Numerical analysis of ship motion coupled with tank sloshing

Xu Li a,b, Tao Zhang a,b, YongOu Zhang a,b, YaXing Wang c

a School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China b Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Huazhong University of Science and Technology, Wuhan 430074, China c School of Engineering, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, U.K. E-mail address: lixu199123@gmail.com (X. Li), zhangt7666@mail.hust.edu.cn (Tao Zhang),

zhangyo1989@gmail.com (Y. O. Zhang).

Abstract— The effects of tank sloshing are usually ignored or linearized in ship motion analysis. Recent studies show that the nonlinearity caused by the coupling of tank sloshing and ship motion can be significant in a certain frequency range. The tank sloshing can be excited by the ship motion, at the same time, the tank sloshing induces an impact load on the tank wall, which has a significant influence on the ship motion. In this paper, a time domain equation has been built to obtain the results of coupling effects which is more accurate than the results obtained from the frequency domain equation considering the effects of damping. In solving of the time domain equation, hydrodynamic coefficients and wave loads are obtained by a program Hydrostar which is based on the potential theory in frequency domain. The radiation force is integrated in time domain by using impulse response function (IRF) approach. The liquid motion in the tank is simulated in time domain by computational fluid dynamics (CFD) program which has high accuracy in simulating tank sloshing. The tank sloshing force and moment computed by CFD program based on volume of fluid (VOF) method are then applied as an external force of the ship motion. The simulated results of ship motion are in turn used as the excitation of tank sloshing and repeated by ensuing time steps. This time domain method is accurate in solving coupling effects of ship motion and tank sloshing, because it combines the advantages of potential flow theory and viscous flow theory. The response amplitude operator (RAO) of ship motion of computation and experiment is proved in good agreement.

Keywords—nonlinearity; tank sloshing; time domain; RAO

NOMENCLATURE A wave amplitude B characteristic breadth of free surface in tank

( )C ω damping matrix of ship body

( )s ωC damping matrix of sloshing fluid

( )44C ω damping of ship body in the 4th - 4th degree of freedom

( )*44C ω modified damping of ship body in the 4th - 4th degree

of freedom ijC wave damping coefficient of ship body in the

th - thi j degree of freedom d water depth

( )wF ω wave exciting force vector on the ship body

( )sF ω sloshing-induced force vector

( )ext tF external force on the surface of ship hull excited by waves and hydrodynamic reactions

( )sa tF sloshing-induced force on the tank wall ( )s tF hydrostatic and hydrodynamic force induced by fluid

motion in tank g acceleration of gravity h vertical distance from free surface to tank bottom K hydrostatic restoring stiffness matrix

sK reduction of hydrostatic restoring stiffness matrix k wave number

44K hydrostatic restoring stiffness of ship body in the 4th - 4th degree of freedom

*44K modified hydrostatic restoring stiffness of ship body

in the 4th - 4th degree of freedom ′44K adjustment of hydrostatic restoring stiffness of ship

body in the 4th - 4th degree of freedom sI second moment of inner-tank free surface with respect

to the axis of rotational motion M mass matrix of ship body

( )a ωM added mass matrix of ship body

( )s ωM mass matrix of inner-tank fluid

( )as ωM added mass matrix of sloshing fluid aijM added mass of ship body in the th - thi j degree of

freedom ( )∞a

ijM added mass of ship body in the th - thi j degree of freedom in infinite frequency

n number of frequency order ( )ijR t retardation function in the th - thi j degree of freedom

( )si z sine integral

liquidV volume of liquid in tank Φ velocity potential η free surface elevation θ incident wave angle

978-1-4799-3646-5/14/$31.00 ©2014 IEEE978-1-4799-3646-5/14/$31.00 ©2014 IEEE This is a DRAFT. As such it may not be cited in other works. The citable Proceedings of the Conference will be published in

IEEE Xplore shortly after the conclusion of the conference.

sρ density of inner-tank fluid γ damping ratio

nω value of the thn -order frequency

I. INTRODUCTION With the development of liquid cargo carrier, like Liquefied

Natural Gas (LNG) Floating Production Storage and Offloading (FPSO) and Floating Liquefied Natural Gas (FLNG), the effects of free surface and sloshing inside the tank on the ship’s motion are of great concern. Besides, the coupling effects become more and more crucial as the size of ship increases with market demand.

