- Email: [email protected]
Roll response of an LNG carrier considering the liquid cargo flow
Ocean Engineering 129 (2017) 83–91
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Roll response of an LNG carrier considering the liquid cargo flow a,⁎
b
Wenhua Zhao , Finlay McPhail a b
crossmark
Faculty of Engineering, Computing and Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009 Australia Shell Global Solutions BV (Shell), Kessler Park 1, 2280 AB, Rijswijk The Netherlands
A R T I C L E I N F O
A BS T RAC T
Keywords: Hydrodynamics Roll response Moss-type LNG carrier Spherical tanks Sloshing
Floating liquefied natural gas (FLNG) facilities are a game changer in offshore hydrocarbon development. One of the key challenges associated is offtake of LNG from FLNG to LNG carriers. Side-by-side offloading is considered a promising method, but one which may be sensitive to LNG carrier roll response. During an offload the sloshing of the cargo at intermediate load conditions could couple with the global vessel response i.e. coupling of sloshing with the higher harmonics (double or triple value) of the roll natural frequency. An experimental study is conducted with a barge-like vessel carrying two spherical tanks. These tanks are filled with water to different volumes to simulate different load conditions. For comparison, same tests were performed for which the tanks were empty, but the vessel properties modified to a ‘frozen’ approximation of an intermediate load condition. The effects of the internal liquid cargo motions on global roll motions are clarified, for the first time, for vessels with spherical tanks. The roll response of the vessel with liquid cargo inside spherical tanks is greater than observed for the equivalent ‘frozen-cargo’ case. This study also provides a foundation for improved numerical modelling with relevancy for side-by-side offloading operations.
1. Introduction
over the years, and a number of studies have been conducted for roll motions. For roll motions with a small amplitude, a linear formulation can provide relatively reliable predictions (Blagoveshchenskiĭ, 1962; Salvesen et al., 1971). However, as the response amplitude becomes large, nonlinearities become significant for roll motions, necessitating a nonlinear formulation (Bhattacharyya, 1978). One possible nonlinear contribution for example comes from ‘water-structure interactions in the splash zone’ which may be accounted for by introducing nonlinear restoring coefficients (Denise, 1983). However, it is unable to match completely with experimental data by only considering nonlinear restoring moments (Denise, 1983; Robinson and Stoddart, 1987). To achieve a complete match, viscous damping needs to be included. In contrast, there is no evidence showing the necessity of introducing nonlinear restoring into a model that has accounted for the nonlinear damping effects. It is generally held that the sources of roll damping include skin friction, eddies by flow separation, dynamic lift (with forward speed), wave radiation, and bilge keels (Ikeda et al., 1978; Himeno, 1981; Downie et al., 1988). Of these sources, radiation damping may be predicted reliably through potential theory, and the effect of skin friction on the hull is small and can be ignored for most practical purposes (Kato, 1958). Predicting other damping terms is still a challenge however. Although damping may be estimated through semi-empirical or empirical methods, the underlying rationale for choosing such a damping coefficient varies from practitioner to
Floating liquefied natural gas (FLNG) (Zhao et al., 2011), a new type of offshore facility, has been developed as a game changer in offshore hydrocarbon development for unlocking stranded gas reserves. The concept of processing gas offshore can reduce costs and environmental impact in key plays. LNG carriers that have been previously berthed against a jetty on the coast, or in ports with breakwaters, are now required to offload cargo in open seas. Shell is pursuing a side-by-side offloading operation between FLNG vessel and LNG carriers as this allows offtake by un-modified vessels, and minimization of new hardware or procedures compared to a tandem operation. Significant challenges remain however and reliable offloading is an important aspect for successful FLNG implementation. In this scenario, the proximity between vessels means that operability is sensitive to the roll response of the LNG carrier and it may dominate the operational capabilities in certain regions for side-by-side offloading. The roll responses at intermediate load conditions represent a complex technical challenge, in particular when considering the liquid cargo motion inside tanks. An emerging need for side-by-side offloading necessitates a scientific understanding of the hydrodynamics governing the interaction of liquid cargo motion in tanks and the roll response of an LNG carrier in the ocean. Roll response of LNG carriers has attracted considerable attention
⁎
Corresponding author. E-mail address: [email protected] (W. Zhao).
http://dx.doi.org/10.1016/j.oceaneng.2016.11.023 Received 20 April 2016; Received in revised form 19 August 2016; Accepted 17 November 2016 0029-8018/ © 2016 Elsevier Ltd. All rights reserved.
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
2. Governing equation of roll response
practitioner (Himeno, 1981). More recently, Taylan (2000) and Surendran et al. (2005) studied the roll dynamics of a vessel by incorporating both nonlinear restoring and damping effects. Due to the complexity of the nonlinear resonance, they suggest conducting experiments to verify the estimated amplitude of roll responses. When it comes to global roll response coupled with liquid cargo motion, the problem becomes more complex due to the inherent nonlinearity of sloshing. There have been classical numerical studies on the coupled problem by ignoring the nonlinearity of the sloshing flows inside the tanks (Molin et al., 2002; Malenica et al., 2003; Newman, 2005). To consider the nonlinear sloshing flows, Lee et al. (2007) and Lee and Kim (2010) conducted time domain simulations using computational fluid dynamics (CFD). In their studies, linear flow is assumed for the external waves and the nonlinear internal sloshing is simulated based on a CFD scheme using a finite difference method. Based on a finite element method for the simulation of nonlinear sloshing, Mitra et al. (2012) went one step further to carry out numerical simulations in time domain where nonlinear ship motion is adopted. These existing studies indicate that methods based on linear assumptions fail to incorporate the strong nonlinearities of internal sloshing, while CFD methods are time consuming. Predicting the hydrodynamics of an LNG carrier considering its internal sloshing remains a challenge. To develop a better understanding of this coupled problem, attempts have been made in this study in developing reliable techniques for scale model tests. Early experiments can be traced back to the study by Mikelis et al. (1984), in which the coupled problem was investigated based on a carrier with prismatic tanks. Francescutto and Contento (1994) experimentally studied the coupling between ship roll motions and the sloshing in a floodable compartment, with the ship subjected to a beam sea condition. Molin et al. (2002) conducted experiments utilising a barge-like vessel with a partially filled rectangular tank on deck, subjecting it to the combined excitations of roll, sway and heave. Nam and Kim (2007) carried out an experimental study on a barge equipped with two partially filled prismatic tanks, showing the importance of a fully-coupled analysis, in particular for roll motions. With a similar experimental set-up, Nasar et al. (2008, 2010) conducted an experimental study on the effects of intermediate load conditions. In their studies, the centre of gravity, the radius of gyration and the total mass are different as load conditions vary. To get a comprehensive understanding and guarantee a unique variable parameter when load conditions change, Zhao et al., (2014a, 2014b) conducted a series of experimental studies where sloshing coupled with both roll only and coupled roll, sway and heave motions with rectangular tanks. To the authors' knowledge, very limited published effort focuses on global roll response coupled with liquid cargo motions inside spherical tanks such as are used on Moss-type LNG carriers. However, the global roll motion with partially-filled spherical tanks may be a limiting criterion for side-by-side offloading operations in open seas. The aim of the present study is to characterise the influence of liquid cargo motions inside spherical tanks on the roll response of an LNG carrier. The results in this study are partly expected to serve as a benchmark for numerical models that may be incorporated into FLNG berth availability studies. To achieve this, a series of scale model tests, including inclination tests, decay tests, and white noise wave tests, are conducted. In the model test, each wave case is repeated with the spherical tanks filled with liquid cargo and equivalent ‘frozen-cargo’ respectively, where the total mass, centre of gravity and radius of gyration hold the same. The global roll response amplitude operators (RAOs) of the vessel, with and without considering the effects of internal sloshing, have been studied, together with the short-term response spectra in swells. Furthermore, the internal sloshing elevations inside the spherical tanks are captured, facilitating understanding of the coupling effect mechanism.
