- Email: [email protected]
On natural frequencies of three-dimensional moonpool of vessels in the fixed and free-floating conditions
Ocean Engineering 195 (2020) 106656
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
On natural frequencies of three-dimensional moonpool of vessels in the fixed and free-floating conditions Xin Xu a, b, Xinshu Zhang a, b, *, Bei Chu a, b, Haiyang Huang a, b a b
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China
A B S T R A C T
The hydrodynamic interactions between the vessel motion and resonant wave response inside the three-dimensional moonpool are studied in the present paper. In particular, the shift of natural frequencies of a three-dimensional moonpool in the fixed and free-floating conditions is computed and examined systematically. Barges with different moonpool dimensions are analyzed to examine how the moonpool configurations affect the shift of resonant frequencies. Detailed analyses are performed for cases with and without recess. Numerical computations are performed for regular waves with wave headings θ ¼ 0∘ , 45∘ and 90∘ . Free-surface ele vations inside the moonpool with respect to the incident wave frequencies are computed for both fixed and free-floating conditions. It is observed that natural frequencies of the moonpool in the free-floating condition are higher than those in the fixed condition. For cases without recess, the shift of resonant frequency is more prominent in piston mode resonance, in particular for shallow or long moonpools. In contrast, for cases with a recess, the shift of resonant frequency in the first sloshing mode is relatively larger than that in piston mode. In addition, the effects of motion RAOs on the free-surface elevation inside the moonpool in the freefloating condition are also examined. It is found that the moonpool resonance can induce a local hump in heave and pitch responses. Moreover, through deriving a modified frozen mode approximation model, we reveal that, comparing to the fixed condition, the vessel motion in the free-floating condition yields a reduction in the added mass due to the fluid underneath the moonpool so that the piston mode resonant frequency increases.
1. Introduction Moonpools are vertical openings in a floating body, frequently adopted in drilling ships for pipeline laying, and in diving support ves sels for launching of diving bells and other subsea equipments (see Faltinsen and Timokha (2015); Yoo et al. (2017)). Resonant water mo tion occurs at the natural modes of the moonpool, including piston mode, where the water inside the moonpool heaves up and down, and sloshing modes, where the water inside the moonpool moves back and forth between the vertical walls. Since the resonant fluid motion may lead to significant impact on the hull structure or equipments on the deck, it is necessary to accurately predict the resonant frequencies to assist the concept design of the floater and marine operation. Most of the previous studies on moonpools are based on conditions where the structure is either fixed in incident waves, or oscillated harmonically in calm water. Under the assumption of infinite length and beam of a barge, Molin (2001) proposed a semi-analytical model to predict the resonant frequencies and modal shapes of a three-dimensional moonpool in infinite water depth by solving eigen value problems. Based on domain decomposition and Galerkin method, Faltinsen et al. (2007) developed a semi-analytical method to compute
the fluid motions in a two-dimensional moonpool and performed ex periments for comparison. Molin et al. (2009) carried out model tests to study the gap resonances between two rectangular barges which were rigidly connected to a carriage. Free-surface elevation derived from tests in irregular waves are compared with numerical solutions using a code based on linear potential flow theory. Zhao et al. (2017) investigated the first and higher harmonic components of the resonant fluid response in the gap between two identical fixed rectangular boxes by performing experiments in a wave basin. Furthermore, they made a prediction of amplitude of gap resonances at different gap widths by scaling the linear damping (Zhao et al., 2018). Zhang and Bandyk (2013, 2014) solved the radiation problem in a two-layer fluid for rectangular bodies restricted to the upper-layer fluid and the surface-piercing rectangular bodies, respectively. Xu et al. (2019) studied the wave diffraction problem of two-dimensional moonpools in a two-layer fluid with finite fluid depth. Recently, Molin et al. (2018) and Zhang et al. (2019) proposed new models to solve the three-dimensional and two-dimensional moonpool problems in finite water depth. However, as discussed in McIver (2005), the natural frequencies of a moonpool in the free-floating condition can be different from the solu tions computed as the structure is fixed (called ‘fixed condition’ in this
* Corresponding author. Innovative Marine Hydrodynamics Laboratory (iMHL), Shanghai Jiao Tong University, China. E-mail address: [email protected] (X. Zhang). https://doi.org/10.1016/j.oceaneng.2019.106656 Received 20 June 2019; Received in revised form 20 September 2019; Accepted 27 October 2019 Available online 26 November 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 1. Sketch of the vessel model without recess in moonpool. ðaÞ: top view; ðbÞ: side view at centerline (y ¼ 0). Table 1 Load matrix and model sizes of vessels without recess in moonpool. Model No.
L (m)
B (m)
l (m)
b (m)
d (m)
D (m)
CP (m)
NP (m)
1 2 3 4 5
180.8 180.8 180.8 180.8 180.8
32.2 32.2 32.2 32.2 32.2
45.6 45.6 45.6 20.0 60.0
11.2 11.2 11.2 11.2 11.2
11 4 20 11 11
22.8 22.8 22.8 22.8 22.8
(0, (0, (0, (0, (0,
(10, 0, 0) (10, 0, 0) (10, 0, 0) (8, 0, 0) (13, 0, 0)
paper) or in harmonic oscillation conditions with a prescribed motion. Focusing on piston mode motion in the moonpool and rigid-body mo tions, Fredriksen et al. (2015) studied regular wave-induced behavior of a two-dimensional floating body with moonpool through experiments and hybrid numerical methods. It was found that the piston mode fre quency in diffraction problem is almost same as that in radiation, but it is larger for the vessel in the free-floating condition when exposing to incident waves. Guo et al. (2016, 2017) performed experiments for a recessed drillship moonpool and measured wave responses inside the moonpool. Molin (2017) extended the theoretical model in Molin (2001) to compute the resonant frequencies for moonpools with a recess, and compared the results with the measurements by Guo. The prediction of the natural frequencies agreed well with the experiment, except for the first sloshing mode frequency. Newman (2018) investigated the reso nant natural modes of the same type of vessel through WAMIT, and used a domain decomposition method with Legendre polynomials, instead of Fourier functions, for computing the natural frequencies and added mass coefficients. It was found that the shift of natural frequencies in the fixed and free-floating condition are small or non-existent for the particular hull and moonpool configuration. However, till now, the mechanism of the shift of the natural frequencies and effect of moonpool configura tions on the shift are still not well understood, which are the motivations of the present study. The present paper aims to explore how the moonpool dimensions affect the shift of moonpool resonant frequencies (including piston and sloshing modes) in the fixed and free-floating conditions and understand the mechanism on the shift by building up a modified frozen mode approximation model with the body motion effects being accounted.
The present paper is organized as follows. Barges with different di mensions of moonpool are modelled. In addition, a modified frozen mode approximation model is developed to explain the physics on the shift of the frequencies by accounting for the body motion. The radiation-diffraction code WAMIT is employed to compute the freesurface elevations inside moonpool in both fixed and free-floating con ditions and the six-degree of freedom motion RAOs. The effects of vessel motions on the moonpool resonances are discussed in detail. 2. Computational cases Barge-type vessels, with or without recess in a moonpool, are modelled with different dimensions. The moonpool without recess is shown in Fig. 1, where the center of the moonpool coincides with that of the barge. The origin of the body coordinates is placed at the center of the waterplane inside moonpool. The global coordinates coincide with the body-fixed coordinates when the vessel is in calm equilibrium. The draft of vessel is denoted as d, and the length and width are denoted as L and B, respectively. The moonpool length and width are denoted by l and b, respectively. The distance from the aft side of moonpool to the midship section is denoted as D. Two numerical wave probes are placed inside the moonpool to measure the free-surface elevation. As shown in Fig. 1 ðaÞ, two wave probes (denoted by CP and NP) are placed inside the moonpool. Both CP and NP are used to identify the piston and symmetric sloshing mode resonances, but the anti-symmetric sloshing modes can only be captured by the elevation at NP inside the moonpool without recess. As illustrated in Table 1, five different sets of moonpool di mensions are chosen for the cases without recess. Model 1 is the base model. Model 2 and Model 3 are used to study the effect of draft, and Model 4 and Model 5 are adopted to study the effect of moonpool length. To simplify the study, the VCG and inertia of all the vessels without recess in moonpool are kept the same, as listed in Table 2. Fig. 2 illustrates the vessel with recessed moonpool. The center of moonpool is located at 0.4 m behind the midship section of the barge. The length and height of recess are denoted as r and h, respectively. As shown in Table 3, five different combinations of dimension are pre sented. Model 6 shares the same size as the physical model in Guo et al.
Table 2 VCG and inertia of vessels without recess in moonpool. Parameters
Symbol
Unit
Value
The vertical center of gravity Roll radius of gyration
VCG Rxx
m m
1.50 11.58
m
54.86
Rzz
m
55.34
Pitch radius of gyration Yaw radius of gyration
Ryy
0, 0) 0, 0) 0, 0) 0, 0) 0, 0)
2
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 2. Sketch of the vessel with recess in moonpool. ðaÞ: top view; ðbÞ: side view at centerline.
(2016, 2017) for comparison. Compared with Model 6, the length of recess is varied in Models 7 and 8, and the recess height is varied in Models 9 and 10. As shown in Fig. 2 ðaÞ, three numerical wave probes are placed inside the moonpool to measure the free-surface elevations and capture the piston and sloshing mode resonances. The wave probe (FP) is located at the forward end of the moonpool. The wave probe (CP) is located at the center of moonpool, and the wave probe (AP) is placed at the aft side of moonpool. The coordinates of FP, CP and AP are ( 21.4 m, 0 m, 0 m), (0 m, 0 m, 0 m) and (22.2 m, 0 m, 0 m), respec tively. As listed in Table 4, the VCG and inertia of all the vessels with a recess in moonpool are kept the same. It should be noted that the water depth is assumed to be infinite in all the present cases. The fixed condition can be considered as a purely wave diffraction problem, while the free-floating condition can be considered as a com bination of the radiation and diffraction problems. When it is in freefloating condition, the vessel is undergoing six-degree of freedom mo tions in regular incident waves. The incident wave amplitude is denoted as A. The incident wave heading θ is defined as the angle between the positive x-axis of the global coordinate system and the direction in which the incident wave propagates. θ ¼ 0∘ , 45∘ and 90∘ are chosen in the present simulations. The wave frequency ω is varied from 0.1 rad/s to 1.4 rad/s. Correspondingly, the free-surface elevation η at specific locations and motion RAOs are obtained. As shown in Fig. 3, about 500 higher-order panels are used to dis cretize the model without recess, and 1000 panels for the recessed model. The panel size is approximately 3 m in length. For simplicity, the mass density of these models are set as homogeneous. The higher-order panel option based on B-splines and direct solver option are adopted. In addition, the irregular frequencies are avoided by adding meshes on the interior free-surface, which is also generated by WAMIT. Numerical convergence is assessed by repeating the computations for increasing number of panels, as shown in Fig. 4. As can be seen, the dotted line represents the free-surface elevation inside the model discretized by 3-meter-long panels, and the accuracy is high enough to capture the sloshing mode frequency even in high-frequency region.