In the past, tank sloshing in the coupling problem is simplified as a free-surface problem or a mass-spring model, which could obtain enough accuracy in a limited field. In recent decades, many studies have been carried out in a larger field for solving this problem, which can be mainly summarized into two categories: the study in frequency domain based on the linear potential flow theory and the study in time domain based on the nonlinear vicious flow theory. Molin et al. [1, 2], Malenica et al. [3] and Newman [4] had investigated the coupling of ship motion and sloshing by assuming linear sloshing flow in frequency domain. The assumption of linear ship motion is necessary and reasonable in the coupling analysis according to past studies. However, the assumption of linear sloshing flow is not reasonable when the viscous force of tank sloshing is much greater than the inertia force. Recently, research on ship motion in time domain has been used frequently, especially an impulse response function (IRF) method has been used to investigate the ship motion in time domain. Cummins [5] developed the IRF approach to transform frequency domain solutions to time domain. Kim et al. [6] extended this approach to analyze the coupling problem of ship motion with tank sloshing and a good agreement was reported between experimental data and simulation results. Rognebakke and Faltinsen [7] investigated two-dimensional experiments of the hull section containing tanks filled with different levels of water swayed by regular waves. The model was simulated using both linear and nonlinear sloshing model and a good agreement was found between calculation results and experimental data. Kim [8] used three-dimensional finite-difference method to simulate sloshing flow and used time-domain panel method to obtain the ship motion. Then these two methods are combined to simulate the coupling problem of ship motion and tank sloshing. Lee et al. [9] investigated the coupling problem of ship motion and tank sloshing with a new time domain simulation scheme. Seung [10] developed a 3D time domain potential-viscous hybrid method. Time domain codes CHARM3D and ABSLO3D have been developed to solve the coupling problem of ship motion and sloshing. In the study by Seung and Kim [11], wave, wind, current, fender, hawser and mooring as well as hydrodynamic interaction between Liquefied Natural Gas Carrier (LNGC) and Floating Terminal (FT) which was arranged side-by-side were considered.

Experiments have always been one of the most effective methods to investigate the coupling effects of tank sloshing and ship motion. Model test results are usually used to verify the

theoretical and numerical results. Kim [8], Gaillarde et al. [12], Molin et al. [2], Clauss et al. [13] and Li et al. [14] carried out a series of model tests to investigate the effects of tank sloshing on the ship motion.

There are many studies on sloshing-induced loads recently. Many numerical and theoretical methods have been proposed to solve sloshing problem. Faltinsen [15] and Faltinsen et al. [16] proposed a discrete infinite-dimensional modal system to describe nonlinear sloshing problem. Kim [17] used a finite difference method to simulate the sloshing flows in the two- and three-dimensional liquid containers. Kim et al. [18] proposed a new two-dimensional finite-element method based on the stream-function theory to analyze the sloshing pressure in time and space. Kim et al. [19] developed a three-dimensional finite-element program to investigate the sloshing-induced loads by an LNG tank and compared their results with the test data and the results calculated by Faltinsen and Timokha [20] based on modal theory. Kim et al. [21] used a finite difference method to simulate violent sloshing flows and impact occurrence in two- and three-dimensional prismatic tanks and then the computational results for two-dimensional tanks are compared with available experimental data and the results for three-dimensional. Loots et al. [22] developed an improved volume of fluid (iVOF) method to calculate sloshing impact loads and the results are compared with sloshing test data. Lee et al. [23] carried out a series of parametric sensitivity studies on unmatched dimensionless scale parameters by using a computational fluid dynamics (CFD) program. Landrini et al. [24], Colagrossi et al. [25] and Zhu et al. [26] investigated the sloshing problem in two-dimensional model based on the smoothed particle hydrodynamics (SPH) method.