To facilitate the analysis, the equation of motion for roll response only is discussed by ignoring the coupling from motions in other degrees of freedom. A typical equation of roll motion can be expressed as follows (Himeno, 1981; Taylan, 2000; Surendran et al., 2005):
Aφ φ̈ + Bφ(φ̇) + Cφ(φ) = Mφ(ωt ),
(1)
where, φ stands for the roll angle and the over-dot denotes differentiation with respect to timet . Aφ represents the transverse virtual mass moment of inertia for the vessel, Bφ the roll damping moment, Cφ the restoring moment and Mφ the exciting moment due to external waves and internal sloshing. To express the nonlinearity of the roll motion, the damping term in Eq. (1) are usually given in the following form:
Bφ(φ̇) = BL φ̇ + BN φ̇ φ̇ ,
(2)
where, BL and BN stand for linear and nonlinear roll damping moment coefficients, respectively. It is a common practice to approximate the nonlinear viscous damping with an equivalent linear damping (e.g. a percentage of critical damping) (Wilson, 1996). As a consequence, the nonlinear damping term Bφ(φ̇) can be expressed through an equivalent linear damping as follows:
Bφ(φ̇) = Beφ̇ ,
(3)
where, Be can be divided into the following components (Himeno, 1981):
Be = BW + BF + BL + BE + BBK ,
(4)
where, BW stands for the wave radiation damping, BF the friction damping from the hull, BL the lift damping of the hull, BE the eddymaking damping and BBK the damping due to the bilge keels. Of these components, it is difficult to distinguish linear damping from nonlinear damping. As an approximation, the radiation damping BW , the friction damping BF in the absence of ship forward speed, the lift damping BL in the presence of ship forward speed and parts of bilge keel effects BBK contribute to the linear term in Eq. (2), while the eddymaking damping BE and parts of bilge keel effects BBK take effect through the quadratic term. Scale model test is a feasible way to estimate these coefficients. Scale effects can be an important problem for the estimation of some forms of roll damping. Of these damping components, only the frictional damping suffers from scale effects (Himeno, 1981). The non-dimensional value of the friction damping is almost proportional to λ−0.75 (λ the scaling coefficient), allowing us to ignore the frictional damping of actual ships. The above literature evidence has shown that it remains a challenge to predict roll motions using only numerical models due to the complexity and nonlinearity. Inclusion of intermediate load conditions increases this complexity as the sloshing inside tanks is inherently nonlinear. Scale model tests are an acceptable and reliable way to handle this complex nonlinear problem. 3. Model particulars Focusing on the global roll response and internal sloshing problem, a barge-like vessel, 200 m long (full-scale equivalent) and beam of 46 m (full-scale equivalent) was constructed (Table 1). This model had an identical cross section in the longitudinal direction and was designed to have roll characteristics representative of a Moss-type LNG carrier. Two spherical tanks are placed on the vessel to model sloshing effects. The model tests were conducted in the Deepwater Wave Basin at Shanghai Jiao Tong University at a scale of 1:60. In this study, the barge-like vessel is designed on the assumption that the spherical tanks are filled with water, because the usage of LNG 84
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Table 1 Particulars of the vessel with two spherical tanks in 50% filling condition. Full scale
Model scale
Length between perpendiculars (m) Breadth (m) Depth (m) Mean Draft (m) Displacement weight (kg) Prior to cargo loading Radius of roll gyration (m) Radius of pitch gyration (m) Water/equivalent solid weights to be filled in the two spherical tanks (kg)
200 46 25.5 11.1 99,297,900
3.333 0.767 0.425 0.185 459.7
19.4 68.5 29,880,015
0.323 1.142 138.3
VCG (m) Kxx (m) Kyy (m)
Target
Achieved
Error
0.23 0.323 1.142
0.231 0.319 1.1466
0.43% −1.24% 0.40%
Beam sea
Designation
Table 2 Calibration results prior to cargo loading.
Head sea
as a fluid in a laboratory would be prohibitively complex and hazardous. Water is approximately twice the density of LNG (465 kg/ m3) so two tanks of water represents a similar mass to four LNG tanks, as may be expected on a typical Moss-type LNG carrier. The tanks are sized and positioned such that they approximate a larger number of tanks filled with LNG as closely as possible. An LNG carrier will experience different intermediate load conditions during a side-by-side offloading operation. In this study, an intermediate load condition of 50% by volume is selected as the reference. The particulars of the vessel in this load condition are listed in Table 1 in both full scale and model scale. The corresponding lines plan (in Fig. 1) shows that the vessel is equipped with both bilge keels and two spherical tanks. The radius of the spherical tanks is 19.25 m in full scale. To clarify the effects of liquid cargo motions on the global roll responses, the two spherical tanks are filled with water for one case (referred to as ‘liquid-cargo’ case), and left empty for a second case for which the quasi-static properties are matched with solid weights (referred to as ‘frozen-cargo’ case). As shown in Table 1, the only difference between the two cases is whether there is liquid flow inside the tanks. The other parameters hold the same such as the total mass, draft, and centre of gravity in vertical direction (VCG). Prior to the model tests, all the parameters shown in Table 1 are calibrated. In the liquid-cargo case, the calibration is conducted based on the parameters prior to cargo loading and then the spherical tanks are filled with water to achieve the target filling level. The measurements for the liquid-cargo case during the calibration are shown in Table 2. In the frozen-cargo case, there is no liquid cargo at all, so the calibration is conducted based on the total value of the parameters in Table 1. The VCG is adjusted to be 0.2775 m with an error of 2.40% relative to the target value (0.271 m), pitch radius is adjusted to be 1.0454 m with an error of −1.38% relative to the target value (1.06 m). The roll radius is achieved through matching the roll period with that obtained in the liquid-cargo case.
Fig. 2. Configuration of the horizontally moored vessel in wave basin.
4. Experimental set-up 4.1. Model configuration in the wave basin In this study, model tests are conducted with the vessel both in calm water and waves. Decay tests in calm water are conducted to obtain the decay curves. In the decay tests, the model is given an initial heel angle and then released; vessel motions are recorded simultaneously. Wave tests are carried out to obtain the RAOs of the roll motions and the internal sloshing. During wave test runs, the vessel model is moored in the wave basin with a horizontal mooring system, as shown in Fig. 2. The horizontal mooring system is designed to be soft enough to avoid influencing the wave frequency motion responses. The natural period of the horizontal moored system is measured to be 163 s in full scale, which is far away from the roll natural period and is out of the interest of this study. In addition to the vessel's motions, the surface elevations inside the spherical tanks are recorded synchronously through wave probes. As shown in Fig. 3, the wave probes are fixed along the arc-shape side walls of the spherical tanks, allowing the measurement of the water surface elevation along the inside wall of the spherical tanks. 4.2. Calibration of waves White noise waves, instead of regular waves, are used to ascertain the RAOs of the vessel and the internal sloshing. The white noise wave spectrum has a bandwidth from 4 to 29 s and significant wave height of
Arc-shape wave probe
Fig. 1. Lines plan of the vessel with two spherical tanks: (a) front view; (b) side view; (c) top view.
Vertical wave probe
Fig. 3. Wave probes in the spherical tanks.
85
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 4. White noise waves in beam seas: (a) response spectrum, (b) representative time series.
Fig. 5. White noise waves in quartering seas: (a) response spectrum, (b) representative time series.
3 m in full scale, so one test run is able to cover both the roll natural period and the sloshing natural periods of the liquid cargo which are a distance apart. Prior to the wave tests, a time series of the white noise wave is generated without the model in the basin. The duration of the white noise waves at full scale is 3 h. In the model tests, the heading of the model was kept unchanged, but the wave makers at different sides of the wave basin were used to generate waves coming in different directions to form the beam and quartering sea states, as shown in Fig. 2. This avoids the re-setting up of the model, and thus avoids potential error due to setting-up of the model. The generated white noise spectra are compared with the target spectra in Figs. 4 and 5, together with a representative time series of the generated wave elevations. It can be seen in Fig. 4(a) and Fig. 5(a) that the generated wave spectra match well with the target spectra.
Table 3 Regular wave parameters in full scale. Regular waves
Wave height (m)
Period (s)
Wave slope
No. No. No. No. No.
3 6 7 9 9
7.00 13.97 15.80 16.10 16.88
0.0393 0.0197 0.0180 0.0223 0.0203
1 2 3 4 5
RAOs can be obtained from regular wave tests, and the repeatability of the results are illustrated in Fig. 6. It can be seen in Fig. 6 that the model test results match well for each duplicated run, confirming the reliability of the model tests.
5. Results and discussion 5.2. Decay test results 5.1. Repeatability study Free decay tests of the model in calm water are conducted with the spherical tanks filled with liquid cargo and corresponding ‘frozen cargo’ respectively, aiming primarily to check the roll natural period and secondly to clarify the effects of the liquid cargo motion on roll motions in calm water. The extinction traces of the model in the two different ballasting cases are compared in Fig. 7. The roll motion in the liquidcargo case is found to decay quicker than that in the frozen-cargo case,
To estimate the performance of the experimental set-up a repeatability study is conducted. A series of duplicated regular wave runs is conducted with the vessel in the ‘frozen cargo’ set-up. The regular wave parameters used in this study are listed in Table 3. The wave slopes (ratio of wave height to wave length) as shown in Table 3 are designed to be less than 0.05, to keep the regular waves as linear as possible. 86
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
incident white noise waves. The roll RAOs of the vessel in two different ballasting conditions are compared in Fig. 8, with (a) for beam seas and (b) for quartering seas. It can be seen in Fig. 8(a) that the peak roll RAO occurs at the roll natural period (15.16 s), and it is slightly larger in the liquid-cargo case (6.79 deg/m) than that in the frozen-cargo case (6.67 deg/m). However, when looking at the comparison results in quartering seas, the internal sloshing exhibits relatively larger effects on the global roll response at the roll natural period. It can be seen that the peak roll RAO is amplified by 8.40% from 3.81 deg/m in the frozen-cargo case to 4.13 deg/m in the liquid-cargo case. We think this amplification is a result of the internal sloshing effect, instead of the parametric roll which may occur in head seas and is very sensitive to vessel heading. As described above, the vessel model used in this study is designed to be generally like a rectangular box, so there is negligible periodic variation of the transverse stability of the vessel in waves, characterised by a decrease of stability when the mid-ship of the vessel is in the wave crest and an increase in the wave trough (France et al., 2003). In addition, the wave height used in this study is small, which makes parametric roll impossible here in quartering or beam seas. As shown in Fig. 8, the roll RAOs are sensitive to the headings of the vessel. The peak RAOs in quartering seas drop down to two thirds of those in beam seas. An interesting phenomenon can be noted in Fig. 8(a) is that an obvious secondary peak roll RAO occurs at 6.39 s for the liquid-cargo case. This is associated with the internal sloshing. As can be seen in the same figure, there is a tiny peak for the roll RAO even in the solid case at around 7.52 s, which is half of the natural roll response period (15.16 s). This indicates that the small peak in frozen-cargo case is induced by the second-order harmonics. However, the peak value (0.17 deg/m) is small compared to that in the liquid-cargo case (1.22 deg/m). As a consequence, the second-order harmonics might make a small contribution, but the coupling between internal sloshing and the second-order harmonics of the main roll motions is very important and leads to the secondary peak roll RAO shown in the liquid case in Fig. 8(a). A physical understanding of the coupling between internal sloshing and higher harmonics of the main roll motions has been given by Zhao et al. (2016). Another interesting phenomenon that may be observed in Fig. 8 is that the sloshing effect on the global roll motion is sensitive to the heading of the vessel. In beam seas, one can observe an obvious secondary peak of roll RAOs, while it entirely absent when the heading changes to quartering seas. As the above discussion, the RAO at the roll natural period is more amplified in quartering seas than that in beam seas. According to previous studies (Kim et al., 2007; Zhao et al., 2014a), this phenomenon can be explained through the phases of the internal sloshing elevations relative to the external waves. The phase of the global roll motions and the internal sloshing are given with respect to the external waves as shown in Figs. 9 and 10. One can see in Fig. 9(a) a rapid change of phase in the liquid-cargo case at the period where a peak RAO (shown in Fig. 8(a)) occurs, while the frozen-cargo case does not show the same phenomenon. When we changed the headings to quartering seas (Fig. 9(b)), the roll phases in the frozen-cargo and liquid-cargo cases are pretty similar around 6.39 s, which is consistent with the similar RAOs as shown in Fig. 8(b). For both beam and quartering seas at the period around 15.16 s, the roll phases in the frozen-cargo and liquid-cargo cases are very similar. The phases of the internal sloshing elevations relative to the external waves are plotted in Fig. 10. If the internal sloshing moves in phase with the incident waves, they will work together to amplify the vessel's global roll response. In contrast, if it is out of phase with the incident waves, the internal sloshing can cancel part of the effects induced by wave forces. It can be observed that the relative phase at the roll natural period (15.16 s) is 17 deg for beam seas (Fig. 10(a)), while it is −7 deg for quartering seas (Fig. 10(b)). The internal liquid cargo motion and external waves at the roll natural period are more in phase
Fig. 6. Repeatability of the model tests.