Table 3 Load matrix and model sizes of vessels with recess in moonpool. Model No.
L (m)
B (m)
l (m)
b (m)
d (m)
D (m)
r (m)
h (m)
6 7 8 9 10
180.8 180.8 180.8 180.8 180.8
32.2 32.2 32.2 32.2 32.2
45.6 45.6 45.6 45.6 45.6
11.2 11.2 11.2 11.2 11.2
11 11 11 11 11
22.4 22.4 22.4 22.4 22.4
16.0 8.0 4.0 16.0 16.0
7.2 7.2 7.2 5.0 3.0
Table 4 VCG and inertia of vessels with recess in moonpool. Parameters
Symbol
Unit
Value
The vertical center of gravity Roll radius of gyration
VCG Rxx
m m
1.50 11.52
m
54.33
Rzz
m
54.80
Pitch radius of gyration Yaw radius of gyration
Ryy
3. Frozen mode approximation model with accounting for the body motion In Molin (2017), the fluid inside the moonpool was assumed as a solid body or frozen restriction, and the influence of the fluid under neath the moonpool was taken as added mass on the dynamic system. Thus, the natural frequency of moonpool can be obtained by analyzing the free heave motion of the frozen restriction. Further, by constraining the heave motion of barge in free-floating condition using WAMIT, the predicted piston mode resonant frequency is 0.735 rad/s and is the same
Fig. 3. Panelization adopted in WAMIT computations. Mesh (blue color) on the interior free surface are adopted to remove the irregular frequencies. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 3
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Figure 4. ðaÞ: Convergence test for Model 1 with respect to number of panels; ðbÞ: Convergence test for Model 6 with respect to number of panels.
Fig. 5. Sketch of the modified frozen mode approximation model by accounting for body motion. ðaÞ: top view; ðbÞ: side view at centerline (y ¼ 0).
as that in fixed condition (see the detailed results in subsection 4.1), which indicates that the shift of the piston mode is mainly attributed to the heave motion in free-floating condition, and the other horizontal modes have little effects on the shift. Therefore, the natural frequency of the moonpool in the free-floating condition can be qualitatively esti mated by accounting for the vessel motion through modifying the body boundary condition. The sketch of the present model and coordinate system is illustrated in Fig. 5. For simplicity, we set l1 ¼ l2 and b1 ¼ b2 in this paper. As shown, the whole fluid domain is divided into two subdomains. We denote the fluid domain inside the moonpool ðz > dÞ as subdomain I, and take the external fluid domain ðz < dÞ as subdomain II.
written as ∞ X
Em
ϕe ¼ m¼1
coshαm ðz þ H þ dÞ sinαm ðx þ L = 2Þ coshðαm HÞ
(1)
where αm ¼ mπ=ðl1 þl þl2 Þ ¼ mπ=L and m ¼ 1; 2; …; ∞ are the integers. In the present model, the velocities at the opening of moonpool and the body motion velocity are different depending on the motion RAO and phase shift. The ratio of body motion velocity to the mean velocity at opening of the moonpool is denoted as μ. By matching the vertical velocity at the common boundary ( L=2 � x � L=2, z ¼ d) of sub domains I and II, we obtain � ∞ X 1; l=2 � x � l=2 Em αm tanhðαm HÞsinαm ðxþL=2Þ¼ μ; L=2 � x � l=2; and l=2 � x � L=2 m¼1
3.1. Two-dimensional case
(2)
To simplify the analyses, the problem is treated in two-dimensional. But the key physics on the shift of the natural frequency should be the similar. In the external fluid domain, the velocity potential can be
By multiplying each side with sinαm ðx þ L=2Þ, and integrating along x direction over [ L=2, L=2], we obtain 4
X. Xu et al.
1 Em αm tanhðαm HÞL ¼ 2
Ocean Engineering 195 (2020) 106656
Z
Z
l=2 L=2
Z
l=2
μsinαm ðx þ L = 2Þdx þ
L=2
sinαm ðx þ L = 2Þdx þ l=2
μsinαm ðx þ L = 2Þdx l=2
(3)
Fig. 6. Model 1. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
Fig. 7. Model 1. Heave and pitch motion RAOs in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: heave, 0∘ ; ðbÞ: heave, 45∘ ; ðcÞ: heave, 90∘ ; ðdÞ: pitch, 0∘ ; ðeÞ: pitch, 45∘ ; ðf Þ: roll, 90∘ .
5
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 8. Model 2. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
Fig. 9. Model 3. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
6
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Then the coefficients Em can be determined as Em ¼ 2
½cosðαm l1 Þ
cosαm ðl1 þ lÞ�ð1 μÞ þ μ½1 α2m tanhðαm HÞL
m
ð 1Þ �
Thus, the heave motion equation of the frozen restriction can be written as
(4)
ðρlbd þ Mal ÞZ€ þ ρglbZ ¼ 0
Thus, the added mass due to the lower fluid domain ðz � dÞ is ob tained as
Z
l
Mal ¼ ρ
ϕe dx ¼ 2ρ 0
cosαm ðl1 þ lÞ�2 ð1
∞ X ½cosðαm l1 Þ n¼1
Then the piston mode frequency can be estimated by
μÞ þ μ½1 ð 1Þm �½cosðαm l1 Þ α3m tanhðαm HÞL
(6)
Thus, the piston mode frequency can be estimated as follow sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρgl ω¼ ρld þ Mal
Emn m¼1 n¼1
coshγmn ðzþd þHÞ sinαm ðxþL=2Þsinβn ðyþB=2Þ coshðγ mn HÞ
4.1. Moonpool without recess
(8)
Fig. 6 illustrates the free-surface elevation with respect to the wave frequency at CP and NP in different wave directions for Model 1. As shown in Fig. 6, the amplitude of piston mode in free-floating condition is larger than those in fixed condition for wave headings θ ¼ 0∘ and 45∘ , but it is smaller in free-floating condition for θ ¼ 90∘ . We check the phase angles of the heave motion of the barge and observe that the phase of heave motion RAO around the piston mode in θ ¼ 90∘ is almost 180� out of phase with those in θ ¼ 0∘ and 45∘ . This may lead to larger crosscoupling damping on the wave motion inside the moonpool in freefloating condition for θ ¼ 90∘ , and further induces less resonant wave motion around the piston mode resonance. Since vortex shedding at the corner of moonpool is neglected, the obtained wave amplitude may be overestimated. However, the present paper focuses on the shift of resonant frequency, which may not be affected significantly by the vortex shedding induced damping (see Faltinsen et al. (2007)). Three local maxima in the free-surface elevation corresponding to three nat ural frequencies can be found in either fixed or free-floating conditions. The piston mode frequency in the fixed and free-floating conditions are 0.735 rad/s and 0.76 rad/s, respectively. The shift is around 3.40%. The first sloshing mode frequency in the fixed condition is 0.913 rad/s, while that in the free-floating is about 0.923 rad/s, with the shift being 1.10%. The second sloshing mode frequency is the same (1.18 rad/s) in both
where αm ¼ mπ=ðl1 þ l þ l2 Þ, βn ¼ nπ=ðb1 þ b þ b2 Þ, γ 2mn ¼ α2m þ β2n and m; n ¼ 1; 2; …; ∞ are the integers. By matching the vertical velocity on the common boundary ( L= 2 � x � L=2, B=2 � y � B=2, z ¼ d) of subdomains I and II, we obtain � ∞ X ∞ X 1; on S1 Emn γmn tanh ðγ mn HÞsinαm ðx þL =2Þsinβn ðyþB =2Þ ¼ (9) μ; on S2 m¼1 n¼1 where S1 represents the opening at z ¼ d, and S2 represents the hull bottom. By multiplying each side of Eqn. (9) with sinαm ðx þ L =2Þsinβn ðy þ B =2Þ, and integrating along x direction over [ L=2, L= 2] and along y direction over [ B=2, B=2], we obtain ZZ 1 Emn γ mn tanhðγmn HÞLB ¼ μ sinαm ðx þ L = 2Þsinβn ðy þ B = 2Þdxdy 4 S2 ZZ þ sinαm ðx þ L = 2Þsinβn ðy þ B = 2Þdxdy (10) S1
Then the coefficients Emn can be determined as Emn ¼
4 μ½ð γ mn tanhðγmn HÞLB
μ½ð
1Þn
1Þm
1 þ cosðαm l1 Þ
cosαm ðl1 þ lÞ� þ cosαm ðl1 þ lÞ
cosðαm l1 Þ
αm
1 þ cosðβn b1 Þ
cosβn ðb1 þ bÞ� þ cosβn ðb1 þ bÞ βn
(14)
The natural frequencies are obtained by searching for the local maxima in the RAO of free-surface elevation inside the moonpool. The first local maximum corresponds to the piston mode, and the others are associated with sloshing mode resonances. It should be noted that the present study focuses on longitudinal sloshing modes, which are more important in the practical application.
(7)
When the model is extended to three-dimensional case, the velocity potential in the external fluid domain can be written as ∞ X ∞ X
(5)
4. Numerical results
3.2. Three-dimensional case
ϕe ðx;y;zÞ¼
cosαm ðl1 þ lÞ�
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρglb ω¼ ρlbd þ Mal
The heave motion equation of the frozen restriction can be written as ðρld þ Mal ÞZ€ þ ρglZ ¼ 0
(13)
cosðβn b1 Þ
(11)
and the added mass due to the lower fluid domain ðz � dÞ is obtained as
Mal ¼ ρ
∞ X ∞ X
Emn tanhðγmn HÞ m¼1 n¼1
cosαm ð l=2 þ L=2Þ
αm
cosαm ðl=2 þ L=2Þ cosβn ð b=2 þ B=2Þ cosβn ðb=2 þ B=2Þ βn
7
(12)