In this paper, IRF approach and CFD program for tank sloshing simulation are used to calculate the coupling effects of ship motion and tank sloshing. A time domain equation is built to obtain the results of coupling effects which is more accurate than the results obtained from the frequency domain equation considering the effects of damping. In the computation, value of the damping ratio is confirmed by comparing the simulation results with experimental data. Finally, response amplitude operators (RAOs) of the ship coupled with tank at different filling levels is obtained both in time domain and frequency domain.

II. MATHEMATICAL FORMULATION

A. Ship Motion Potential theory is an effective approach to calculate ship

motion in induced waves. In potential theory, fluid is assumed as irrotational and inviscid flow, so velocity potential can be used to represent the entire flow field as shown in (1).

2Φ = 0∇ (1)

Boundary value problem (BVP) is one of the most used approaches to solve velocity potential. Three of boundary conditions should be defined to solve (1), which are bottom

boundary condition, dynamic and kinematic boundary condition of the free surface, namely:

Φ 0

z= d

on the bottomz −

∂ =∂

(2)

Φ 0∂ ∂ ∂ ∂+ + − =

∂ ∂ ∂ ∂ z=η

η η ηu υ on the free surfacet x y z (3)

( )Φ 1 Φ Φ 02

∂ + ∇ ∇ + =∂

iz=η

gz on the free surfacet

(4)

It is difficult to solve velocity potential directly with these boundary conditions due to the nonlinearity of dynamic and kinematic boundary conditions. The most common approach to obtain acceptable approximated solution is the perturbation method, which assumes that the wave has small amplitude compared with water depth and wave length. The first-order and second-order velocity potential is obtained by using the perturbation method, which can be written as follows:

( ) ( ) ( )1 cos sincoshΦ Re

coshi kx θ+ky θ ωtk z + digA e

ω kd−⎡ ⎤−= ⎢ ⎥

⎣ ⎦ (5)

( ) ( ) ( )2 2 cos 2 sin 24

cosh 23Φ Re8 sinh

i kx θ ky θ ωt2 k z + dωA e

kd+ −⎡ ⎤

= ⎢ ⎥⎣ ⎦

(6)

It is also difficult to solve the velocity potential of entire flow field directly considering the complex surface of ship hull. Velocity potential is divided into the incident potential, diffraction potential and radiation potential and solved separately based on the diffraction and radiation theory. The force on the ship caused by wave can be solved after the total velocity potential obtained. The ship body is assumed as a rigid body, so the ship motion equation can be written as follows:

( )w t+ + =M C K Fς ς ς (7)

B. Sloshing Natural Frequency The natural frequency of tank with a water depth is derived

by the wave dispersion relation:

( )2 tanh=ω kg kh (8)

k is wave number which is related to wave length by:

2= πkL

(9)

Relation between wave length and the characteristic breadth of free surface is:

2=L Bn

(10)

where B represents the characteristic breadth of free surface. The characteristic breadth of free surface represents the breadth of tank for transverse mode and the length of tank for longitudinal mode at the free surface.

Equation (9) and (10) are substituted into (8), then the natural frequency of tank for each mode is obtained:

tanh⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

nnπ πnhω gB B

(11)

C. Coupling Problem in Frequency Domain When taking tank sloshing into account, the sloshing-

induced loads should be added as the external excitation. The equation of ship motion with inner-tank sloshing loads in six degrees of freedom can be written as follows:

( ) ( ) ( ) ( )aw sω ω ω ω⎡ ⎤+ + + = +⎣ ⎦M M C K F Fς ς ς (12)

In (12), the mass matrix M includes the mass of fluid in the tank. By ignoring the inertia force of tank, the sloshing-induced force is given by:

( ) ( ) ( )as s ss ω ω ω= + +F M C Kς ς ς (13)

where sK is the reduction of restoring stiffness, which can be written as follows:

s s sI ρ g=K (14)

For the roll motion, the effect of viscosity is important and dominant, so it is necessary to revise the damping by adding the linear equivalent damping coefficient:

( ) ( )2 + ( )=* a44 44 44 44C ω γ M M ω K (15)

where γ depends on the shape of ship hull, so it needs to be discussed in the particular condition.