Fig. 7. Roll decay curves of the vessel with the spherical tanks filled with liquid cargo and fixed cargo.
and the damping coefficient is slightly amplified from 0.0226 (frozencargo case) to 0.0238 (liquid-cargo case). The slight increase of damping in the liquid-cargo case may be associated with the energy dissipation induced by the liquid cargo motions inside the tanks. This phenomenon indicates that the cargo sloshing adds a small amount of damping, which is close to negligible at the natural response period of the vessel. However, if we consider the wave-structure interactions, the coupling between roll response of the vessel and the internal liquid cargo motions will be another scenario. This will be discussed in detail in the following section. 5.3. RAOs To investigate the sloshing effect of the liquid cargo on the global roll response in waves, white noise wave tests in both beam and quartering seas (shown in Fig. 2) were conducted. The spherical tanks are filled to 50% of volume for the liquid cargo tests, while the vessel is ballasted with solid steel weights for the ‘frozen-cargo’ tests. Time traces of the roll motions and internal sloshing elevations are measured. Spectral analyses are then conducted to obtain the power spectrum density function, which can further be used to ascertain the RAOs through the following equation:
H (ω ) =
Sxy(ω) Sxx (ω)
,
(5)
where, H (ω) is the RAO, Sxy(ω) the cross spectral density function of the responses, and Sxx (ω) the auto spectral density function of the 87
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 8. Roll RAOs of the vessel in different heading conditions: (a) beam seas, (b) quartering seas.
lateral direction of the vessel, without components in the longitudinal direction. As shown in Table 4, the measured sloshing periods show very good agreement with the analogous solutions by Faltinsen and Timokha (2012). It is not difficult to understand why the sloshing waves in the lateral direction dominate in both beam and quartering seas. This is due to the fact that even in quartering seas, the pitch motion is far smaller than roll motion, therefore it cannot induce such significant waves in the longitudinal direction as those in the lateral direction. Significant sloshing RAOs can be found in Fig. 11 at different resonant periods, the roll responses at the corresponding periods are negligible as shown in Fig. 8 (except the secondary RAO peak for liquid-cargo case in Fig. 8(a)). Through the comparison of the results in the two figures, one can observe that only the first mode of sloshing has coupling effects on the global roll motions, and the higher modes of sloshing seem to have very little influence on the global roll response.
in quartering seas. As a consequence, the internal liquid cargo motion shows more amplification effects at the roll natural period in quartering seas than that in beam seas. A similar phenomenon is seen at the sloshing period: the relative phase at the sloshing period is −75 deg (in phase) in the beam sea, amplifying the wave force effect; while it is −120 deg (out of phase) in the quartering sea, cancelling the wave force. This explains why a secondary response peak occurs in beam seas at the sloshing period, but disappears in quartering seas. To deepen the understanding of the sloshing flows inside spherical tanks, the RAOs in different heading conditions are calculated and shown in Fig. 11. One can observe in Fig. 11 that in addition to the fundamental sloshing peak, higher modes of sloshing peaks occur in both beam seas and quartering seas. It is interesting to observe that these sloshing peaks occur at the same periods in the two different heading conditions. This can be explained through the sloshing characteristics inside spherical tanks. Sloshing inside spherical tanks can support free surface oscillations with integral azimuthal wave numbers m (including zero) and for each m , there is an infinite sequence of discrete oscillation frequencies (McIVER, 1989; Faltinsen and Timokha, 2012). Therefore, sloshing inside spherical tanks consists of waves in both lateral and longitudinal directions of the vessel. It can be deduced that for both beam and quartering seas, the sloshing waves in the lateral direction of the vessel dominate and the sloshing waves in longitudinal direction can be negligible. To prove this deduction, the measured sloshing periods in the two heading conditions are compared with the periods of the sloshing modes only in
5.4. Short-term analyses During side-by-side offloading, an LNG carrier is generally not expected to experience wind-sea on the beam because most FLNG designs are turret-moored, allowing them to ‘weathervane’ into prevailing conditions. However, in some locations such as the North West Australian Shelf and Western Africa, swells may come in beam or quartering seas, which may induce critical roll motions for an LNG carrier. To estimate the effects of the liquid cargo motion, two typical swell conditions from the North West Australian Shelf are taken as the reference: one is a short period swell with a period equal to the sloshing
Fig. 9. Relative phases of roll motions to external waves: (a) beam seas; (b) quartering seas.
88
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 10. Relative phases of the internal sloshing to external waves: (a) beam seas; (b) quartering seas.
Fig. 11. Sloshing RAOs of the liquid cargo in different modes: (a) beam seas, (b)quartering seas.
One can see in Fig. 12 and Table 6 that the response peak and the Table 4 Sloshing periods of spherical tanks in half-filled condition.
Table 5 Swell conditions for short-term analyses.
Methods
1st
2nd
3rd
4th
Analogous solution (s) Present (s) Discrepancy
6.38 6.39 0.19%
4.67 4.55 −2.63%
3.88 3.89 0.40%
3.38 3.35 −1.00%
Hs (m)
Tp (s)
Short period Long period
0.9 0.9
6.39 15.16
most probable maximum in 3 h in the frozen-cargo case are 0.03 deg2/ s and 0.14 deg, while they reach up to 1.53 deg2/s and 0.86 deg in the liquid-cargo case. The extreme roll response is amplified due to the internal sloshing. This phenomenon indicates that for a short period swell coming in beam seas, the internal sloshing may affect the global roll response to a certain degree. However, around the period of 15.16 s when vessel roll motions are important, the roll responses are indistinguishable in the liquid- and frozen-cargo cases (shown in Fig. 12(b)). A possible reason for this phenomenon might be associated with the fact that the period at which the sloshing-induced response occurs is far away from the roll natural period. According to the McCarty and Stephens (1960) study, the first mode of sloshing period in a spherical tank will not be larger than 7 s, therefore the sloshing period will not be too close to the roll natural period. At reduced load conditions, the internal sloshing can be severer (Zhao et al., 2014b); in contrast, the weight of the fluid inside tanks will decrease, but it is unclear which
period and the other is a long period swell with a period equal to the roll natural period of the vessel. The parameters of the two swell conditions are listed in Table 5. A Gaussian spectrum in the following form is used:
⎡ (ω − ω ) 2 ⎤ ⎛ H ⎞2 1 0 ⎥, Sη(ω) = ⎜ s ⎟ ⋅ ⋅exp⎢ − ⎢⎣ 2⋅σ 2⋅ω0 2 ⎥⎦ ⎝ 4 ⎠ σ⋅ 2π ⋅ω0
Swells
(6)
where, ω0 stands for the peak frequency of the spectrum, σ the dimensionless shape parameter. In this study, the value of σ is selected as 0.02Hz . Short-term analyses have been conducted based on the above swell conditions and the RAOs obtained in beam and quartering seas. The roll response spectra of the vessel are obtained and shown in Figs. 12 and 13. Furthermore, the most probable maxima of the roll response within 3 h are estimated and listed in Table 6. 89
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 12. Response spectrum of the roll motion in beam seas: (a) short swell with period of 6.39 s, (b) long swell with period of 15.16 s. Note the different scales on the two figures.