X. Xu et al.
Ocean Engineering 195 (2020) 106656
frequency in the fixed and free-floating conditions are 0.943 rad/s and 1.002 rad/s, respectively. The shift is around 6.26%. The first sloshing mode frequency in the fixed condition is 1.090 rad/s, while that in the free-floating condition is 1.097 rad/s, with the shift being 0.64%. The second sloshing mode occurs at 1.283 rad/s, while that in the freefloating condition is 1.285 rad/s and lead to a shift being only 0.16%. Compared with the natural frequencies in Model 1, piston mode fre quency turns to be larger on account of the decrease of the draft, which is consistent with the conclusions in Zhang et al. (2019). Further, due to shallower draft, the shift of piston mode frequency in Model 2 tends to be larger than that in Model 1, while the shifts in sloshing mode fre quencies are both very small in Models 1 and 2. The draft of Model 3 is increased to 20 m, and Fig. 9 shows the freesurface elevation. Three resonant frequencies are found for both fixed and free-floating conditions, as expected. The piston mode frequency in the fixed and free-floating conditions are 0.598 rad/s and 0.613 rad/s, respectively. The shift is 2.51%, which is smaller than those in Model 1 and Model 2. The first sloshing mode frequency in the fixed condition is 0.848 rad/s, while that in the free-floating condition is 0.858 rad/s, with the shift being 1.18%. The second sloshing mode frequencies are the same in both fixed and free-floating conditions (1.175 rad/s). The shifts in piston mode of Models 1, 2 and 3 are listed in Table 5. As the draft decreases, the shift in piston mode frequencies increases. Compared with Model 1, the moonpool length in Model 4 is reduced to 20 m. Fig. 10 illustrates the free-surface elevation, and only two resonant frequencies are found for the short moonpool. The piston mode frequency in the fixed condition is at 0.765 rad/s, while that in the freefloating condition is at 0.778 rad/s. The shift is 1.70%, which is smaller than that in Model 1. The first sloshing mode frequency in the fixed and free-floating conditions are 1.255 rad/s and 1.260 rad/s, respectively, with the shift being 0.40%. The shifts of the sloshing resonant fre quencies are both very small in Models 1 and 4. In Model 5, the moonpool length is increased to 60 m, and Fig. 11 shows the free-surface elevation. Four resonant frequencies are found for fixed and free-floating conditions, respectively. The piston mode frequency in the fixed condition is 0.728 rad/s, while that in the freefloating condition is at 0.760 rad/s. The shift is 4.40%, larger than those in Model 1 and Model 4. The first sloshing mode frequency in the
Table 5 Shifts of piston mode frequencies in the fixed and free-floating conditions for � FF � �ω ωFC � Models 1, 2 and 3. f0 is the shift in piston mode: f0 ¼ 0 FC 0 � 100%,
ω0
FC where ωFF 0 and ω0 represent the piston mode frequencies in the free-floating condition and in the fixed condition, respectively.
Model No.
d (m)
f0
3 1 2
20 11 4
2.51% 3.40% 6.26%
conditions. Fig. 7 presents the heave, pitch and roll motion RAOs denoted by X3 , X5 and X4 , respectively in different wave directions for Model 1. As shown in the figure, there is a local maximum at 0.76 rad/s in heave RAO, which corresponds to the piston mode frequency in the freefloating condition. This suggests that the piston mode resonance can affect heave RAO, which is similar to the conclusions for a twodimensional moonpool in Fredriksen et al. (2015). In addition, it is observed that there is a local maximum at 0.923 rad/s in pitch, which corresponds to the first sloshing mode inside moonpool. However, the local maximum corresponding to the second sloshing mode does not appear in heave RAO. This may be attributed to the fact that the free-surface elevation around the second sloshing mode is too small to affect the heave motion. Furthermore, there are two local maxima around 0.66 rad/s in Fig. 7 ðaÞ and 0.645 rad/s in ðcÞ, which correspond to the heave natural frequencies in θ ¼ 0∘ and 90∘ , respectively. Both the heave natural frequencies are smaller than the piston mode frequency. The difference of natural periods in the two incident directions may be attributed to the fact that the water motion inside moonpool induces negative added mass for the vessel, but its magnitude relies on the amplitude of water motion, which differs in the two wave headings. Fig. 7 ðfÞ shows the roll motion RAO in θ ¼ 90∘ with additional linear damping being 5% of the critical damping. Compared with Model 1, the draft in Model 2 is reduced from 11 m to 4 m. Fig. 8 gives the free-surface elevation at CP and NP in different wave directions for Model 2. Again, three resonant frequencies are found for both fixed and free-floating conditions. The piston mode
Fig. 10. Model 4. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ . 8
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 11. Model 5. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
Table 6 Shifts of piston mode frequencies in the fixed and free-floating conditions for Models 1, 4 and 5. Model No.
l (m)
f0
4 1 5
20 45.6 60
1.70% 3.40% 4.40%
Fig. 13. Added mass Mal due to lower layer fluid underneath moonpool with respect to velocity ratio μ. Mal0 is the added mass when the barge is fixed (i.e. μ ¼ 0). Geometry parameters are l1 ¼ l2 ¼ 67:6 m, l ¼ 45:6 m, b ¼ 11:2 m, b1 ¼ b2 ¼ 10:5 m, d ¼ 11 m, H ¼ 1000 m.
condition is smaller than that in the free-floating condition. In order to better understand this phenomenon, the modified frozen mode approximation, proposed in Section 3, is adopted for further analyses. First, Fig. 12 shows the phase difference between the barge motion and motion of the frozen restriction inside the moonpool. As shown in the figure, the motion of the frozen restriction and barge are almost 180� out of phase (phase shift close to π) around the piston mode frequency. This means that the velocity ratio μ should be negative. Fig. 13 illustrates the added mass due to the lower fluid with respect to μ in three-dimensional cases. The added mass is non-dimensionalized by Mal0 when the body is fixed, i.e. μ ¼ 0. As can be seen, as μ is negative, the added mass due to the lower fluid is smaller than that in the fixed condition (μ ¼ 0), which can induce larger piston mode frequency based on the frozen mode approximation model given in Eqn. (14). Hence, the physics on the shift
Fig. 12. Phase difference between frozen restriction motion and barge motion for moonpool without recess in vertical direction. The piston mode frequency in free-floating condition is 0.76 rad/s.
fixed and free-floating conditions occur at 0.850 rad/s and 0.858 rad/s, respectively, with the shift being 0.94%. The frequencies of the second sloshing mode in both fixed and free-floating condition are 1.048 rad/s. The shifts in piston mode of Models 1, 4 and 5 are listed in Table 6, and are shown to increase with the growth of moonpool length. For moonpool without recess, the piston mode frequency in the fixed 9
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Figure 14. ðaÞ: Variation of free-surface elevation at FP with respect to incident wave frequency ω; ðbÞ: Variation of free-surface elevation at AP with respect to ω. The present results are compared with the semi-analytical solution in Molin (2017) and the experimental data in Guo et al. (2016).
4.2. Moonpool with a recess For moonpool with a recess, the present results are compared with those given in Molin (2017) and Guo et al. (2016), which studied a similar problem by eigenfunction matching method and model test, respectively. Model 6 has the same moonpool dimension with that by Molin (2017). Fig. 14 illustrates the variation of the free-surface eleva tions at FP, CP and AP. It should be noted that the present coordinate system is different from that in Guo et al. (2016), where wave heading 180∘ is equivalent to 0∘ in the present study. As can be seen, the resonant frequencies in the fixed condition are almost the same as that in the free-floating condition for the piston mode, the second sloshing mode, and the third sloshing mode, respectively. However, there is some dif ference between the resonant frequencies of the first sloshing mode. It should be noted that case studied by Molin can be considered as in the fixed condition, while the measurements in Guo et al. (2016) can be considered as in the free-floating condition. As shown, the present re sults are in excellent agreement with that in Molin (2017) and Guo et al. (2016), except that the predicted first sloshing mode frequency in the fixed condition is still slightly larger than that by Molin. In order to find out the reason, we run an additional case with the same dimension as Model 6 except for a larger beam, which is the same as the barge length (L ¼ B). Fig. 15 illustrates the comparison of the free-surface elevation RAOs between the two models at the same wave probe CP. As shown, the obtained first sloshing mode resonant frequency using square barge model is around 0.78 rad/s which is almost same as the solution by Molin (2017). It suggests that the beam of the barge can substantially affect the first longitudinal sloshing mode frequency. Moreover, it can be clarified and confirmed that difference between Molin’s solution and results from WAMIT in the fixed condition can be attributed to assumption on infinite beam and length in that paper. More detailed results are illustrated in Fig. 16. The resonant fre quencies of piston mode in both fixed and free-floating conditions are 0.415 rad/s. The first sloshing mode frequency in the fixed and free-
Fig. 15. Comparison of free-surface elevations at CP inside moonpool of Model 6 and a square barge model (L ¼ B) with wave heading θ ¼ 0∘ in the fixed condition.
of the piston mode frequency is clarified. In fact, the added mass with respect to μ in the two-dimensional case shows a similar trend, which can be adopted to explain the shift shown in Fig. 5 in Fredriksen et al. (2015). Further, it can also help us to understand why the magnitude of shift in shallow draft is relatively larger since the ‘real mass’ in frozen restriction model is less while the reduction in the added mass from the lower fluid is almost the same. It should be noted that, although μ may varies at different draft, the effect of the mass inside the moonpool is dominant.
