solved in AQWA are corrected by introducing additional viscous damping. A reference on estimating the viscous damping components can be made to Himeno (1981). For heave and pitch motions, it is seen that the CFD results agree well with PF predictions. The estimated comparison difference varies between 0.27% and 7.00%. For roll motion, the peak response frequency correlates well for the FLNG. Discrepancies in magnitude are observed between the CFD and PF results for low frequency wave cases. Similar to the findings from computations in head sea waves, full scale motion transfer functions of the FLNG and LNG individually in oblique sea waves are close to model scale simulation results. The largest comparison difference is estimated to be 3.70% which is close to the numerical uncertainty for the current computations. Again, this indicates scale effects have minor influence for the studied oblique sea cases.

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Fig. 13. Model scale time history predictions of wave induced forces/moments on FLNG/LNG by CFD: (a) surge force in 0.808 Hz oblique waves; (b) surge force in 0.747 Hz oblique waves; (c) heave force in 0.808 Hz oblique waves; (d) heave force in 0.747 Hz oblique waves; (e) pitch moment in 0.808 Hz oblique waves; (f) pitch moment in 0.747 Hz oblique waves; (g) yaw moment in 0.808 Hz oblique waves; (h) yaw moment in 0.747 Hz oblique waves.

Full scale contours of wave diffraction around ship hulls after eight encountered wave periods are illustrated in Fig. 17. It can be seen that as the wave length increases, diffraction becomes less noticeable.

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(a)

Fig. 14. EFD, PF and CFD predictions of wave induced force/moment transfer functions for FLNG/LNG fixed in 6 DoF in oblique sea waves: (a) surge force transfer function of FLNG; (b) surge force transfer function of LNG; (c) sway force transfer function of FLNG; (d) sway force transfer function of LNG; (e) heave force transfer function of FLNG; (f) heave force transfer function of LNG; (g) pitch moment transfer function of FLNG; (h) pitch moment transfer function of LNG; (i) yaw moment transfer function of FLNG; (j) yaw moment transfer function of LNG.

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(b) Fig. 14. (continued)

7. Concluding remarks A generic FLNG and a LNG vessel are investigated individually in regular head and oblique sea waves at zero forward speed using commercial URANS solver, Star-CCM+. Predictions on the wave induced loads and motions for varying wave frequencies are reported in this paper. First, a verification and validation study for the computations is carried out for a particular wave case, comparing against results from physical model scale experiment. The validation uncertainty for the case is found to be greater than the comparison error, implying that the current computational setup is validated. Based on the same numerical setup, a systematic investigation is conducted for predicting wave induced loads and motions with changing sea environment. In head sea waves, the FLNG/LNG is studied for both 6 DoF motion being fixed and free to heave and pitch conditions at MTB water depth (d/TFLNG = 3.3 or d/TLNG = 8.0) and in deeper water depth (d/TFLNG = 33.3 or d/TLNG = 80.0). The results are compared with EFD and PF solutions. In addition, full scale CFD computations are performed for the tested wave frequencies in deep water condition (d/TFLNG = 33.3 or d/TLNG = 80.0) to investigate possible scale effects. From the obtained wave loads and ship motions, several interesting observations can be made:

• Model scale CFD computations correlate well with experimental data. With the ship being fixed in 6 DoF, the average comparison errors of the wave force/moment transfer functions against EFD are 14.70% and 11.10% for FLNG and LNG respectively. For the free heave and pitch condition, the average comparison error of the ship motion transfer functions is 9.10% for the FLNG. • CFD predictions demonstrate slightly better accuracy when compared against PF results. For the condition when the ship is fixed in 6 DoF, the average comparison errors for PF predictions on wave force/moment transfer functions

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Fig. 15. Model scale and full scale time history predictions of wave induced loads and motions in 0.747 Hz deep water oblique sea waves by CFD: (a) model scale heave motion; (b) full scale heave motion; (c) model scale roll motion; (d) full scale roll motion; (e) model scale pitch motion; (f) full scale pitch motion.