Fig. 13. Response spectrum of the roll motion in quartering seas: (a) short swell with period of 6.39 s, (b) long swell with period of 15.16 s. Note the different scales on the two figures.
effect dominates. To make a decision on whether short period waves (6–7 s) should be included in analyses during design, further study is needed for reduced load conditions. It can be further found in Fig. 12(b) that the roll response in long period swells holds almost the same in both the liquid- and the frozen-cargo cases. This indicates that if the long swell comes in beam seas, the sloshing effects on the global roll response are relatively small. It can be observed in Fig. 13 and Table 6 that if a short period swell comes in a quartering sea, the roll responses in the liquid- and frozencargo cases are 0.18 deg and 0.05 deg, respectively, which are very small in absolute value and their difference is negligible. However, if a long period swell comes in quartering seas, the most probable largest roll response will be amplified by 6.83% from 6.50 to 6.97 deg due to the internal sloshing effects. As the load conditions become lower, the internal sloshing can become more severe (Zhao et al., 2014b), which may increase the amplification effect. Therefore, it is worthy of further study to clarify whether the internal sloshing could induce more significant effects on roll response at reduced load conditions. Through the comparison between Figs. 12 and 13, one can observe that if a short period swell occurs in beam seas, the roll response of the vessel may be affected by the internal sloshing; while the roll responses in both the liquid- and frozen-cargo cases are too small in short period swells coming in quartering seas, thereby the sloshing effect can be negligible. For a long period swell in beam seas, even if the swell period is equal to roll natural period, the sloshing effects on the global roll response can be negligible; while internal sloshing will amplify the roll
Table 6 Most probable maximums of the roll response within 3 h. Headings
Swells
Liquid case (deg)
‘Frozen’ case (deg)
Beam seas
Short period Long period Short period Long period
1.91 11.87 0.18 6.97
0.31 11.98 0.05 6.50
Quartering seas
response slightly for long period quartering swells. 6. Conclusion To investigate the coupling between roll motions and internal sloshing, a series of scale model tests are conducted. To examine the effect of sloshing, both a liquid-cargo test and a ‘frozen-cargo’ test are conducted. The effects of liquid cargo motions in spherical tanks on the global roll response of a barge-like vessel at an intermediate load condition are examined for the first time. The main conclusions drawn are as follows:
• • 90
For beam seas, the roll response of the LNG carrier with liquid cargo shows the same roll period as that with frozen cargo for spherical tanks. The internal sloshing induces a secondary spectral peak for roll motions in beam seas, but it does not seemingly induce significant
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
• •
•
lashing systems. Mar. Technol. 40 (1), 1–19. Francescutto, A., Contento, G., 1994. An experimental study of the coupling between roll motion and sloshing in a compartment, In: Proceedings of the Fourth International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers. Himeno, Y., 1981 Prediction of ship roll damping. A State of the Art. DTIC Document. Ikeda, Y., Himeno, Y., Tanaka, N., 1978. A prediction method for ship roll damping. Report of the Department of Naval Architecture, University of Osaka Prefecture (00405). Kato, H., 1958. On the frictional resistance to the rolling of ships. J. Soc. Nav. Archit. Jpn. 102, (115-112). Kim, Y., Nam, B.W., Kim, D.W., Kim, Y.S., 2007. Study on coupling effects of ship motion and sloshing. Ocean Eng. 34 (16), 2176–2187. Lee, S., Kim, M., 2010. The effects of inner-liquid motion on LNG vessel responses. J. Offshore Mech. Arct. Eng. 132 (2), 021101. Lee, S., Kim, M., Lee, D., Kim, J., Kim, Y., 2007. The effects of LNG-tank sloshing on the global motions of LNG carriers. Ocean Eng. 34 (1), 10–20. Malenica, S., Zalar, M., Chen, X., 2003. Dynamic coupling of seakeeping and sloshing, In: Proceedings of the 13th international offshore and polar engineering conference, pp. 484–490. McCarty, J.L., Stephens, D.G., 1960. Investigation of the natural frequencies of fluids in spherical and cylindrical tanks. NASA, Technical Note, D-252. McIVER, P., 1989. Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. J. Fluid Mech. 201, 243–257. Mikelis, N., Miller, J., Taylor, K., 1984. Sloshing in partially filled liquid tanks and its effect on ship motions: numerical simulations and experimental verification. R. Inst. Nav. Archit. Trans., 126. Mitra, S., Wang, C., Reddy, J., Khoo, B., 2012. A 3D fully coupled analysis of nonlinear sloshing and ship motion. Ocean Eng. 39, 1–13. Molin, B., Remy, F., Rigaud, S., De Jouette, C., 2002. LNG-FPSO’s: frequency domain, coupled analysis of support and liquid cargo motion, In: Proceedings of the 10th International Maritime Association of the Mediterranean (IMAM) Conference, Rethymnon, Greece. Nam, B.W., Kim, Y., 2007. Effects of Sloshing on the Motion Response of LNG-FPSO in Waves, In: Proceedings of the 22nd Workshop onWater Waves and Floating Bodies, Plitvice, Croatia. Nasar, T., Sannasiraj, S., Sundar, V., 2008. Experimental study of liquid sloshing dynamics in a barge carrying tank. Fluid Dyn. Res. 40 (6), 427–458. Nasar, T., Sannasiraj, S., Sundar, V., 2010. Motion responses of barge carrying liquid tank. Ocean Eng. 37 (10), 935–946. Newman, J., 2005. Wave effects on vessels with internal tanks, In: Proceedings of the 20th Workshop on Water Waves and Floating Bodies, Spitsbergen, Norway. Robinson, R., Stoddart, A., 1987. An engineering assessment of the role of non-linearities in transportation barge roll response. Nav. Archit.. Salvesen, N., Tuck, E., Faltinsen, O., 1971. Ship motions and sea loads. Det norske Veritas, Oslo. Surendran, S., Lee, S., Reddy, J., Lee, G., 2005. Non-linear roll dynamics of a Ro-Ro ship in waves. Ocean Eng. 32 (14), 1818–1828. Taylan, M., 2000. The effect of nonlinear damping and restoring in ship rolling. Ocean Eng. 27 (9), 921–932. Wilson, E.L., 1996. Three Dimensional Static and Dynamic Analysis of Structures. Computers and Structures, Inc., Berkeley, California, USA. Zhao, W., Yang, J., Hu, Z., Wei, Y., 2011. Recent developments on the hydrodynamics of floating liquid natural gas (FLNG). Ocean Eng. 38 (14), 1555–1567. Zhao, W., Yang, J., Hu, Z., Tao, L., 2014a. Coupling between roll motions of an FLNG vessel and internal sloshing. J. Offshore Mech. Arct. Eng. 136 (2), 021102. Zhao, W., Yang, J., Hu, Z., Xiao, L., Tao, L., 2014b. Hydrodynamics of a 2D vessel including internal sloshing flows. Ocean Eng. 84, 45–53. Zhao, W., Wolgamot, H., Eatock TaylorR., Taylor, P.H., 2016. Nonlinear harmonics in the roll motion of a moored barge coupled to sloshing in partially filled spherical tanks. In: Proceedings of the 31th International Workshop on Water Waves and Floating Bodies. Plymouth, MI, USA. April 2016. pp. 3–6 〈http://www.iwwwfb.org/ Abstracts/iwwwfb31/iwwwfb31_53.pdf〉.
roll response. The roll natural frequency is much lower than the sloshing frequency, so the coupling between these two phenomena is through the higher harmonics of the roll motions, i.e. the double frequency of the main roll motion. For quartering seas, the maximum roll RAO in the liquid-cargo case is amplified by 8.40% compared to that in the frozen-cargo case. This effect may further prompt explicit inclusion of liquid cargo dynamics in availability studies for side-by-side offloading. The general results of this study tend to indicate that the internal sloshing inside spherical tanks could amplify the global roll motions but not significantly, at least for this specific (half volume) load condition. When the filling level of the tanks is reduced, the internal sloshing may become severer, increasing the influence on global roll motions. On the other hand, the weight of the liquid cargo inside tanks decrease at reduced load conditions, decreasing the sloshing effect. It is however unclear which effect will dominate. Further study is merited to examine whether sloshing-induced response peaks matter at reduced load conditions, so as to make a decision on whether short period waves (6–7 s) should be included.
This study does not attempt to solve the coupling problem of roll responses and liquid cargo sloshing completely. It was initiated as a derisking exercise to investigate what effect internal liquid cargo sloshing in spherical tanks may have on global vessel motions and to serve as a basis for numerical work. Assessments of side-by-side operability should expect to undertake significant additional numerical work to ensure the full range of motions is captured and understood for a project specific environment. Acknowledgement This work was funded by Shell Global Solutions BV and conducted in the Deepwater Offshore Basin at Shanghai Jiao Tong University as a joint program. The first author is supported by the Shell EMI offshore engineering initiative at the University of Western Australia. Shell would like to acknowledge the support and dedication of the staff at the Shanghai Jiao Tong University Deepwater Offshore Basin. They proved to be professional, diligent, and accommodating to requirements throughout the campaign of model tests. References Bhattacharyya, R., 1978. Dynamics of Marine Vehicles. Wiley, New York. Blagoveshchenskiĭ, S., 1962. Theory of Ship Motions. Dover Publications, New York. Denise, J., 1983. On the roll motion of barges. Downie, M., Bearman, P., Graham, J., 1988. Effect of vortex shedding on the coupled roll response of bodies in waves. J. Fluid Mech. 189, 243–261. Faltinsen, O.M., Timokha, A.N., 2012. Analytically approcximate natural sloshing modes for a spherical tanks shape. J. Fluid Mech. 703, 391–401. France, W.N., Levadou, M., Treakle, T.W., Paulling, J.R., Michel, R.K., Moore, Colin, 2003. An investigation of head-sea parametric rolling and its influence on container
91
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Roll response of an LNG carrier considering the liquid cargo flow a,⁎
b
Wenhua Zhao , Finlay McPhail a b
crossmark
Faculty of Engineering, Computing and Mathematics, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009 Australia Shell Global Solutions BV (Shell), Kessler Park 1, 2280 AB, Rijswijk The Netherlands
A R T I C L E I N F O
A BS T RAC T
Keywords: Hydrodynamics Roll response Moss-type LNG carrier Spherical tanks Sloshing
Floating liquefied natural gas (FLNG) facilities are a game changer in offshore hydrocarbon development. One of the key challenges associated is offtake of LNG from FLNG to LNG carriers. Side-by-side offloading is considered a promising method, but one which may be sensitive to LNG carrier roll response. During an offload the sloshing of the cargo at intermediate load conditions could couple with the global vessel response i.e. coupling of sloshing with the higher harmonics (double or triple value) of the roll natural frequency. An experimental study is conducted with a barge-like vessel carrying two spherical tanks. These tanks are filled with water to different volumes to simulate different load conditions. For comparison, same tests were performed for which the tanks were empty, but the vessel properties modified to a ‘frozen’ approximation of an intermediate load condition. The effects of the internal liquid cargo motions on global roll motions are clarified, for the first time, for vessels with spherical tanks. The roll response of the vessel with liquid cargo inside spherical tanks is greater than observed for the equivalent ‘frozen-cargo’ case. This study also provides a foundation for improved numerical modelling with relevancy for side-by-side offloading operations.