10
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 16. Model 6. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
Fig. 17. Model 6. Heave and pitch motion RAOs in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: heave, 0∘ ; ðbÞ: heave, 45∘ ; ðcÞ: heave, 90∘ ; ðdÞ: pitch, 0∘ ; ðeÞ: pitch, 45∘ ; ðf Þ: roll, 90∘ . 11
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 18. Model 7. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
floating conditions are 0.803 rad/s and 0.820 rad/s, respectively, with the shift being about 2.18%. The second sloshing mode frequency in the fixed condition is 1.075 rad/s, while that in the free-floating condition is 1.077 rad/s. The shift is 0.19%, much less than that of the first sloshing mode resonance. Fig. 17 illustrates the heave, pitch and roll motion RAOs in different wave directions for Model 6. There is a local maximum at 0.415 rad/s in heave motion, which is the piston frequency in the free-floating condi tion. Different from the cases without recess, the local maximum at 0.820 rad/s corresponding the first sloshing mode resonance in the freefloating condition also occurs in the heave RAO. This implies that the first sloshing mode in the free-floating condition has substantial effects on heave motion due to the recess. In fact, when the heave motion of barge is constrained in free-floating condition using WAMIT, the first sloshing mode resonant frequency is 0.805 rad/s, which is very close to that in fixed condition. This indicates that the shift of the first sloshing mode resonant frequency for the present case with recess is mainly caused by the heave motion of the barge. This may be because the hy drodynamic pressure along the longitudinal direction of the moonpool is not anti-symmetric any more in the presence of a recess. Due to similar mechanism, the pitch motion is also affected by the piston mode reso nance. In addition, there are another two local maxima around 0.67 rad/ s in Fig. 17 ðaÞ and 0.65 rad/s in ðcÞ, which correspond to the heave natural frequencies in θ ¼ 0∘ and 90∘ , respectively. Different from the cases without recess, heave natural frequencies in both wave directions are higher than the piston mode frequency. Compared with Model 6, the length of recess in Model 7 is reduced to 8 m. Fig. 18 gives the free-surface elevation for Model 9. As can be found
in the plots, the piston mode frequency in the fixed and free-floating conditions are 0.593 rad/s and 0.598 rad/s, respectively, with the shift being 0.843%. The first sloshing mode frequency in the fixed condition is 0.825 rad/s, while that in the free-floating condition is 0.843 rad/s, with the shift being 2.12%. The second sloshing mode frequency in the fixed condition is 1.11 rad/s, almost the same as that in the free-floating condition. In Model 8, the length of recess is reduced to 4 m. Fig. 19 illustrates the free-surface elevation with respect to wave frequency for Model 8. As can be seen, the piston mode frequency in the fixed condition is at 0.690 rad/s, while that of free-floating condition is 0.705 rad/s. The shift of the frequency is 2.17%. The first sloshing mode frequency in the fixed condition is 0.863 rad/s, while that in the free-floating condition is 0.875 rad/s. The shift of the frequency is 1.40%. The resonant fre quencies of the second sloshing mode in both fixed and free-floating conditions are 1.14 rad/s. The shifts in the first sloshing mode for these three models are summarized in Table 7. By comparing the shifts in these three models, it can be concluded that the longer the recess length, the larger the shift in the first sloshing mode frequencies in the fixed and free-floating condition. Compared with Model 6, the height of recess h is reduced to 5 m in Model 9. Fig. 20 illustrates the free-surface elevation with respect to wave frequency for Model 9. The piston mode frequency in the fixed condition is 0.470 rad/s, while that in the free-floating condition is 0.473 rad/s. The shift is 0.64%. The first sloshing mode frequency in the fixed condition is 0.825 rad/s, while that in the free-floating condition is 0.845 rad/s. The shift is 2.42%. The natural frequencies of the second sloshing mode in both fixed and free-floating conditions are 1.123 rad/s. 12
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 19. Model 8. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
5. Conclusions
Table 7 Shifts of the first sloshing mode frequencies in the fixed condition and freefloating condition for Models 6, 7 and 8. f1 is the shift in the first sloshing � FF � �ω ωFC � FC mode: f1 ¼ 1 FC 1 � 100%, where ωFF 1 and ω1 are the first sloshing mode
Barges with different moonpool dimensions are modelled for WAMIT computations. Cases with and without recess are both adopted for an alyses. Free-surface elevations, heave, pitch motion and roll RAOs are computed in both fixed and free-floating conditions. The main objective of the present study is to explore how the moonpool dimensions affect the shift of the resonant frequencies in the fixed and free-floating vessels, and the relation between the moonpool resonances and vessel motion responses in the free-floating condition. The predicted natural frequencies and shifts in the fixed and freefloating conditions of all models are summarized in Table A.1 in Appendix A. By systematic computations and analyses, it is observed that the natural frequencies of the moonpool in a free-floating condition are higher than that in a fixed condition for both piston and longitudinal sloshing resonance modes. In addition, it is found that for vessel without recess in moonpool, the shift of resonant frequency in the piston mode is larger than that in sloshing modes, and the shift in the first sloshing mode is larger than that in the higher-order sloshing modes. For vessels with a recess, the shift of frequency in piston mode is less than that in the first sloshing mode. It is found that the shift can be affected by the draft of vessel. For moonpool without recess, as the draft is reduced, the shift in piston mode increases. As the length of moonpool increases, the shift in piston mode resonance increases, but the shift in sloshing modes is almost less
ω1
frequency in the free-floating condition and in the fixed condition, respectively. Model No.
r (m)
f1
6 7 8
16 8 4
2.18% 2.12% 1.40%
In Model 10, the height of recess is reduced to 3 m. Fig. 21 illustrates the free-surface elevation with respect to wave frequency for Model 10. The piston mode frequency in the fixed condition is 0.508 rad/s, while that in the free-floating condition is 0.510 rad/s. The shift is 0.39%. The first sloshing mode frequency in the fixed condition is at 0.838 rad/s, while that in the free-floating condition is 0.858 rad/s. The shift is 2.39%. The natural frequencies of the second sloshing mode in both fixed and free-floating conditions are 1.145 rad/s. The shifts in the first sloshing mode for these three models are summarized in Table 8. As expected, the shift of the first sloshing mode frequency is not sensitive to recess height.
13
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 20. Model 9. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
14
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 21. Model 10. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
the first sloshing mode have effects on both heave and pitch motions. The water motion in a piston mode may yield antisymmetric hydrody namic pressure along the longitudinal direction of the moonpool, which leads to the local maximum in heave motion RAO. Similarly, the water motion in the first sloshing mode causes the local maximum in pitch motion RAO. It should be noted that damping due to flow separation is important near the resonant frequency, which needs to be investigated in future study. The present study is based on the potential flow theory and focuses on the natural frequency instead of response amplitude. The nonlinear free surface effects and vortex shedding were not taken into consider ation, but should be studied in the future. We believe the present study can offer offshore engineers some guidance or hints when they make decision on the moonpool configuration considering the difference in the natural frequencies between the fixed and free-floating conditions.
Table 8 Shifts of the first sloshing mode frequencies in the fixed condition and freefloating condition for Models 6, 9 and 10. Model No.
h (m)
f1
6 9 10
7.2 5 3
2.18% 2.42% 2.39%
than 1%. Moreover, a modified frozen mode approximation is proposed to examine the effects of the body motion. It is found body motion causes the reduction in the added mass from the fluid underneath the moonpool and yield an increase in the piston mode frequency. Moreover, it is found the infinite-beam assumption in Molin’s paper may lead to the under estimation of the first longitudinal sloshing mode resonant frequency for the model adopted in the experiments. It is found that moonpool resonance can induce local hump in heave and pitch motion RAOs. For vessels without recess in moonpool, the resonances in the piston mode and the second sloshing mode have ef fects on the heave motion, while the resonance in the first sloshing mode can only affect the pitch motion. For vessels with recess in moonpool, resonances in piston mode and
Acknowledgements Special thanks to Prof. Bernard Molin and Prof. Rodney Eatock Taylor for inspiring discussions and comments with one of the authors Xinshu Zhang during the IWWWFB 2019. Moreover we feel very grateful to Prof. Harry Bingham for his help on using the WAMIT.
15
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Appendix A. Summary of the results for all cases Table A.1 Summary of all the cases on the shifts of the piston and the first sloshing mode frequencies in the fixed and free-floating conditions.
Moonpool without recess
Moonpool with a recess
Model No.
L (m)
B (m)
l (m)
b (m)
d (m)
D (m)
r (m)
h(m)
ωFC 0
ωFF 0
f0
ωFC 1
ωFF 1
f1
1 2 3 4 5 6 7 8 9 10
180.8 180.8 180.8 180.8 180.8 180.8 180.8 180.8 180.8 180.8
32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2
45.6 45.6 45.6 20.0 60.0 45.6 45.6 45.6 45.6 45.6
11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2
11 4 20 11 11 11 11 11 11 11
22.8 22.8 22.8 22.8 22.8 22.4 22.4 22.4 22.4 22.4
– – – – – 16.0 8.0 4.0 16.0 16.0
– – – – – 7.2 7.2 7.2 5.0 3.0
0.735 0.943 0.598 0.765 0.728 0.415 0.593 0.690 0.470 0.508
0.760 1.002 0.613 0.778 0.760 0.415 0.598 0.705 0.473 0.510
3.40% 6.26% 2.51% 1.70% 4.40% 0.00% 0.84% 2.17% 0.53% 0.49%
0.913 1.090 0.848 1.255 0.850 0.8025 0.825 0.863 0.825 0.838
0.923 1.097 0.858 1.260 0.858 0.820 0.8425 0.875 0.845 0.858
1.10% 0.64% 1.18% 0.40% 0.94% 2.18% 2.12% 1.40% 2.42% 2.39%
References
Molin, B., Zhang, X., Huang, H., Remy, F., 2018. On natural modes in moonpools and gaps in finite depth. J. Fluid Mech. 840, 530–554. Newman, J.N., 2018. Resonant response of a moonpool with a recess. Appl. Ocean Res. 76, 98–109. Xu, X., Song, X., Zhang, X., Yuan, Z., 2019. On wave diffraction of two-dimensional moonpools in a two-layer fluid with finite depth. Ocean Eng. 173, 571–586. Yoo, S.O., Kim, H.J., Lee, D.Y., Kim, B., Yeung, S.H., 2017. Experimental and numerical study on the flow reduction in the moonpool of floating offshore structure. In: Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. OMAE, Trondheim, Norway. Zhang, X., Bandyk, P., 2013. On two-dimensional moonpool resonance for twin bodies in a two-layer fluid. Appl. Ocean Res. 40, 1–13. Zhang, X., Bandyk, P., 2014. Two-dimensional moonpool resonances for interface and surface-piercing twin bodies in a two-layer fluid. Appl. Ocean Res. 47, 204–218. Zhang, X., Huang, H., Song, X., 2019. On natural frequencies and modal shapes in twodimensional asymmetric and symmetric moonpools in finite water depth. Appl. Ocean Res. 82, 117–129. Zhao, W., Taylor, P.H., Wolgamot, H.A., Taylor, R.E., 2018. Linear viscous damping in random wave excited gap resonance at laboratory scale–Newwave analysis and reciprocity. J. Fluid Mech. 80, 59–76. Zhao, W., Wolgamot, H.A., Taylor, P.H., Taylor, R. Eatock, 2017. Gap resonance and higher harmonics driven by focused transient wave groups. J. Fluid Mech. 812, 905–939.
Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N., 2007. Two-dimensional resonant piston-like sloshing in a moonpool. J. Fluid Mech. 575, 359–397. Faltinsen, O.M., Timokha, A.N., 2015. On damping of two-dimensional piston-mode sloshing in a rectangular moonpool under forced heave motions. J. Fluid Mech. 772, R1. Fredriksen, A.G., Kristiansen, T., Faltinsen, O.M., 2015. Wave-induced response of a floating two-dimensional body with a moonpool. Phil. Trans. R. Soc. A 373, 20140109. Guo, X., Lu, H., Yang, J., Peng, T., 2016. Study on hydrodynamics performances of a deep-water drillship and water motions inside its rectangular moonpool. In: Proceedings of the 26th International Ocean and Polar Engineering Conference. ISOPE, Rhodes, Greece. Guo, X., Lu, H., Yang, J., Peng, T., 2017. Resonant water motions within a recessing type moonpool in a drilling vessel. Ocean Eng. 129, 228–239. McIver, P., 2005. Complex resonances in the water-wave problem for a floating structure. J. Fluid Mech. 536, 423–443. Molin, B., 2001. On the piston and sloshing modes in moonpools. J. Fluid Mech. 430 (5), 27–50. Molin, B., 2017. On natural modes in moonpools with recesses. Appl. Ocean Res. 67, 1–8. Molin, B., Remy, F., Camhi, A., Ledoux, A., 2009. Experimental and numerical study of the gap resonances in-between two rectangular barges. In: 13th International Congress of the International Maritime Association of the Mediterranean. IMAM, Istanbul, Turkey.