Equation (13) and (15) are substituted into (12), then the modified coupling equation of ship motion and tank sloshing can be written as:

( ) ( ) ( ) ( ) ( )a as + +* s44ω ω ω C ω C ω⎡ ⎤ ⎡ ⎤+ − +⎣ ⎦ ⎣ ⎦M M M Cς ς

( )sw ω⎡ ⎤+ − =⎣ ⎦K K Fς (16)

The motion and force vectors of ship body can be written as:

w

Re 1,2,3,4,5,6

Re 1,2,3,4,5,6

iωtj

iωtwj

= ς e j

= F e j

⎧ =⎪⎨

=⎪⎩F

ς (17)

Equation (17) is substituted into (16), then the coupling equation of motion in frequency domain can be rewritten as:

( ) ( ) ( ) ( ) ( )a as 2 s+ +*44ω ω ω i ω C ω ω ω⎡ ⎡ ⎤ ⎡ ⎤− + − +⎣ ⎦ ⎣ ⎦⎣ M M M C C

sj wjς⎤⎡ ⎤+ − =⎣ ⎦⎦K K F (18)

D. IRF Approach for Ship Motion According to radiation theory, the first-order radiation force

of ship body in frequency domain can be written as:

( ) ( ) ( )( )1 1 1

Ω

ΦRe Φ Re

B

jiωt aRij j j ij j ij jF ρ ς e dS M ς C ς

n−

∂

⎡ ⎤∂= − ∂ = +⎢ ⎥

∂⎢ ⎥⎣ ⎦∫∫ (19)

The added mass matrix and wave damping matrix are symmetrical and are only related to the frequency of ship motion.

All of the hydrodynamic coefficients like added mass and damping coefficient are calculated in frequency domain, so the corresponding forces need to be converted into time domain. After derivation according to IRF approach, the first-order radiation force can be rewritten as follows:

( ) ( ) ( ) ( ) ( )1

−∞

= − ∞ − −∫t

aRij ij j ij jF t M ς R t τ ς τ dτ (20)

Where the convolution integral represents the memory effects of the wave force on the ship hull. Retardation function

( )ijR t is related to the radiation damping in frequency domain. The equations for ( )∞a

ijM and ( )ijR t are given by:

( ) ( ) ( )0

sin( )∞

∞ = + ∫a aij ij ij

ωtM M ω R t dtω

(21)

( ) ( ) ( )0

2 cos∞

= ∫ij ijR t C ω ω dωπ

(22)

Where a truncation error is inevitable because of the integral of (22), which is calculated in a finite frequency range. A special improvement from Kim [6] is carried out and the retardation function can be rewritten as:

( ) ( ) ( ) ( )Γ

0

2 cos= +∫ij ij ijR t C ω ωt dω υ tπ

(23)

where

( ) ( ) ( )Γ

2 cosij ijυ t C ω ωt dωπ

∞

= ∫

( ) ( )2

Γ

cos2 0ij

ωtR dω

π ω

∞

′≈ − ∫

( ) ( ) ( )cos Γ Γ si Γ2 0Γ+

′= − ij

t t tR

π (24)

where ( )si z is the sine integral:

( ) sinsi∞

= −∫z

tz dtt

(25)

Fig. 1 shows the effect of cut off frequency and truncation error on heave-heave retardation function.

Fig. 1. Effects of cut off frequency and truncation error on heave-heave retardation function (L: ship length)

(a)

(b)

E. Coupling Problem in Time Domain When the tank sloshing load is considered, the force acted

on the ship body should be rewritten as follows:

( ) ( ) ( )ext sat t t= +F F F (26)

For calculation purposes, the inertia force should be extracted from sloshing-induced force and moment.

( ) ( ) ssa st t= +F F M ς (27)

In the experiment, the displacement of FPSO remains the same by adjusting ballast with different tank filling level. Consequently, the vertical mass center changes with the tank filling level and should be modified.