1. Introduction
over the years, and a number of studies have been conducted for roll motions. For roll motions with a small amplitude, a linear formulation can provide relatively reliable predictions (Blagoveshchenskiĭ, 1962; Salvesen et al., 1971). However, as the response amplitude becomes large, nonlinearities become significant for roll motions, necessitating a nonlinear formulation (Bhattacharyya, 1978). One possible nonlinear contribution for example comes from ‘water-structure interactions in the splash zone’ which may be accounted for by introducing nonlinear restoring coefficients (Denise, 1983). However, it is unable to match completely with experimental data by only considering nonlinear restoring moments (Denise, 1983; Robinson and Stoddart, 1987). To achieve a complete match, viscous damping needs to be included. In contrast, there is no evidence showing the necessity of introducing nonlinear restoring into a model that has accounted for the nonlinear damping effects. It is generally held that the sources of roll damping include skin friction, eddies by flow separation, dynamic lift (with forward speed), wave radiation, and bilge keels (Ikeda et al., 1978; Himeno, 1981; Downie et al., 1988). Of these sources, radiation damping may be predicted reliably through potential theory, and the effect of skin friction on the hull is small and can be ignored for most practical purposes (Kato, 1958). Predicting other damping terms is still a challenge however. Although damping may be estimated through semi-empirical or empirical methods, the underlying rationale for choosing such a damping coefficient varies from practitioner to
Floating liquefied natural gas (FLNG) (Zhao et al., 2011), a new type of offshore facility, has been developed as a game changer in offshore hydrocarbon development for unlocking stranded gas reserves. The concept of processing gas offshore can reduce costs and environmental impact in key plays. LNG carriers that have been previously berthed against a jetty on the coast, or in ports with breakwaters, are now required to offload cargo in open seas. Shell is pursuing a side-by-side offloading operation between FLNG vessel and LNG carriers as this allows offtake by un-modified vessels, and minimization of new hardware or procedures compared to a tandem operation. Significant challenges remain however and reliable offloading is an important aspect for successful FLNG implementation. In this scenario, the proximity between vessels means that operability is sensitive to the roll response of the LNG carrier and it may dominate the operational capabilities in certain regions for side-by-side offloading. The roll responses at intermediate load conditions represent a complex technical challenge, in particular when considering the liquid cargo motion inside tanks. An emerging need for side-by-side offloading necessitates a scientific understanding of the hydrodynamics governing the interaction of liquid cargo motion in tanks and the roll response of an LNG carrier in the ocean. Roll response of LNG carriers has attracted considerable attention
⁎
Corresponding author. E-mail address: [email protected] (W. Zhao).
http://dx.doi.org/10.1016/j.oceaneng.2016.11.023 Received 20 April 2016; Received in revised form 19 August 2016; Accepted 17 November 2016 0029-8018/ © 2016 Elsevier Ltd. All rights reserved.
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
2. Governing equation of roll response
practitioner (Himeno, 1981). More recently, Taylan (2000) and Surendran et al. (2005) studied the roll dynamics of a vessel by incorporating both nonlinear restoring and damping effects. Due to the complexity of the nonlinear resonance, they suggest conducting experiments to verify the estimated amplitude of roll responses. When it comes to global roll response coupled with liquid cargo motion, the problem becomes more complex due to the inherent nonlinearity of sloshing. There have been classical numerical studies on the coupled problem by ignoring the nonlinearity of the sloshing flows inside the tanks (Molin et al., 2002; Malenica et al., 2003; Newman, 2005). To consider the nonlinear sloshing flows, Lee et al. (2007) and Lee and Kim (2010) conducted time domain simulations using computational fluid dynamics (CFD). In their studies, linear flow is assumed for the external waves and the nonlinear internal sloshing is simulated based on a CFD scheme using a finite difference method. Based on a finite element method for the simulation of nonlinear sloshing, Mitra et al. (2012) went one step further to carry out numerical simulations in time domain where nonlinear ship motion is adopted. These existing studies indicate that methods based on linear assumptions fail to incorporate the strong nonlinearities of internal sloshing, while CFD methods are time consuming. Predicting the hydrodynamics of an LNG carrier considering its internal sloshing remains a challenge. To develop a better understanding of this coupled problem, attempts have been made in this study in developing reliable techniques for scale model tests. Early experiments can be traced back to the study by Mikelis et al. (1984), in which the coupled problem was investigated based on a carrier with prismatic tanks. Francescutto and Contento (1994) experimentally studied the coupling between ship roll motions and the sloshing in a floodable compartment, with the ship subjected to a beam sea condition. Molin et al. (2002) conducted experiments utilising a barge-like vessel with a partially filled rectangular tank on deck, subjecting it to the combined excitations of roll, sway and heave. Nam and Kim (2007) carried out an experimental study on a barge equipped with two partially filled prismatic tanks, showing the importance of a fully-coupled analysis, in particular for roll motions. With a similar experimental set-up, Nasar et al. (2008, 2010) conducted an experimental study on the effects of intermediate load conditions. In their studies, the centre of gravity, the radius of gyration and the total mass are different as load conditions vary. To get a comprehensive understanding and guarantee a unique variable parameter when load conditions change, Zhao et al., (2014a, 2014b) conducted a series of experimental studies where sloshing coupled with both roll only and coupled roll, sway and heave motions with rectangular tanks. To the authors' knowledge, very limited published effort focuses on global roll response coupled with liquid cargo motions inside spherical tanks such as are used on Moss-type LNG carriers. However, the global roll motion with partially-filled spherical tanks may be a limiting criterion for side-by-side offloading operations in open seas. The aim of the present study is to characterise the influence of liquid cargo motions inside spherical tanks on the roll response of an LNG carrier. The results in this study are partly expected to serve as a benchmark for numerical models that may be incorporated into FLNG berth availability studies. To achieve this, a series of scale model tests, including inclination tests, decay tests, and white noise wave tests, are conducted. In the model test, each wave case is repeated with the spherical tanks filled with liquid cargo and equivalent ‘frozen-cargo’ respectively, where the total mass, centre of gravity and radius of gyration hold the same. The global roll response amplitude operators (RAOs) of the vessel, with and without considering the effects of internal sloshing, have been studied, together with the short-term response spectra in swells. Furthermore, the internal sloshing elevations inside the spherical tanks are captured, facilitating understanding of the coupling effect mechanism.