16
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
On natural frequencies of three-dimensional moonpool of vessels in the fixed and free-floating conditions Xin Xu a, b, Xinshu Zhang a, b, *, Bei Chu a, b, Haiyang Huang a, b a b
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai, 200240, China
A B S T R A C T
The hydrodynamic interactions between the vessel motion and resonant wave response inside the three-dimensional moonpool are studied in the present paper. In particular, the shift of natural frequencies of a three-dimensional moonpool in the fixed and free-floating conditions is computed and examined systematically. Barges with different moonpool dimensions are analyzed to examine how the moonpool configurations affect the shift of resonant frequencies. Detailed analyses are performed for cases with and without recess. Numerical computations are performed for regular waves with wave headings θ ¼ 0∘ , 45∘ and 90∘ . Free-surface ele vations inside the moonpool with respect to the incident wave frequencies are computed for both fixed and free-floating conditions. It is observed that natural frequencies of the moonpool in the free-floating condition are higher than those in the fixed condition. For cases without recess, the shift of resonant frequency is more prominent in piston mode resonance, in particular for shallow or long moonpools. In contrast, for cases with a recess, the shift of resonant frequency in the first sloshing mode is relatively larger than that in piston mode. In addition, the effects of motion RAOs on the free-surface elevation inside the moonpool in the freefloating condition are also examined. It is found that the moonpool resonance can induce a local hump in heave and pitch responses. Moreover, through deriving a modified frozen mode approximation model, we reveal that, comparing to the fixed condition, the vessel motion in the free-floating condition yields a reduction in the added mass due to the fluid underneath the moonpool so that the piston mode resonant frequency increases.
1. Introduction Moonpools are vertical openings in a floating body, frequently adopted in drilling ships for pipeline laying, and in diving support ves sels for launching of diving bells and other subsea equipments (see Faltinsen and Timokha (2015); Yoo et al. (2017)). Resonant water mo tion occurs at the natural modes of the moonpool, including piston mode, where the water inside the moonpool heaves up and down, and sloshing modes, where the water inside the moonpool moves back and forth between the vertical walls. Since the resonant fluid motion may lead to significant impact on the hull structure or equipments on the deck, it is necessary to accurately predict the resonant frequencies to assist the concept design of the floater and marine operation. Most of the previous studies on moonpools are based on conditions where the structure is either fixed in incident waves, or oscillated harmonically in calm water. Under the assumption of infinite length and beam of a barge, Molin (2001) proposed a semi-analytical model to predict the resonant frequencies and modal shapes of a three-dimensional moonpool in infinite water depth by solving eigen value problems. Based on domain decomposition and Galerkin method, Faltinsen et al. (2007) developed a semi-analytical method to compute
the fluid motions in a two-dimensional moonpool and performed ex periments for comparison. Molin et al. (2009) carried out model tests to study the gap resonances between two rectangular barges which were rigidly connected to a carriage. Free-surface elevation derived from tests in irregular waves are compared with numerical solutions using a code based on linear potential flow theory. Zhao et al. (2017) investigated the first and higher harmonic components of the resonant fluid response in the gap between two identical fixed rectangular boxes by performing experiments in a wave basin. Furthermore, they made a prediction of amplitude of gap resonances at different gap widths by scaling the linear damping (Zhao et al., 2018). Zhang and Bandyk (2013, 2014) solved the radiation problem in a two-layer fluid for rectangular bodies restricted to the upper-layer fluid and the surface-piercing rectangular bodies, respectively. Xu et al. (2019) studied the wave diffraction problem of two-dimensional moonpools in a two-layer fluid with finite fluid depth. Recently, Molin et al. (2018) and Zhang et al. (2019) proposed new models to solve the three-dimensional and two-dimensional moonpool problems in finite water depth. However, as discussed in McIver (2005), the natural frequencies of a moonpool in the free-floating condition can be different from the solu tions computed as the structure is fixed (called ‘fixed condition’ in this
* Corresponding author. Innovative Marine Hydrodynamics Laboratory (iMHL), Shanghai Jiao Tong University, China. E-mail address: [email protected] (X. Zhang). https://doi.org/10.1016/j.oceaneng.2019.106656 Received 20 June 2019; Received in revised form 20 September 2019; Accepted 27 October 2019 Available online 26 November 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 1. Sketch of the vessel model without recess in moonpool. ðaÞ: top view; ðbÞ: side view at centerline (y ¼ 0). Table 1 Load matrix and model sizes of vessels without recess in moonpool. Model No.
L (m)
B (m)
l (m)
b (m)
d (m)
D (m)
CP (m)
NP (m)
1 2 3 4 5
180.8 180.8 180.8 180.8 180.8
32.2 32.2 32.2 32.2 32.2
45.6 45.6 45.6 20.0 60.0
11.2 11.2 11.2 11.2 11.2
11 4 20 11 11
22.8 22.8 22.8 22.8 22.8
(0, (0, (0, (0, (0,
(10, 0, 0) (10, 0, 0) (10, 0, 0) (8, 0, 0) (13, 0, 0)
paper) or in harmonic oscillation conditions with a prescribed motion. Focusing on piston mode motion in the moonpool and rigid-body mo tions, Fredriksen et al. (2015) studied regular wave-induced behavior of a two-dimensional floating body with moonpool through experiments and hybrid numerical methods. It was found that the piston mode fre quency in diffraction problem is almost same as that in radiation, but it is larger for the vessel in the free-floating condition when exposing to incident waves. Guo et al. (2016, 2017) performed experiments for a recessed drillship moonpool and measured wave responses inside the moonpool. Molin (2017) extended the theoretical model in Molin (2001) to compute the resonant frequencies for moonpools with a recess, and compared the results with the measurements by Guo. The prediction of the natural frequencies agreed well with the experiment, except for the first sloshing mode frequency. Newman (2018) investigated the reso nant natural modes of the same type of vessel through WAMIT, and used a domain decomposition method with Legendre polynomials, instead of Fourier functions, for computing the natural frequencies and added mass coefficients. It was found that the shift of natural frequencies in the fixed and free-floating condition are small or non-existent for the particular hull and moonpool configuration. However, till now, the mechanism of the shift of the natural frequencies and effect of moonpool configura tions on the shift are still not well understood, which are the motivations of the present study. The present paper aims to explore how the moonpool dimensions affect the shift of moonpool resonant frequencies (including piston and sloshing modes) in the fixed and free-floating conditions and understand the mechanism on the shift by building up a modified frozen mode approximation model with the body motion effects being accounted.
The present paper is organized as follows. Barges with different di mensions of moonpool are modelled. In addition, a modified frozen mode approximation model is developed to explain the physics on the shift of the frequencies by accounting for the body motion. The radiation-diffraction code WAMIT is employed to compute the freesurface elevations inside moonpool in both fixed and free-floating con ditions and the six-degree of freedom motion RAOs. The effects of vessel motions on the moonpool resonances are discussed in detail. 2. Computational cases Barge-type vessels, with or without recess in a moonpool, are modelled with different dimensions. The moonpool without recess is shown in Fig. 1, where the center of the moonpool coincides with that of the barge. The origin of the body coordinates is placed at the center of the waterplane inside moonpool. The global coordinates coincide with the body-fixed coordinates when the vessel is in calm equilibrium. The draft of vessel is denoted as d, and the length and width are denoted as L and B, respectively. The moonpool length and width are denoted by l and b, respectively. The distance from the aft side of moonpool to the midship section is denoted as D. Two numerical wave probes are placed inside the moonpool to measure the free-surface elevation. As shown in Fig. 1 ðaÞ, two wave probes (denoted by CP and NP) are placed inside the moonpool. Both CP and NP are used to identify the piston and symmetric sloshing mode resonances, but the anti-symmetric sloshing modes can only be captured by the elevation at NP inside the moonpool without recess. As illustrated in Table 1, five different sets of moonpool di mensions are chosen for the cases without recess. Model 1 is the base model. Model 2 and Model 3 are used to study the effect of draft, and Model 4 and Model 5 are adopted to study the effect of moonpool length. To simplify the study, the VCG and inertia of all the vessels without recess in moonpool are kept the same, as listed in Table 2. Fig. 2 illustrates the vessel with recessed moonpool. The center of moonpool is located at 0.4 m behind the midship section of the barge. The length and height of recess are denoted as r and h, respectively. As shown in Table 3, five different combinations of dimension are pre sented. Model 6 shares the same size as the physical model in Guo et al.
Table 2 VCG and inertia of vessels without recess in moonpool. Parameters
Symbol
Unit
Value
The vertical center of gravity Roll radius of gyration
VCG Rxx
m m
1.50 11.58
m
54.86
Rzz
m
55.34
Pitch radius of gyration Yaw radius of gyration
Ryy
0, 0) 0, 0) 0, 0) 0, 0) 0, 0)
2
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 2. Sketch of the vessel with recess in moonpool. ðaÞ: top view; ðbÞ: side view at centerline.
(2016, 2017) for comparison. Compared with Model 6, the length of recess is varied in Models 7 and 8, and the recess height is varied in Models 9 and 10. As shown in Fig. 2 ðaÞ, three numerical wave probes are placed inside the moonpool to measure the free-surface elevations and capture the piston and sloshing mode resonances. The wave probe (FP) is located at the forward end of the moonpool. The wave probe (CP) is located at the center of moonpool, and the wave probe (AP) is placed at the aft side of moonpool. The coordinates of FP, CP and AP are ( 21.4 m, 0 m, 0 m), (0 m, 0 m, 0 m) and (22.2 m, 0 m, 0 m), respec tively. As listed in Table 4, the VCG and inertia of all the vessels with a recess in moonpool are kept the same. It should be noted that the water depth is assumed to be infinite in all the present cases. The fixed condition can be considered as a purely wave diffraction problem, while the free-floating condition can be considered as a com bination of the radiation and diffraction problems. When it is in freefloating condition, the vessel is undergoing six-degree of freedom mo tions in regular incident waves. The incident wave amplitude is denoted as A. The incident wave heading θ is defined as the angle between the positive x-axis of the global coordinate system and the direction in which the incident wave propagates. θ ¼ 0∘ , 45∘ and 90∘ are chosen in the present simulations. The wave frequency ω is varied from 0.1 rad/s to 1.4 rad/s. Correspondingly, the free-surface elevation η at specific locations and motion RAOs are obtained. As shown in Fig. 3, about 500 higher-order panels are used to dis cretize the model without recess, and 1000 panels for the recessed model. The panel size is approximately 3 m in length. For simplicity, the mass density of these models are set as homogeneous. The higher-order panel option based on B-splines and direct solver option are adopted. In addition, the irregular frequencies are avoided by adding meshes on the interior free-surface, which is also generated by WAMIT. Numerical convergence is assessed by repeating the computations for increasing number of panels, as shown in Fig. 4. As can be seen, the dotted line represents the free-surface elevation inside the model discretized by 3-meter-long panels, and the accuracy is high enough to capture the sloshing mode frequency even in high-frequency region.
Table 3 Load matrix and model sizes of vessels with recess in moonpool. Model No.