In Fig. 2, where mG represents the original mass center of ship body and ballast, sg represents the mass center of inner-tank fluid.

The roll restoring stiffness can be modified as follows:

*44 44 44K K K ′= − (28)

44 liquid GgK ρgV L′ = (29)

where

= m sGgL KG Kg− (30)

Considering the above derivation and adjustment, the coupling equation of ship motion and sloshing in time domain can be written as:

( ) ( ) ( ) ( )a s44

t

ω t τ τ dτ K−∞

′⎡ ⎤+ − + − + −⎣ ⎦ ∫M M M R Kς ς ς

( ) ( )ext st t= +F F (31)

III. NUMERICAL COMPUTATION

A. Simulation Procedure In this paper, coupling problem between ship motion and

tank sloshing is solved both in frequency domain and time domain. Fig. 3 illustrates the related programs are used to solve these problems and the corresponding steps.

In frequency domain, the coupling between ship motion and tank sloshing can be solved directly. In solving the time domain equation, hydrodynamic coefficients and wave loads are obtained by Hydrostar which is a hydrodynamic software developed in Bureau Veritas. Then the radiation force is obtained in time domain by using the IRF method. The liquid

Fig. 2. Modification of roll restoring stiffness

Fig. 3. Flow chart of solving ship motion and tank sloshing coupling

motion in the tank is simulated in time domain by a CFD program based on the VOF method which has already been proved to be accurate in simulating tank sloshing. The tank sloshing force and moment computed by CFD program are then applied as an external force on the ship. The simulated ship motion is in turn used as the excitation of tank sloshing.

B. FPSO Model In this paper, an FPSO experiment carried out by MARINE

is simulated to investigate the coupling problem of ship motion and tank sloshing. Fig. 4 shows the geometry of ship hull and tanks. The length, breadth and draft of FPSO are 285.0m, 63.0m and 13.0m, respectively. The displacement volume of FPSO is 220,017.6 m3. Fig. 5 shows the mesh of FPSO with 5024 elements and tanks with 56% filling level in Hydrostar. Two tanks inside the ship filled with fresh water are tested in three different filling levels (18%, 37% and 56%). The length, breadth and height of the rear tank and the front tank are 49.68m, 46.92m, 32.23m and 56.62m, 46.92m, 32.23m respectively. The distance from the bottom of tank to the keel line is 3.3m.

Fig. 4. Sketch of FPSO with internal tanks

Fig. 5. Mesh of FPSO and tanks (56% filling level)

C. Natural Frequency, Added Mass and Damping Coefficients

Table I shows the natural frequencies of transverse mode for three different filling levels 18%, 37% and 56%. Except for the length, the geometry of two tanks is the same, therefore, the transverse natural frequency of two tanks is equal, and the value of longitudinal natural frequency is different according to (11). Fig. 6 and Fig. 7 show the roll and pitch added mass, damping coefficients of the front tank and the rear tank.

TABLE I. NATURAL FREQUENCIES OF TANKS

Tank filling level

Element number for each tank

Transverse mode

(rad/s)

Longitudinal mode(rad/s)

Front tank Rear tank

18% 400 0.49 0.41 0.47

37% 600 0.66 0.56 0.63

56% 800 0.74 0.64 0.71 Fig. 6. Roll and pitch added mass, damping coefficients of the front tank

(a)

(b)

(c)

(d)

(a)

(b)

Fig. 7. Roll and pitch added mass, damping coefficients of the rear tank

D. Analysis of Damping Ratio Roll motion has always been one of the most concerned

problems and is more complicate when the effects of tanks are considered. It is not difficult to obtain the RAOs of ship coupled with tanks by solving (18). Fig. 8 shows the RAOs of roll motion obtained by calculation and experiment at three different filling levels in the beam sea condition.