To facilitate the analysis, the equation of motion for roll response only is discussed by ignoring the coupling from motions in other degrees of freedom. A typical equation of roll motion can be expressed as follows (Himeno, 1981; Taylan, 2000; Surendran et al., 2005):
Aφ φ̈ + Bφ(φ̇) + Cφ(φ) = Mφ(ωt ),
(1)
where, φ stands for the roll angle and the over-dot denotes differentiation with respect to timet . Aφ represents the transverse virtual mass moment of inertia for the vessel, Bφ the roll damping moment, Cφ the restoring moment and Mφ the exciting moment due to external waves and internal sloshing. To express the nonlinearity of the roll motion, the damping term in Eq. (1) are usually given in the following form:
Bφ(φ̇) = BL φ̇ + BN φ̇ φ̇ ,
(2)
where, BL and BN stand for linear and nonlinear roll damping moment coefficients, respectively. It is a common practice to approximate the nonlinear viscous damping with an equivalent linear damping (e.g. a percentage of critical damping) (Wilson, 1996). As a consequence, the nonlinear damping term Bφ(φ̇) can be expressed through an equivalent linear damping as follows:
Bφ(φ̇) = Beφ̇ ,
(3)
where, Be can be divided into the following components (Himeno, 1981):
Be = BW + BF + BL + BE + BBK ,
(4)
where, BW stands for the wave radiation damping, BF the friction damping from the hull, BL the lift damping of the hull, BE the eddymaking damping and BBK the damping due to the bilge keels. Of these components, it is difficult to distinguish linear damping from nonlinear damping. As an approximation, the radiation damping BW , the friction damping BF in the absence of ship forward speed, the lift damping BL in the presence of ship forward speed and parts of bilge keel effects BBK contribute to the linear term in Eq. (2), while the eddymaking damping BE and parts of bilge keel effects BBK take effect through the quadratic term. Scale model test is a feasible way to estimate these coefficients. Scale effects can be an important problem for the estimation of some forms of roll damping. Of these damping components, only the frictional damping suffers from scale effects (Himeno, 1981). The non-dimensional value of the friction damping is almost proportional to λ−0.75 (λ the scaling coefficient), allowing us to ignore the frictional damping of actual ships. The above literature evidence has shown that it remains a challenge to predict roll motions using only numerical models due to the complexity and nonlinearity. Inclusion of intermediate load conditions increases this complexity as the sloshing inside tanks is inherently nonlinear. Scale model tests are an acceptable and reliable way to handle this complex nonlinear problem. 3. Model particulars Focusing on the global roll response and internal sloshing problem, a barge-like vessel, 200 m long (full-scale equivalent) and beam of 46 m (full-scale equivalent) was constructed (Table 1). This model had an identical cross section in the longitudinal direction and was designed to have roll characteristics representative of a Moss-type LNG carrier. Two spherical tanks are placed on the vessel to model sloshing effects. The model tests were conducted in the Deepwater Wave Basin at Shanghai Jiao Tong University at a scale of 1:60. In this study, the barge-like vessel is designed on the assumption that the spherical tanks are filled with water, because the usage of LNG 84
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Table 1 Particulars of the vessel with two spherical tanks in 50% filling condition. Full scale
Model scale
Length between perpendiculars (m) Breadth (m) Depth (m) Mean Draft (m) Displacement weight (kg) Prior to cargo loading Radius of roll gyration (m) Radius of pitch gyration (m) Water/equivalent solid weights to be filled in the two spherical tanks (kg)
200 46 25.5 11.1 99,297,900
3.333 0.767 0.425 0.185 459.7
19.4 68.5 29,880,015
0.323 1.142 138.3
VCG (m) Kxx (m) Kyy (m)
Target
Achieved
Error
0.23 0.323 1.142
0.231 0.319 1.1466
0.43% −1.24% 0.40%
Beam sea
Designation
Table 2 Calibration results prior to cargo loading.
Head sea
as a fluid in a laboratory would be prohibitively complex and hazardous. Water is approximately twice the density of LNG (465 kg/ m3) so two tanks of water represents a similar mass to four LNG tanks, as may be expected on a typical Moss-type LNG carrier. The tanks are sized and positioned such that they approximate a larger number of tanks filled with LNG as closely as possible. An LNG carrier will experience different intermediate load conditions during a side-by-side offloading operation. In this study, an intermediate load condition of 50% by volume is selected as the reference. The particulars of the vessel in this load condition are listed in Table 1 in both full scale and model scale. The corresponding lines plan (in Fig. 1) shows that the vessel is equipped with both bilge keels and two spherical tanks. The radius of the spherical tanks is 19.25 m in full scale. To clarify the effects of liquid cargo motions on the global roll responses, the two spherical tanks are filled with water for one case (referred to as ‘liquid-cargo’ case), and left empty for a second case for which the quasi-static properties are matched with solid weights (referred to as ‘frozen-cargo’ case). As shown in Table 1, the only difference between the two cases is whether there is liquid flow inside the tanks. The other parameters hold the same such as the total mass, draft, and centre of gravity in vertical direction (VCG). Prior to the model tests, all the parameters shown in Table 1 are calibrated. In the liquid-cargo case, the calibration is conducted based on the parameters prior to cargo loading and then the spherical tanks are filled with water to achieve the target filling level. The measurements for the liquid-cargo case during the calibration are shown in Table 2. In the frozen-cargo case, there is no liquid cargo at all, so the calibration is conducted based on the total value of the parameters in Table 1. The VCG is adjusted to be 0.2775 m with an error of 2.40% relative to the target value (0.271 m), pitch radius is adjusted to be 1.0454 m with an error of −1.38% relative to the target value (1.06 m). The roll radius is achieved through matching the roll period with that obtained in the liquid-cargo case.
Fig. 2. Configuration of the horizontally moored vessel in wave basin.
4. Experimental set-up 4.1. Model configuration in the wave basin In this study, model tests are conducted with the vessel both in calm water and waves. Decay tests in calm water are conducted to obtain the decay curves. In the decay tests, the model is given an initial heel angle and then released; vessel motions are recorded simultaneously. Wave tests are carried out to obtain the RAOs of the roll motions and the internal sloshing. During wave test runs, the vessel model is moored in the wave basin with a horizontal mooring system, as shown in Fig. 2. The horizontal mooring system is designed to be soft enough to avoid influencing the wave frequency motion responses. The natural period of the horizontal moored system is measured to be 163 s in full scale, which is far away from the roll natural period and is out of the interest of this study. In addition to the vessel's motions, the surface elevations inside the spherical tanks are recorded synchronously through wave probes. As shown in Fig. 3, the wave probes are fixed along the arc-shape side walls of the spherical tanks, allowing the measurement of the water surface elevation along the inside wall of the spherical tanks. 4.2. Calibration of waves White noise waves, instead of regular waves, are used to ascertain the RAOs of the vessel and the internal sloshing. The white noise wave spectrum has a bandwidth from 4 to 29 s and significant wave height of
Arc-shape wave probe
Fig. 1. Lines plan of the vessel with two spherical tanks: (a) front view; (b) side view; (c) top view.
Vertical wave probe
Fig. 3. Wave probes in the spherical tanks.
85
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 4. White noise waves in beam seas: (a) response spectrum, (b) representative time series.
Fig. 5. White noise waves in quartering seas: (a) response spectrum, (b) representative time series.
3 m in full scale, so one test run is able to cover both the roll natural period and the sloshing natural periods of the liquid cargo which are a distance apart. Prior to the wave tests, a time series of the white noise wave is generated without the model in the basin. The duration of the white noise waves at full scale is 3 h. In the model tests, the heading of the model was kept unchanged, but the wave makers at different sides of the wave basin were used to generate waves coming in different directions to form the beam and quartering sea states, as shown in Fig. 2. This avoids the re-setting up of the model, and thus avoids potential error due to setting-up of the model. The generated white noise spectra are compared with the target spectra in Figs. 4 and 5, together with a representative time series of the generated wave elevations. It can be seen in Fig. 4(a) and Fig. 5(a) that the generated wave spectra match well with the target spectra.
Table 3 Regular wave parameters in full scale. Regular waves
Wave height (m)
Period (s)
Wave slope
No. No. No. No. No.
3 6 7 9 9
7.00 13.97 15.80 16.10 16.88
0.0393 0.0197 0.0180 0.0223 0.0203
1 2 3 4 5
RAOs can be obtained from regular wave tests, and the repeatability of the results are illustrated in Fig. 6. It can be seen in Fig. 6 that the model test results match well for each duplicated run, confirming the reliability of the model tests.
5. Results and discussion 5.2. Decay test results 5.1. Repeatability study Free decay tests of the model in calm water are conducted with the spherical tanks filled with liquid cargo and corresponding ‘frozen cargo’ respectively, aiming primarily to check the roll natural period and secondly to clarify the effects of the liquid cargo motion on roll motions in calm water. The extinction traces of the model in the two different ballasting cases are compared in Fig. 7. The roll motion in the liquidcargo case is found to decay quicker than that in the frozen-cargo case,
To estimate the performance of the experimental set-up a repeatability study is conducted. A series of duplicated regular wave runs is conducted with the vessel in the ‘frozen cargo’ set-up. The regular wave parameters used in this study are listed in Table 3. The wave slopes (ratio of wave height to wave length) as shown in Table 3 are designed to be less than 0.05, to keep the regular waves as linear as possible. 86
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
incident white noise waves. The roll RAOs of the vessel in two different ballasting conditions are compared in Fig. 8, with (a) for beam seas and (b) for quartering seas. It can be seen in Fig. 8(a) that the peak roll RAO occurs at the roll natural period (15.16 s), and it is slightly larger in the liquid-cargo case (6.79 deg/m) than that in the frozen-cargo case (6.67 deg/m). However, when looking at the comparison results in quartering seas, the internal sloshing exhibits relatively larger effects on the global roll response at the roll natural period. It can be seen that the peak roll RAO is amplified by 8.40% from 3.81 deg/m in the frozen-cargo case to 4.13 deg/m in the liquid-cargo case. We think this amplification is a result of the internal sloshing effect, instead of the parametric roll which may occur in head seas and is very sensitive to vessel heading. As described above, the vessel model used in this study is designed to be generally like a rectangular box, so there is negligible periodic variation of the transverse stability of the vessel in waves, characterised by a decrease of stability when the mid-ship of the vessel is in the wave crest and an increase in the wave trough (France et al., 2003). In addition, the wave height used in this study is small, which makes parametric roll impossible here in quartering or beam seas. As shown in Fig. 8, the roll RAOs are sensitive to the headings of the vessel. The peak RAOs in quartering seas drop down to two thirds of those in beam seas. An interesting phenomenon can be noted in Fig. 8(a) is that an obvious secondary peak roll RAO occurs at 6.39 s for the liquid-cargo case. This is associated with the internal sloshing. As can be seen in the same figure, there is a tiny peak for the roll RAO even in the solid case at around 7.52 s, which is half of the natural roll response period (15.16 s). This indicates that the small peak in frozen-cargo case is induced by the second-order harmonics. However, the peak value (0.17 deg/m) is small compared to that in the liquid-cargo case (1.22 deg/m). As a consequence, the second-order harmonics might make a small contribution, but the coupling between internal sloshing and the second-order harmonics of the main roll motions is very important and leads to the secondary peak roll RAO shown in the liquid case in Fig. 8(a). A physical understanding of the coupling between internal sloshing and higher harmonics of the main roll motions has been given by Zhao et al. (2016). Another interesting phenomenon that may be observed in Fig. 8 is that the sloshing effect on the global roll motion is sensitive to the heading of the vessel. In beam seas, one can observe an obvious secondary peak of roll RAOs, while it entirely absent when the heading changes to quartering seas. As the above discussion, the RAO at the roll natural period is more amplified in quartering seas than that in beam seas. According to previous studies (Kim et al., 2007; Zhao et al., 2014a), this phenomenon can be explained through the phases of the internal sloshing elevations relative to the external waves. The phase of the global roll motions and the internal sloshing are given with respect to the external waves as shown in Figs. 9 and 10. One can see in Fig. 9(a) a rapid change of phase in the liquid-cargo case at the period where a peak RAO (shown in Fig. 8(a)) occurs, while the frozen-cargo case does not show the same phenomenon. When we changed the headings to quartering seas (Fig. 9(b)), the roll phases in the frozen-cargo and liquid-cargo cases are pretty similar around 6.39 s, which is consistent with the similar RAOs as shown in Fig. 8(b). For both beam and quartering seas at the period around 15.16 s, the roll phases in the frozen-cargo and liquid-cargo cases are very similar. The phases of the internal sloshing elevations relative to the external waves are plotted in Fig. 10. If the internal sloshing moves in phase with the incident waves, they will work together to amplify the vessel's global roll response. In contrast, if it is out of phase with the incident waves, the internal sloshing can cancel part of the effects induced by wave forces. It can be observed that the relative phase at the roll natural period (15.16 s) is 17 deg for beam seas (Fig. 10(a)), while it is −7 deg for quartering seas (Fig. 10(b)). The internal liquid cargo motion and external waves at the roll natural period are more in phase
Fig. 6. Repeatability of the model tests.