L (m)
B (m)
l (m)
b (m)
d (m)
D (m)
r (m)
h (m)
6 7 8 9 10
180.8 180.8 180.8 180.8 180.8
32.2 32.2 32.2 32.2 32.2
45.6 45.6 45.6 45.6 45.6
11.2 11.2 11.2 11.2 11.2
11 11 11 11 11
22.4 22.4 22.4 22.4 22.4
16.0 8.0 4.0 16.0 16.0
7.2 7.2 7.2 5.0 3.0
Table 4 VCG and inertia of vessels with recess in moonpool. Parameters
Symbol
Unit
Value
The vertical center of gravity Roll radius of gyration
VCG Rxx
m m
1.50 11.52
m
54.33
Rzz
m
54.80
Pitch radius of gyration Yaw radius of gyration
Ryy
3. Frozen mode approximation model with accounting for the body motion In Molin (2017), the fluid inside the moonpool was assumed as a solid body or frozen restriction, and the influence of the fluid under neath the moonpool was taken as added mass on the dynamic system. Thus, the natural frequency of moonpool can be obtained by analyzing the free heave motion of the frozen restriction. Further, by constraining the heave motion of barge in free-floating condition using WAMIT, the predicted piston mode resonant frequency is 0.735 rad/s and is the same
Fig. 3. Panelization adopted in WAMIT computations. Mesh (blue color) on the interior free surface are adopted to remove the irregular frequencies. (For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.) 3
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Figure 4. ðaÞ: Convergence test for Model 1 with respect to number of panels; ðbÞ: Convergence test for Model 6 with respect to number of panels.
Fig. 5. Sketch of the modified frozen mode approximation model by accounting for body motion. ðaÞ: top view; ðbÞ: side view at centerline (y ¼ 0).
as that in fixed condition (see the detailed results in subsection 4.1), which indicates that the shift of the piston mode is mainly attributed to the heave motion in free-floating condition, and the other horizontal modes have little effects on the shift. Therefore, the natural frequency of the moonpool in the free-floating condition can be qualitatively esti mated by accounting for the vessel motion through modifying the body boundary condition. The sketch of the present model and coordinate system is illustrated in Fig. 5. For simplicity, we set l1 ¼ l2 and b1 ¼ b2 in this paper. As shown, the whole fluid domain is divided into two subdomains. We denote the fluid domain inside the moonpool ðz > dÞ as subdomain I, and take the external fluid domain ðz < dÞ as subdomain II.
written as ∞ X
Em
ϕe ¼ m¼1
coshαm ðz þ H þ dÞ sinαm ðx þ L = 2Þ coshðαm HÞ
(1)
where αm ¼ mπ=ðl1 þl þl2 Þ ¼ mπ=L and m ¼ 1; 2; …; ∞ are the integers. In the present model, the velocities at the opening of moonpool and the body motion velocity are different depending on the motion RAO and phase shift. The ratio of body motion velocity to the mean velocity at opening of the moonpool is denoted as μ. By matching the vertical velocity at the common boundary ( L=2 � x � L=2, z ¼ d) of sub domains I and II, we obtain � ∞ X 1; l=2 � x � l=2 Em αm tanhðαm HÞsinαm ðxþL=2Þ¼ μ; L=2 � x � l=2; and l=2 � x � L=2 m¼1
3.1. Two-dimensional case
(2)
To simplify the analyses, the problem is treated in two-dimensional. But the key physics on the shift of the natural frequency should be the similar. In the external fluid domain, the velocity potential can be
By multiplying each side with sinαm ðx þ L=2Þ, and integrating along x direction over [ L=2, L=2], we obtain 4
X. Xu et al.
1 Em αm tanhðαm HÞL ¼ 2
Ocean Engineering 195 (2020) 106656
Z
Z
l=2 L=2
Z
l=2
μsinαm ðx þ L = 2Þdx þ
L=2
sinαm ðx þ L = 2Þdx þ l=2
μsinαm ðx þ L = 2Þdx l=2
(3)
Fig. 6. Model 1. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
Fig. 7. Model 1. Heave and pitch motion RAOs in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: heave, 0∘ ; ðbÞ: heave, 45∘ ; ðcÞ: heave, 90∘ ; ðdÞ: pitch, 0∘ ; ðeÞ: pitch, 45∘ ; ðf Þ: roll, 90∘ .
5
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 8. Model 2. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
Fig. 9. Model 3. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
6
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Then the coefficients Em can be determined as Em ¼ 2
½cosðαm l1 Þ
cosαm ðl1 þ lÞ�ð1 μÞ þ μ½1 α2m tanhðαm HÞL
m
ð 1Þ �
Thus, the heave motion equation of the frozen restriction can be written as
(4)
ðρlbd þ Mal ÞZ€ þ ρglbZ ¼ 0
Thus, the added mass due to the lower fluid domain ðz � dÞ is ob tained as
Z
l
Mal ¼ ρ
ϕe dx ¼ 2ρ 0
cosαm ðl1 þ lÞ�2 ð1
∞ X ½cosðαm l1 Þ n¼1
Then the piston mode frequency can be estimated by
μÞ þ μ½1 ð 1Þm �½cosðαm l1 Þ α3m tanhðαm HÞL
(6)
Thus, the piston mode frequency can be estimated as follow sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρgl ω¼ ρld þ Mal
Emn m¼1 n¼1
coshγmn ðzþd þHÞ sinαm ðxþL=2Þsinβn ðyþB=2Þ coshðγ mn HÞ
4.1. Moonpool without recess
(8)
Fig. 6 illustrates the free-surface elevation with respect to the wave frequency at CP and NP in different wave directions for Model 1. As shown in Fig. 6, the amplitude of piston mode in free-floating condition is larger than those in fixed condition for wave headings θ ¼ 0∘ and 45∘ , but it is smaller in free-floating condition for θ ¼ 90∘ . We check the phase angles of the heave motion of the barge and observe that the phase of heave motion RAO around the piston mode in θ ¼ 90∘ is almost 180� out of phase with those in θ ¼ 0∘ and 45∘ . This may lead to larger crosscoupling damping on the wave motion inside the moonpool in freefloating condition for θ ¼ 90∘ , and further induces less resonant wave motion around the piston mode resonance. Since vortex shedding at the corner of moonpool is neglected, the obtained wave amplitude may be overestimated. However, the present paper focuses on the shift of resonant frequency, which may not be affected significantly by the vortex shedding induced damping (see Faltinsen et al. (2007)). Three local maxima in the free-surface elevation corresponding to three nat ural frequencies can be found in either fixed or free-floating conditions. The piston mode frequency in the fixed and free-floating conditions are 0.735 rad/s and 0.76 rad/s, respectively. The shift is around 3.40%. The first sloshing mode frequency in the fixed condition is 0.913 rad/s, while that in the free-floating is about 0.923 rad/s, with the shift being 1.10%. The second sloshing mode frequency is the same (1.18 rad/s) in both
where αm ¼ mπ=ðl1 þ l þ l2 Þ, βn ¼ nπ=ðb1 þ b þ b2 Þ, γ 2mn ¼ α2m þ β2n and m; n ¼ 1; 2; …; ∞ are the integers. By matching the vertical velocity on the common boundary ( L= 2 � x � L=2, B=2 � y � B=2, z ¼ d) of subdomains I and II, we obtain � ∞ X ∞ X 1; on S1 Emn γmn tanh ðγ mn HÞsinαm ðx þL =2Þsinβn ðyþB =2Þ ¼ (9) μ; on S2 m¼1 n¼1 where S1 represents the opening at z ¼ d, and S2 represents the hull bottom. By multiplying each side of Eqn. (9) with sinαm ðx þ L =2Þsinβn ðy þ B =2Þ, and integrating along x direction over [ L=2, L= 2] and along y direction over [ B=2, B=2], we obtain ZZ 1 Emn γ mn tanhðγmn HÞLB ¼ μ sinαm ðx þ L = 2Þsinβn ðy þ B = 2Þdxdy 4 S2 ZZ þ sinαm ðx þ L = 2Þsinβn ðy þ B = 2Þdxdy (10) S1
Then the coefficients Emn can be determined as Emn ¼
4 μ½ð γ mn tanhðγmn HÞLB
μ½ð
1Þn
1Þm
1 þ cosðαm l1 Þ
cosαm ðl1 þ lÞ� þ cosαm ðl1 þ lÞ
cosðαm l1 Þ
αm
1 þ cosðβn b1 Þ
cosβn ðb1 þ bÞ� þ cosβn ðb1 þ bÞ βn
(14)
The natural frequencies are obtained by searching for the local maxima in the RAO of free-surface elevation inside the moonpool. The first local maximum corresponds to the piston mode, and the others are associated with sloshing mode resonances. It should be noted that the present study focuses on longitudinal sloshing modes, which are more important in the practical application.
(7)
When the model is extended to three-dimensional case, the velocity potential in the external fluid domain can be written as ∞ X ∞ X
(5)
4. Numerical results
3.2. Three-dimensional case
ϕe ðx;y;zÞ¼
cosαm ðl1 þ lÞ�
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρglb ω¼ ρlbd þ Mal
The heave motion equation of the frozen restriction can be written as ðρld þ Mal ÞZ€ þ ρglZ ¼ 0
(13)
cosðβn b1 Þ
(11)
and the added mass due to the lower fluid domain ðz � dÞ is obtained as
Mal ¼ ρ
∞ X ∞ X
Emn tanhðγmn HÞ m¼1 n¼1
cosαm ð l=2 þ L=2Þ
αm
cosαm ðl=2 þ L=2Þ cosβn ð b=2 þ B=2Þ cosβn ðb=2 þ B=2Þ βn
7
(12)
X. Xu et al.
Ocean Engineering 195 (2020) 106656
frequency in the fixed and free-floating conditions are 0.943 rad/s and 1.002 rad/s, respectively. The shift is around 6.26%. The first sloshing mode frequency in the fixed condition is 1.090 rad/s, while that in the free-floating condition is 1.097 rad/s, with the shift being 0.64%. The second sloshing mode occurs at 1.283 rad/s, while that in the freefloating condition is 1.285 rad/s and lead to a shift being only 0.16%. Compared with the natural frequencies in Model 1, piston mode fre quency turns to be larger on account of the decrease of the draft, which is consistent with the conclusions in Zhang et al. (2019). Further, due to shallower draft, the shift of piston mode frequency in Model 2 tends to be larger than that in Model 1, while the shifts in sloshing mode fre quencies are both very small in Models 1 and 2. The draft of Model 3 is increased to 20 m, and Fig. 9 shows the freesurface elevation. Three resonant frequencies are found for both fixed and free-floating conditions, as expected. The piston mode frequency in the fixed and free-floating conditions are 0.598 rad/s and 0.613 rad/s, respectively. The shift is 2.51%, which is smaller than those in Model 1 and Model 2. The first sloshing mode frequency in the fixed condition is 0.848 rad/s, while that in the free-floating condition is 0.858 rad/s, with the shift being 1.18%. The second sloshing mode frequencies are the same in both fixed and free-floating conditions (1.175 rad/s). The shifts in piston mode of Models 1, 2 and 3 are listed in Table 5. As the draft decreases, the shift in piston mode frequencies increases. Compared with Model 1, the moonpool length in Model 4 is reduced to 20 m. Fig. 10 illustrates the free-surface elevation, and only two resonant frequencies are found for the short moonpool. The piston mode frequency in the fixed condition is at 0.765 rad/s, while that in the freefloating condition is at 0.778 rad/s. The shift is 1.70%, which is smaller than that in Model 1. The first sloshing mode frequency in the fixed and free-floating conditions are 1.255 rad/s and 1.260 rad/s, respectively, with the shift being 0.40%. The shifts of the sloshing resonant fre quencies are both very small in Models 1 and 4. In Model 5, the moonpool length is increased to 60 m, and Fig. 11 shows the free-surface elevation. Four resonant frequencies are found for fixed and free-floating conditions, respectively. The piston mode frequency in the fixed condition is 0.728 rad/s, while that in the freefloating condition is at 0.760 rad/s. The shift is 4.40%, larger than those in Model 1 and Model 4. The first sloshing mode frequency in the
Table 5 Shifts of piston mode frequencies in the fixed and free-floating conditions for � FF � �ω ωFC � Models 1, 2 and 3. f0 is the shift in piston mode: f0 ¼ 0 FC 0 � 100%,
ω0
FC where ωFF 0 and ω0 represent the piston mode frequencies in the free-floating condition and in the fixed condition, respectively.