against EFD are 17.80% and 9.20% for FLNG and LNG respectively. For the free heave and pitch condition, the average comparison error of the ship motion transfer functions is 10.90% for the FLNG. • The computed wave loads and ship motions at the MTB water depth (d/TFLNG = 3.3 or d/TLNG = 8.0) are different from that in deeper water depth (d/TFLNG = 33.3 or d/TLNG = 80.0) for low frequency waves (0.625 Hz–0.808 Hz). • Scale effects are not evident in the predictions of wave loads and ship motions in this study focusing on single ship hydrodynamic properties. The difference between model scale and full scale simulation results varies from 0.10% to 2.78% which are less than the estimated numerical uncertainties from the validation study. In oblique sea waves, the FLNG/LNG is initially studied for 6 DoF motion constrained condition at the MTB water depth d = 0.8 m (d/TFLNG = 3.3 or d/TLNG = 8.0) and compared with model scale EFD results and PF simulations. Transfer functions of the wave induced loads are presented for the tested wave frequencies. Computations are also performed in deeper water depth d = 8.0 m (d/TFLNG = 33.3 or d/TLNG = 80.0) oblique sea waves when the FLNG/LNG is free to heave, roll and pitch. Comparisons are made between model scale and full scale simulation results. The following conclusions can be drawn from this part of the study:

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Fig. 16. EFD, PF and CFD predictions of wave induced force/moment transfer functions for FLNG/LNG free to heave, roll and pitch in oblique sea waves: (a) heave motion transfer function of FLNG; (b) heave motion transfer function of LNG; (c) roll motion transfer function of FLNG; (d) roll motion transfer function of LNG; (e) pitch motion transfer function of FLNG; (f) pitch motion transfer function of LNG.

• For 6 DoF motion constrained condition, the average comparison errors between CFD and EFD results on the wave force/moment transfer functions are 15.60% and 9.80% for the FLNG and LNG respectively. PF results show a similar level of accuracy with a comparison error of 14.80% for the FLNG and 9.40% for the LNG.

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Fig. 17. Full scale CFD contour representations of wave diffractions around the ships for different wave frequencies (a) FLNG in 0.808 Hz oblique sea waves; (b) FLNG in 0.747 Hz oblique sea waves; (c) FLNG in 0.699 Hz oblique sea waves; (d) FLNG in 0.625 Hz oblique sea waves.

• For free heave, roll and pitch condition, CFD predictions on the ship motion transfer functions correlate well with the PF results.

• The influence of scale effects on the predictions of ship motions in oblique sea is negligible. The discrepancy between model and full scale predictions varies from 0.10% to 3.70%. In conclusion, the paper has presented predictions of the wave induced loads and motions for a conceptual FLNG and a LNG vessel individually. The work is regarded as a starting point for further investigation of the FLNG–LNG hydrodynamic interactions in waves. The presented single ship results are regarded as the baseline comparison data and the impact of having another vessel in close proximity on the ship’s hydrodynamic performance can be demonstrated and quantified in the future study. Acknowledgements The authors acknowledge the scholarship support provided by the Australian Maritime College, University of Tasmania on the present work. Computations were performed on the Katabatic and Vortex clusters funded by the Tasmanian Partnership for Advanced Computing (TPAC) program. References ANSYS 2013. Aqwa Theory Manual. USA. Bennett, S.S., Hudson, D.A., Temarel, P., 2013. The influence of forward speed on ship motions in abnormal waves: Experimental measurements and numerical predictions. J. Fluids Struct. 39, 154–172. Bhushan, S., Xing, T., Carrica, P., Stern, F., 2009. Model- and full-scale URANS simulations of athena resistance, powering, seakeeping, and 5415 maneuvering. J. Ship Res. 53, 179–198. DNV 2010. Environmental conditions and environmental loads. Det Norske Veritas. Duffy, J.T., 2008. Modelling of ship-bank interaction and ship squat for ship-handling simulation, 2008.

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