Fig. 7. Roll decay curves of the vessel with the spherical tanks filled with liquid cargo and fixed cargo.
and the damping coefficient is slightly amplified from 0.0226 (frozencargo case) to 0.0238 (liquid-cargo case). The slight increase of damping in the liquid-cargo case may be associated with the energy dissipation induced by the liquid cargo motions inside the tanks. This phenomenon indicates that the cargo sloshing adds a small amount of damping, which is close to negligible at the natural response period of the vessel. However, if we consider the wave-structure interactions, the coupling between roll response of the vessel and the internal liquid cargo motions will be another scenario. This will be discussed in detail in the following section. 5.3. RAOs To investigate the sloshing effect of the liquid cargo on the global roll response in waves, white noise wave tests in both beam and quartering seas (shown in Fig. 2) were conducted. The spherical tanks are filled to 50% of volume for the liquid cargo tests, while the vessel is ballasted with solid steel weights for the ‘frozen-cargo’ tests. Time traces of the roll motions and internal sloshing elevations are measured. Spectral analyses are then conducted to obtain the power spectrum density function, which can further be used to ascertain the RAOs through the following equation:
H (ω ) =
Sxy(ω) Sxx (ω)
,
(5)
where, H (ω) is the RAO, Sxy(ω) the cross spectral density function of the responses, and Sxx (ω) the auto spectral density function of the 87
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 8. Roll RAOs of the vessel in different heading conditions: (a) beam seas, (b) quartering seas.
lateral direction of the vessel, without components in the longitudinal direction. As shown in Table 4, the measured sloshing periods show very good agreement with the analogous solutions by Faltinsen and Timokha (2012). It is not difficult to understand why the sloshing waves in the lateral direction dominate in both beam and quartering seas. This is due to the fact that even in quartering seas, the pitch motion is far smaller than roll motion, therefore it cannot induce such significant waves in the longitudinal direction as those in the lateral direction. Significant sloshing RAOs can be found in Fig. 11 at different resonant periods, the roll responses at the corresponding periods are negligible as shown in Fig. 8 (except the secondary RAO peak for liquid-cargo case in Fig. 8(a)). Through the comparison of the results in the two figures, one can observe that only the first mode of sloshing has coupling effects on the global roll motions, and the higher modes of sloshing seem to have very little influence on the global roll response.
in quartering seas. As a consequence, the internal liquid cargo motion shows more amplification effects at the roll natural period in quartering seas than that in beam seas. A similar phenomenon is seen at the sloshing period: the relative phase at the sloshing period is −75 deg (in phase) in the beam sea, amplifying the wave force effect; while it is −120 deg (out of phase) in the quartering sea, cancelling the wave force. This explains why a secondary response peak occurs in beam seas at the sloshing period, but disappears in quartering seas. To deepen the understanding of the sloshing flows inside spherical tanks, the RAOs in different heading conditions are calculated and shown in Fig. 11. One can observe in Fig. 11 that in addition to the fundamental sloshing peak, higher modes of sloshing peaks occur in both beam seas and quartering seas. It is interesting to observe that these sloshing peaks occur at the same periods in the two different heading conditions. This can be explained through the sloshing characteristics inside spherical tanks. Sloshing inside spherical tanks can support free surface oscillations with integral azimuthal wave numbers m (including zero) and for each m , there is an infinite sequence of discrete oscillation frequencies (McIVER, 1989; Faltinsen and Timokha, 2012). Therefore, sloshing inside spherical tanks consists of waves in both lateral and longitudinal directions of the vessel. It can be deduced that for both beam and quartering seas, the sloshing waves in the lateral direction of the vessel dominate and the sloshing waves in longitudinal direction can be negligible. To prove this deduction, the measured sloshing periods in the two heading conditions are compared with the periods of the sloshing modes only in
5.4. Short-term analyses During side-by-side offloading, an LNG carrier is generally not expected to experience wind-sea on the beam because most FLNG designs are turret-moored, allowing them to ‘weathervane’ into prevailing conditions. However, in some locations such as the North West Australian Shelf and Western Africa, swells may come in beam or quartering seas, which may induce critical roll motions for an LNG carrier. To estimate the effects of the liquid cargo motion, two typical swell conditions from the North West Australian Shelf are taken as the reference: one is a short period swell with a period equal to the sloshing
Fig. 9. Relative phases of roll motions to external waves: (a) beam seas; (b) quartering seas.
88
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 10. Relative phases of the internal sloshing to external waves: (a) beam seas; (b) quartering seas.
Fig. 11. Sloshing RAOs of the liquid cargo in different modes: (a) beam seas, (b)quartering seas.
One can see in Fig. 12 and Table 6 that the response peak and the Table 4 Sloshing periods of spherical tanks in half-filled condition.
Table 5 Swell conditions for short-term analyses.
Methods
1st
2nd
3rd
4th
Analogous solution (s) Present (s) Discrepancy
6.38 6.39 0.19%
4.67 4.55 −2.63%
3.88 3.89 0.40%
3.38 3.35 −1.00%
Hs (m)
Tp (s)
Short period Long period
0.9 0.9
6.39 15.16
most probable maximum in 3 h in the frozen-cargo case are 0.03 deg2/ s and 0.14 deg, while they reach up to 1.53 deg2/s and 0.86 deg in the liquid-cargo case. The extreme roll response is amplified due to the internal sloshing. This phenomenon indicates that for a short period swell coming in beam seas, the internal sloshing may affect the global roll response to a certain degree. However, around the period of 15.16 s when vessel roll motions are important, the roll responses are indistinguishable in the liquid- and frozen-cargo cases (shown in Fig. 12(b)). A possible reason for this phenomenon might be associated with the fact that the period at which the sloshing-induced response occurs is far away from the roll natural period. According to the McCarty and Stephens (1960) study, the first mode of sloshing period in a spherical tank will not be larger than 7 s, therefore the sloshing period will not be too close to the roll natural period. At reduced load conditions, the internal sloshing can be severer (Zhao et al., 2014b); in contrast, the weight of the fluid inside tanks will decrease, but it is unclear which
period and the other is a long period swell with a period equal to the roll natural period of the vessel. The parameters of the two swell conditions are listed in Table 5. A Gaussian spectrum in the following form is used:
⎡ (ω − ω ) 2 ⎤ ⎛ H ⎞2 1 0 ⎥, Sη(ω) = ⎜ s ⎟ ⋅ ⋅exp⎢ − ⎢⎣ 2⋅σ 2⋅ω0 2 ⎥⎦ ⎝ 4 ⎠ σ⋅ 2π ⋅ω0
Swells
(6)
where, ω0 stands for the peak frequency of the spectrum, σ the dimensionless shape parameter. In this study, the value of σ is selected as 0.02Hz . Short-term analyses have been conducted based on the above swell conditions and the RAOs obtained in beam and quartering seas. The roll response spectra of the vessel are obtained and shown in Figs. 12 and 13. Furthermore, the most probable maxima of the roll response within 3 h are estimated and listed in Table 6. 89
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
Fig. 12. Response spectrum of the roll motion in beam seas: (a) short swell with period of 6.39 s, (b) long swell with period of 15.16 s. Note the different scales on the two figures.