Model No.
d (m)
f0
3 1 2
20 11 4
2.51% 3.40% 6.26%
conditions. Fig. 7 presents the heave, pitch and roll motion RAOs denoted by X3 , X5 and X4 , respectively in different wave directions for Model 1. As shown in the figure, there is a local maximum at 0.76 rad/s in heave RAO, which corresponds to the piston mode frequency in the freefloating condition. This suggests that the piston mode resonance can affect heave RAO, which is similar to the conclusions for a twodimensional moonpool in Fredriksen et al. (2015). In addition, it is observed that there is a local maximum at 0.923 rad/s in pitch, which corresponds to the first sloshing mode inside moonpool. However, the local maximum corresponding to the second sloshing mode does not appear in heave RAO. This may be attributed to the fact that the free-surface elevation around the second sloshing mode is too small to affect the heave motion. Furthermore, there are two local maxima around 0.66 rad/s in Fig. 7 ðaÞ and 0.645 rad/s in ðcÞ, which correspond to the heave natural frequencies in θ ¼ 0∘ and 90∘ , respectively. Both the heave natural frequencies are smaller than the piston mode frequency. The difference of natural periods in the two incident directions may be attributed to the fact that the water motion inside moonpool induces negative added mass for the vessel, but its magnitude relies on the amplitude of water motion, which differs in the two wave headings. Fig. 7 ðfÞ shows the roll motion RAO in θ ¼ 90∘ with additional linear damping being 5% of the critical damping. Compared with Model 1, the draft in Model 2 is reduced from 11 m to 4 m. Fig. 8 gives the free-surface elevation at CP and NP in different wave directions for Model 2. Again, three resonant frequencies are found for both fixed and free-floating conditions. The piston mode
Fig. 10. Model 4. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ . 8
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 11. Model 5. Free-surface elevation at CP and NP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: CP, 0∘ ; ðbÞ: CP, 45∘ ; ðcÞ: CP, 90∘ ; ðdÞ: NP, 0∘ ; ðeÞ: NP, 45∘ ; ðf Þ: NP, 90∘ .
Table 6 Shifts of piston mode frequencies in the fixed and free-floating conditions for Models 1, 4 and 5. Model No.
l (m)
f0
4 1 5
20 45.6 60
1.70% 3.40% 4.40%
Fig. 13. Added mass Mal due to lower layer fluid underneath moonpool with respect to velocity ratio μ. Mal0 is the added mass when the barge is fixed (i.e. μ ¼ 0). Geometry parameters are l1 ¼ l2 ¼ 67:6 m, l ¼ 45:6 m, b ¼ 11:2 m, b1 ¼ b2 ¼ 10:5 m, d ¼ 11 m, H ¼ 1000 m.
condition is smaller than that in the free-floating condition. In order to better understand this phenomenon, the modified frozen mode approximation, proposed in Section 3, is adopted for further analyses. First, Fig. 12 shows the phase difference between the barge motion and motion of the frozen restriction inside the moonpool. As shown in the figure, the motion of the frozen restriction and barge are almost 180� out of phase (phase shift close to π) around the piston mode frequency. This means that the velocity ratio μ should be negative. Fig. 13 illustrates the added mass due to the lower fluid with respect to μ in three-dimensional cases. The added mass is non-dimensionalized by Mal0 when the body is fixed, i.e. μ ¼ 0. As can be seen, as μ is negative, the added mass due to the lower fluid is smaller than that in the fixed condition (μ ¼ 0), which can induce larger piston mode frequency based on the frozen mode approximation model given in Eqn. (14). Hence, the physics on the shift
Fig. 12. Phase difference between frozen restriction motion and barge motion for moonpool without recess in vertical direction. The piston mode frequency in free-floating condition is 0.76 rad/s.
fixed and free-floating conditions occur at 0.850 rad/s and 0.858 rad/s, respectively, with the shift being 0.94%. The frequencies of the second sloshing mode in both fixed and free-floating condition are 1.048 rad/s. The shifts in piston mode of Models 1, 4 and 5 are listed in Table 6, and are shown to increase with the growth of moonpool length. For moonpool without recess, the piston mode frequency in the fixed 9
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Figure 14. ðaÞ: Variation of free-surface elevation at FP with respect to incident wave frequency ω; ðbÞ: Variation of free-surface elevation at AP with respect to ω. The present results are compared with the semi-analytical solution in Molin (2017) and the experimental data in Guo et al. (2016).
4.2. Moonpool with a recess For moonpool with a recess, the present results are compared with those given in Molin (2017) and Guo et al. (2016), which studied a similar problem by eigenfunction matching method and model test, respectively. Model 6 has the same moonpool dimension with that by Molin (2017). Fig. 14 illustrates the variation of the free-surface eleva tions at FP, CP and AP. It should be noted that the present coordinate system is different from that in Guo et al. (2016), where wave heading 180∘ is equivalent to 0∘ in the present study. As can be seen, the resonant frequencies in the fixed condition are almost the same as that in the free-floating condition for the piston mode, the second sloshing mode, and the third sloshing mode, respectively. However, there is some dif ference between the resonant frequencies of the first sloshing mode. It should be noted that case studied by Molin can be considered as in the fixed condition, while the measurements in Guo et al. (2016) can be considered as in the free-floating condition. As shown, the present re sults are in excellent agreement with that in Molin (2017) and Guo et al. (2016), except that the predicted first sloshing mode frequency in the fixed condition is still slightly larger than that by Molin. In order to find out the reason, we run an additional case with the same dimension as Model 6 except for a larger beam, which is the same as the barge length (L ¼ B). Fig. 15 illustrates the comparison of the free-surface elevation RAOs between the two models at the same wave probe CP. As shown, the obtained first sloshing mode resonant frequency using square barge model is around 0.78 rad/s which is almost same as the solution by Molin (2017). It suggests that the beam of the barge can substantially affect the first longitudinal sloshing mode frequency. Moreover, it can be clarified and confirmed that difference between Molin’s solution and results from WAMIT in the fixed condition can be attributed to assumption on infinite beam and length in that paper. More detailed results are illustrated in Fig. 16. The resonant fre quencies of piston mode in both fixed and free-floating conditions are 0.415 rad/s. The first sloshing mode frequency in the fixed and free-
Fig. 15. Comparison of free-surface elevations at CP inside moonpool of Model 6 and a square barge model (L ¼ B) with wave heading θ ¼ 0∘ in the fixed condition.
of the piston mode frequency is clarified. In fact, the added mass with respect to μ in the two-dimensional case shows a similar trend, which can be adopted to explain the shift shown in Fig. 5 in Fredriksen et al. (2015). Further, it can also help us to understand why the magnitude of shift in shallow draft is relatively larger since the ‘real mass’ in frozen restriction model is less while the reduction in the added mass from the lower fluid is almost the same. It should be noted that, although μ may varies at different draft, the effect of the mass inside the moonpool is dominant.
10
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 16. Model 6. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
Fig. 17. Model 6. Heave and pitch motion RAOs in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: heave, 0∘ ; ðbÞ: heave, 45∘ ; ðcÞ: heave, 90∘ ; ðdÞ: pitch, 0∘ ; ðeÞ: pitch, 45∘ ; ðf Þ: roll, 90∘ . 11
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 18. Model 7. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
floating conditions are 0.803 rad/s and 0.820 rad/s, respectively, with the shift being about 2.18%. The second sloshing mode frequency in the fixed condition is 1.075 rad/s, while that in the free-floating condition is 1.077 rad/s. The shift is 0.19%, much less than that of the first sloshing mode resonance. Fig. 17 illustrates the heave, pitch and roll motion RAOs in different wave directions for Model 6. There is a local maximum at 0.415 rad/s in heave motion, which is the piston frequency in the free-floating condi tion. Different from the cases without recess, the local maximum at 0.820 rad/s corresponding the first sloshing mode resonance in the freefloating condition also occurs in the heave RAO. This implies that the first sloshing mode in the free-floating condition has substantial effects on heave motion due to the recess. In fact, when the heave motion of barge is constrained in free-floating condition using WAMIT, the first sloshing mode resonant frequency is 0.805 rad/s, which is very close to that in fixed condition. This indicates that the shift of the first sloshing mode resonant frequency for the present case with recess is mainly caused by the heave motion of the barge. This may be because the hy drodynamic pressure along the longitudinal direction of the moonpool is not anti-symmetric any more in the presence of a recess. Due to similar mechanism, the pitch motion is also affected by the piston mode reso nance. In addition, there are another two local maxima around 0.67 rad/ s in Fig. 17 ðaÞ and 0.65 rad/s in ðcÞ, which correspond to the heave natural frequencies in θ ¼ 0∘ and 90∘ , respectively. Different from the cases without recess, heave natural frequencies in both wave directions are higher than the piston mode frequency. Compared with Model 6, the length of recess in Model 7 is reduced to 8 m. Fig. 18 gives the free-surface elevation for Model 9. As can be found
in the plots, the piston mode frequency in the fixed and free-floating conditions are 0.593 rad/s and 0.598 rad/s, respectively, with the shift being 0.843%. The first sloshing mode frequency in the fixed condition is 0.825 rad/s, while that in the free-floating condition is 0.843 rad/s, with the shift being 2.12%. The second sloshing mode frequency in the fixed condition is 1.11 rad/s, almost the same as that in the free-floating condition. In Model 8, the length of recess is reduced to 4 m. Fig. 19 illustrates the free-surface elevation with respect to wave frequency for Model 8. As can be seen, the piston mode frequency in the fixed condition is at 0.690 rad/s, while that of free-floating condition is 0.705 rad/s. The shift of the frequency is 2.17%. The first sloshing mode frequency in the fixed condition is 0.863 rad/s, while that in the free-floating condition is 0.875 rad/s. The shift of the frequency is 1.40%. The resonant fre quencies of the second sloshing mode in both fixed and free-floating conditions are 1.14 rad/s. The shifts in the first sloshing mode for these three models are summarized in Table 7. By comparing the shifts in these three models, it can be concluded that the longer the recess length, the larger the shift in the first sloshing mode frequencies in the fixed and free-floating condition. Compared with Model 6, the height of recess h is reduced to 5 m in Model 9. Fig. 20 illustrates the free-surface elevation with respect to wave frequency for Model 9. The piston mode frequency in the fixed condition is 0.470 rad/s, while that in the free-floating condition is 0.473 rad/s. The shift is 0.64%. The first sloshing mode frequency in the fixed condition is 0.825 rad/s, while that in the free-floating condition is 0.845 rad/s. The shift is 2.42%. The natural frequencies of the second sloshing mode in both fixed and free-floating conditions are 1.123 rad/s. 12
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 19. Model 8. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
5. Conclusions
Table 7 Shifts of the first sloshing mode frequencies in the fixed condition and freefloating condition for Models 6, 7 and 8. f1 is the shift in the first sloshing � FF � �ω ωFC � FC mode: f1 ¼ 1 FC 1 � 100%, where ωFF 1 and ω1 are the first sloshing mode
Barges with different moonpool dimensions are modelled for WAMIT computations. Cases with and without recess are both adopted for an alyses. Free-surface elevations, heave, pitch motion and roll RAOs are computed in both fixed and free-floating conditions. The main objective of the present study is to explore how the moonpool dimensions affect the shift of the resonant frequencies in the fixed and free-floating vessels, and the relation between the moonpool resonances and vessel motion responses in the free-floating condition. The predicted natural frequencies and shifts in the fixed and freefloating conditions of all models are summarized in Table A.1 in Appendix A. By systematic computations and analyses, it is observed that the natural frequencies of the moonpool in a free-floating condition are higher than that in a fixed condition for both piston and longitudinal sloshing resonance modes. In addition, it is found that for vessel without recess in moonpool, the shift of resonant frequency in the piston mode is larger than that in sloshing modes, and the shift in the first sloshing mode is larger than that in the higher-order sloshing modes. For vessels with a recess, the shift of frequency in piston mode is less than that in the first sloshing mode. It is found that the shift can be affected by the draft of vessel. For moonpool without recess, as the draft is reduced, the shift in piston mode increases. As the length of moonpool increases, the shift in piston mode resonance increases, but the shift in sloshing modes is almost less
ω1
frequency in the free-floating condition and in the fixed condition, respectively. Model No.
r (m)
f1
6 7 8
16 8 4
2.18% 2.12% 1.40%
In Model 10, the height of recess is reduced to 3 m. Fig. 21 illustrates the free-surface elevation with respect to wave frequency for Model 10. The piston mode frequency in the fixed condition is 0.508 rad/s, while that in the free-floating condition is 0.510 rad/s. The shift is 0.39%. The first sloshing mode frequency in the fixed condition is at 0.838 rad/s, while that in the free-floating condition is 0.858 rad/s. The shift is 2.39%. The natural frequencies of the second sloshing mode in both fixed and free-floating conditions are 1.145 rad/s. The shifts in the first sloshing mode for these three models are summarized in Table 8. As expected, the shift of the first sloshing mode frequency is not sensitive to recess height.
13
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 20. Model 9. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
14
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Fig. 21. Model 10. Free-surface elevation at FP, CP and AP in θ ¼ 0∘ , 45∘ and 90∘ . ðaÞ: FP, 0∘ ; ðbÞ: FP, 45∘ ; ðcÞ: FP, 90∘ ; ðdÞ: CP, 0∘ ; ðeÞ: CP, 45∘ ; ðf Þ: CP, 90∘ ; ðgÞ: AP, 0∘ ; ðhÞ: AP, 45∘ ; ðiÞ: AP, 90∘ .
the first sloshing mode have effects on both heave and pitch motions. The water motion in a piston mode may yield antisymmetric hydrody namic pressure along the longitudinal direction of the moonpool, which leads to the local maximum in heave motion RAO. Similarly, the water motion in the first sloshing mode causes the local maximum in pitch motion RAO. It should be noted that damping due to flow separation is important near the resonant frequency, which needs to be investigated in future study. The present study is based on the potential flow theory and focuses on the natural frequency instead of response amplitude. The nonlinear free surface effects and vortex shedding were not taken into consider ation, but should be studied in the future. We believe the present study can offer offshore engineers some guidance or hints when they make decision on the moonpool configuration considering the difference in the natural frequencies between the fixed and free-floating conditions.
Table 8 Shifts of the first sloshing mode frequencies in the fixed condition and freefloating condition for Models 6, 9 and 10. Model No.
h (m)
f1
6 9 10
7.2 5 3
2.18% 2.42% 2.39%
than 1%. Moreover, a modified frozen mode approximation is proposed to examine the effects of the body motion. It is found body motion causes the reduction in the added mass from the fluid underneath the moonpool and yield an increase in the piston mode frequency. Moreover, it is found the infinite-beam assumption in Molin’s paper may lead to the under estimation of the first longitudinal sloshing mode resonant frequency for the model adopted in the experiments. It is found that moonpool resonance can induce local hump in heave and pitch motion RAOs. For vessels without recess in moonpool, the resonances in the piston mode and the second sloshing mode have ef fects on the heave motion, while the resonance in the first sloshing mode can only affect the pitch motion. For vessels with recess in moonpool, resonances in piston mode and
Acknowledgements Special thanks to Prof. Bernard Molin and Prof. Rodney Eatock Taylor for inspiring discussions and comments with one of the authors Xinshu Zhang during the IWWWFB 2019. Moreover we feel very grateful to Prof. Harry Bingham for his help on using the WAMIT.
15
X. Xu et al.
Ocean Engineering 195 (2020) 106656
Appendix A. Summary of the results for all cases Table A.1 Summary of all the cases on the shifts of the piston and the first sloshing mode frequencies in the fixed and free-floating conditions.
Moonpool without recess
Moonpool with a recess
Model No.
L (m)
B (m)
l (m)
b (m)
d (m)
D (m)
r (m)
h(m)
ωFC 0
ωFF 0
f0
ωFC 1
ωFF 1
f1
1 2 3 4 5 6 7 8 9 10
180.8 180.8 180.8 180.8 180.8 180.8 180.8 180.8 180.8 180.8
32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2 32.2
45.6 45.6 45.6 20.0 60.0 45.6 45.6 45.6 45.6 45.6
11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2
11 4 20 11 11 11 11 11 11 11
22.8 22.8 22.8 22.8 22.8 22.4 22.4 22.4 22.4 22.4
– – – – – 16.0 8.0 4.0 16.0 16.0
– – – – – 7.2 7.2 7.2 5.0 3.0
0.735 0.943 0.598 0.765 0.728 0.415 0.593 0.690 0.470 0.508
0.760 1.002 0.613 0.778 0.760 0.415 0.598 0.705 0.473 0.510
3.40% 6.26% 2.51% 1.70% 4.40% 0.00% 0.84% 2.17% 0.53% 0.49%
0.913 1.090 0.848 1.255 0.850 0.8025 0.825 0.863 0.825 0.838
0.923 1.097 0.858 1.260 0.858 0.820 0.8425 0.875 0.845 0.858
1.10% 0.64% 1.18% 0.40% 0.94% 2.18% 2.12% 1.40% 2.42% 2.39%
References
Molin, B., Zhang, X., Huang, H., Remy, F., 2018. On natural modes in moonpools and gaps in finite depth. J. Fluid Mech. 840, 530–554. Newman, J.N., 2018. Resonant response of a moonpool with a recess. Appl. Ocean Res. 76, 98–109. Xu, X., Song, X., Zhang, X., Yuan, Z., 2019. On wave diffraction of two-dimensional moonpools in a two-layer fluid with finite depth. Ocean Eng. 173, 571–586. Yoo, S.O., Kim, H.J., Lee, D.Y., Kim, B., Yeung, S.H., 2017. Experimental and numerical study on the flow reduction in the moonpool of floating offshore structure. In: Proceedings of the ASME 2017 36th International Conference on Ocean, Offshore and Arctic Engineering. OMAE, Trondheim, Norway. Zhang, X., Bandyk, P., 2013. On two-dimensional moonpool resonance for twin bodies in a two-layer fluid. Appl. Ocean Res. 40, 1–13. Zhang, X., Bandyk, P., 2014. Two-dimensional moonpool resonances for interface and surface-piercing twin bodies in a two-layer fluid. Appl. Ocean Res. 47, 204–218. Zhang, X., Huang, H., Song, X., 2019. On natural frequencies and modal shapes in twodimensional asymmetric and symmetric moonpools in finite water depth. Appl. Ocean Res. 82, 117–129. Zhao, W., Taylor, P.H., Wolgamot, H.A., Taylor, R.E., 2018. Linear viscous damping in random wave excited gap resonance at laboratory scale–Newwave analysis and reciprocity. J. Fluid Mech. 80, 59–76. Zhao, W., Wolgamot, H.A., Taylor, P.H., Taylor, R. Eatock, 2017. Gap resonance and higher harmonics driven by focused transient wave groups. J. Fluid Mech. 812, 905–939.
Faltinsen, O.M., Rognebakke, O.F., Timokha, A.N., 2007. Two-dimensional resonant piston-like sloshing in a moonpool. J. Fluid Mech. 575, 359–397. Faltinsen, O.M., Timokha, A.N., 2015. On damping of two-dimensional piston-mode sloshing in a rectangular moonpool under forced heave motions. J. Fluid Mech. 772, R1. Fredriksen, A.G., Kristiansen, T., Faltinsen, O.M., 2015. Wave-induced response of a floating two-dimensional body with a moonpool. Phil. Trans. R. Soc. A 373, 20140109. Guo, X., Lu, H., Yang, J., Peng, T., 2016. Study on hydrodynamics performances of a deep-water drillship and water motions inside its rectangular moonpool. In: Proceedings of the 26th International Ocean and Polar Engineering Conference. ISOPE, Rhodes, Greece. Guo, X., Lu, H., Yang, J., Peng, T., 2017. Resonant water motions within a recessing type moonpool in a drilling vessel. Ocean Eng. 129, 228–239. McIver, P., 2005. Complex resonances in the water-wave problem for a floating structure. J. Fluid Mech. 536, 423–443. Molin, B., 2001. On the piston and sloshing modes in moonpools. J. Fluid Mech. 430 (5), 27–50. Molin, B., 2017. On natural modes in moonpools with recesses. Appl. Ocean Res. 67, 1–8. Molin, B., Remy, F., Camhi, A., Ledoux, A., 2009. Experimental and numerical study of the gap resonances in-between two rectangular barges. In: 13th International Congress of the International Maritime Association of the Mediterranean. IMAM, Istanbul, Turkey.
16