Fig. 13. Response spectrum of the roll motion in quartering seas: (a) short swell with period of 6.39 s, (b) long swell with period of 15.16 s. Note the different scales on the two figures.
effect dominates. To make a decision on whether short period waves (6–7 s) should be included in analyses during design, further study is needed for reduced load conditions. It can be further found in Fig. 12(b) that the roll response in long period swells holds almost the same in both the liquid- and the frozen-cargo cases. This indicates that if the long swell comes in beam seas, the sloshing effects on the global roll response are relatively small. It can be observed in Fig. 13 and Table 6 that if a short period swell comes in a quartering sea, the roll responses in the liquid- and frozencargo cases are 0.18 deg and 0.05 deg, respectively, which are very small in absolute value and their difference is negligible. However, if a long period swell comes in quartering seas, the most probable largest roll response will be amplified by 6.83% from 6.50 to 6.97 deg due to the internal sloshing effects. As the load conditions become lower, the internal sloshing can become more severe (Zhao et al., 2014b), which may increase the amplification effect. Therefore, it is worthy of further study to clarify whether the internal sloshing could induce more significant effects on roll response at reduced load conditions. Through the comparison between Figs. 12 and 13, one can observe that if a short period swell occurs in beam seas, the roll response of the vessel may be affected by the internal sloshing; while the roll responses in both the liquid- and frozen-cargo cases are too small in short period swells coming in quartering seas, thereby the sloshing effect can be negligible. For a long period swell in beam seas, even if the swell period is equal to roll natural period, the sloshing effects on the global roll response can be negligible; while internal sloshing will amplify the roll
Table 6 Most probable maximums of the roll response within 3 h. Headings
Swells
Liquid case (deg)
‘Frozen’ case (deg)
Beam seas
Short period Long period Short period Long period
1.91 11.87 0.18 6.97
0.31 11.98 0.05 6.50
Quartering seas
response slightly for long period quartering swells. 6. Conclusion To investigate the coupling between roll motions and internal sloshing, a series of scale model tests are conducted. To examine the effect of sloshing, both a liquid-cargo test and a ‘frozen-cargo’ test are conducted. The effects of liquid cargo motions in spherical tanks on the global roll response of a barge-like vessel at an intermediate load condition are examined for the first time. The main conclusions drawn are as follows:
• • 90
For beam seas, the roll response of the LNG carrier with liquid cargo shows the same roll period as that with frozen cargo for spherical tanks. The internal sloshing induces a secondary spectral peak for roll motions in beam seas, but it does not seemingly induce significant
Ocean Engineering 129 (2017) 83–91
W. Zhao, F. McPhail
• •
•
lashing systems. Mar. Technol. 40 (1), 1–19. Francescutto, A., Contento, G., 1994. An experimental study of the coupling between roll motion and sloshing in a compartment, In: Proceedings of the Fourth International Offshore and Polar Engineering Conference. International Society of Offshore and Polar Engineers. Himeno, Y., 1981 Prediction of ship roll damping. A State of the Art. DTIC Document. Ikeda, Y., Himeno, Y., Tanaka, N., 1978. A prediction method for ship roll damping. Report of the Department of Naval Architecture, University of Osaka Prefecture (00405). Kato, H., 1958. On the frictional resistance to the rolling of ships. J. Soc. Nav. Archit. Jpn. 102, (115-112). Kim, Y., Nam, B.W., Kim, D.W., Kim, Y.S., 2007. Study on coupling effects of ship motion and sloshing. Ocean Eng. 34 (16), 2176–2187. Lee, S., Kim, M., 2010. The effects of inner-liquid motion on LNG vessel responses. J. Offshore Mech. Arct. Eng. 132 (2), 021101. Lee, S., Kim, M., Lee, D., Kim, J., Kim, Y., 2007. The effects of LNG-tank sloshing on the global motions of LNG carriers. Ocean Eng. 34 (1), 10–20. Malenica, S., Zalar, M., Chen, X., 2003. Dynamic coupling of seakeeping and sloshing, In: Proceedings of the 13th international offshore and polar engineering conference, pp. 484–490. McCarty, J.L., Stephens, D.G., 1960. Investigation of the natural frequencies of fluids in spherical and cylindrical tanks. NASA, Technical Note, D-252. McIVER, P., 1989. Sloshing frequencies for cylindrical and spherical containers filled to an arbitrary depth. J. Fluid Mech. 201, 243–257. Mikelis, N., Miller, J., Taylor, K., 1984. Sloshing in partially filled liquid tanks and its effect on ship motions: numerical simulations and experimental verification. R. Inst. Nav. Archit. Trans., 126. Mitra, S., Wang, C., Reddy, J., Khoo, B., 2012. A 3D fully coupled analysis of nonlinear sloshing and ship motion. Ocean Eng. 39, 1–13. Molin, B., Remy, F., Rigaud, S., De Jouette, C., 2002. LNG-FPSO’s: frequency domain, coupled analysis of support and liquid cargo motion, In: Proceedings of the 10th International Maritime Association of the Mediterranean (IMAM) Conference, Rethymnon, Greece. Nam, B.W., Kim, Y., 2007. Effects of Sloshing on the Motion Response of LNG-FPSO in Waves, In: Proceedings of the 22nd Workshop onWater Waves and Floating Bodies, Plitvice, Croatia. Nasar, T., Sannasiraj, S., Sundar, V., 2008. Experimental study of liquid sloshing dynamics in a barge carrying tank. Fluid Dyn. Res. 40 (6), 427–458. Nasar, T., Sannasiraj, S., Sundar, V., 2010. Motion responses of barge carrying liquid tank. Ocean Eng. 37 (10), 935–946. Newman, J., 2005. Wave effects on vessels with internal tanks, In: Proceedings of the 20th Workshop on Water Waves and Floating Bodies, Spitsbergen, Norway. Robinson, R., Stoddart, A., 1987. An engineering assessment of the role of non-linearities in transportation barge roll response. Nav. Archit.. Salvesen, N., Tuck, E., Faltinsen, O., 1971. Ship motions and sea loads. Det norske Veritas, Oslo. Surendran, S., Lee, S., Reddy, J., Lee, G., 2005. Non-linear roll dynamics of a Ro-Ro ship in waves. Ocean Eng. 32 (14), 1818–1828. Taylan, M., 2000. The effect of nonlinear damping and restoring in ship rolling. Ocean Eng. 27 (9), 921–932. Wilson, E.L., 1996. Three Dimensional Static and Dynamic Analysis of Structures. Computers and Structures, Inc., Berkeley, California, USA. Zhao, W., Yang, J., Hu, Z., Wei, Y., 2011. Recent developments on the hydrodynamics of floating liquid natural gas (FLNG). Ocean Eng. 38 (14), 1555–1567. Zhao, W., Yang, J., Hu, Z., Tao, L., 2014a. Coupling between roll motions of an FLNG vessel and internal sloshing. J. Offshore Mech. Arct. Eng. 136 (2), 021102. Zhao, W., Yang, J., Hu, Z., Xiao, L., Tao, L., 2014b. Hydrodynamics of a 2D vessel including internal sloshing flows. Ocean Eng. 84, 45–53. Zhao, W., Wolgamot, H., Eatock TaylorR., Taylor, P.H., 2016. Nonlinear harmonics in the roll motion of a moored barge coupled to sloshing in partially filled spherical tanks. In: Proceedings of the 31th International Workshop on Water Waves and Floating Bodies. Plymouth, MI, USA. April 2016. pp. 3–6 〈http://www.iwwwfb.org/ Abstracts/iwwwfb31/iwwwfb31_53.pdf〉.
roll response. The roll natural frequency is much lower than the sloshing frequency, so the coupling between these two phenomena is through the higher harmonics of the roll motions, i.e. the double frequency of the main roll motion. For quartering seas, the maximum roll RAO in the liquid-cargo case is amplified by 8.40% compared to that in the frozen-cargo case. This effect may further prompt explicit inclusion of liquid cargo dynamics in availability studies for side-by-side offloading. The general results of this study tend to indicate that the internal sloshing inside spherical tanks could amplify the global roll motions but not significantly, at least for this specific (half volume) load condition. When the filling level of the tanks is reduced, the internal sloshing may become severer, increasing the influence on global roll motions. On the other hand, the weight of the liquid cargo inside tanks decrease at reduced load conditions, decreasing the sloshing effect. It is however unclear which effect will dominate. Further study is merited to examine whether sloshing-induced response peaks matter at reduced load conditions, so as to make a decision on whether short period waves (6–7 s) should be included.
This study does not attempt to solve the coupling problem of roll responses and liquid cargo sloshing completely. It was initiated as a derisking exercise to investigate what effect internal liquid cargo sloshing in spherical tanks may have on global vessel motions and to serve as a basis for numerical work. Assessments of side-by-side operability should expect to undertake significant additional numerical work to ensure the full range of motions is captured and understood for a project specific environment. Acknowledgement This work was funded by Shell Global Solutions BV and conducted in the Deepwater Offshore Basin at Shanghai Jiao Tong University as a joint program. The first author is supported by the Shell EMI offshore engineering initiative at the University of Western Australia. Shell would like to acknowledge the support and dedication of the staff at the Shanghai Jiao Tong University Deepwater Offshore Basin. They proved to be professional, diligent, and accommodating to requirements throughout the campaign of model tests. References Bhattacharyya, R., 1978. Dynamics of Marine Vehicles. Wiley, New York. Blagoveshchenskiĭ, S., 1962. Theory of Ship Motions. Dover Publications, New York. Denise, J., 1983. On the roll motion of barges. Downie, M., Bearman, P., Graham, J., 1988. Effect of vortex shedding on the coupled roll response of bodies in waves. J. Fluid Mech. 189, 243–261. Faltinsen, O.M., Timokha, A.N., 2012. Analytically approcximate natural sloshing modes for a spherical tanks shape. J. Fluid Mech. 703, 391–401. France, W.N., Levadou, M., Treakle, T.W., Paulling, J.R., Michel, R.K., Moore, Colin, 2003. An investigation of head-sea parametric rolling and its influence on container
91