The figure shows that the first-order natural frequency of ship in roll without the tank is around 0.5 rad/s. The peak value of computational result is greater than experimental data at the roll natural frequency, because viscosity was not taken into account in the simulation. The damping ratio γ is used to reduce the viscous effects. Regardless of the filling ratio,

0.10 ~ 0.15γ = can lead good computational results in frequency domain. For the 18% filling level case, the roll natural frequency of tank is close to the roll natural frequency of ship. Computational results show that the peak value of roll RAOs is moved to around 0.6 rad/s and a new trough occurs at around 0.42 rad/s. In 37% and 56% filling levels, the natural frequency of tanks (0.66 rad/s, 0.74 rad/s respectively) is much higher than the roll natural frequency of ship(0.5 rad/s), and the second peaks are at around 0.76 rad/s and 0.8 rad/s. The first peak value continues to increase and the second peak value continues to decrease with the increasing of filling level. Tanks actually play a role of anti-roll tank around the natural frequency of ship. In 18% filling level, the viscous effect is much greater than the inertia effect, therefore, bigger difference

Fig. 8. Comparison of roll RAOs of experimental and computational results at different filling levels (beam sea)

(c)

(d)

(a)

(b)

(c)

(d)

can be found between computational results and experimental data in frequency domain.

E. Coupling Results Fig. 9 shows the pitch motion RAOs of ship at different

filling levels in head sea condition. Under this condition, the computational results of pitch motion shows a good agreement with experimental data, and there is no big difference in pitch motion RAOs among three filling levels. Therefore, a clear conclusion can be obtained that the effects of tank sloshing are small for the ship in pitch motion.

For considering the viscous of tank fluid, it is necessary to calculate the ship motion with liquid cargo in time domain, and the RAOs of ship motion can be obtained by Fourier analysis in order to compare with the previous results.

Fig. 10 compares the results of roll RAOs at three filling levels in the beam sea condition between experimental data and computational results in time domain and frequency domain. Large overestimation in some frequency points are further reduced by using time domain method which considers the nonlinearity of sloshing flow. For 18% filling level, the sloshing resonance frequency (0.49 rad/s) is close to the ship hull natural frequency (0.5rad/s). The position of peak is moved from 0.5rad/s to 0.6rad/s and the peak value almost remain unchanged. For 37% and 56% filling levels, the split of peaks can be clearly seen as pointed out in Fig. 8. Tanks actually play a role of anti-roll tank when the sloshing resonance frequency is far from the ship hull natural frequency, and the anti-roll effect is more significant as the increasing of difference between the sloshing resonance frequency and the hull natural frequency. In lower filling level, the roll RAOs calculated in time domain is much closer to experimental results for taking the viscosity and nonlinear free-surface effects into account. For higher filling level, the inertia effect is dominant compared with the viscous effect, so the time domain method do not show a significant advantage.

IV. CONCLUSIONS In this paper, numerical simulation of ship motion coupled

with tank sloshing both in time domain and frequency domain are presented. IRF approach and CFD program for tank sloshing simulation are used to solve this problem. The salient conclusions obtained from the study are presented below:

Some parameters affect the accuracy of computation, such as the cut off frequency and the damping coefficient. To minimize truncation error, a modified method from Kim [6] is carried out in calculating the retardation function and proved to be useful.

The value of roll RAOs is sensitive to the damping ratio γ , which is difficult to be confirmed without experiments. The values of γ are basically the same, and do not change with the filling levels of tanks and have nothing to do with whether the ship has tanks or not.

The viscous effect of tank sloshing plays an important role in the coupling effects. In lower filling level, the roll RAOs calculated in time domain is much closer to experimental data

Fig. 9. Comparison of pitch RAOs of experimental and computational results at different filling levels (head sea)

(a)

(b)

(c)

(d)

Fig. 10. Comparison of roll RAOs of experimental and computational results (frequency domain and time domain) at different filling levels (beam sea)

than computational results in frequency domain, since the viscous effect is more dominant than inertia effect in lower filling level. However, in higher filling level, the time domain method has the same accuracy with frequency domain method as the inertia effect is more significant than viscous effect.

ACKNOWLEDGMENT Authors wish to express their gratitude to Dr. Xiaobo Chen,

Chairman of BV (Bureau Veritas) for his support. We also want to thank to Bin Song, Haixia Xu working in Engineer of Advanced Technology Section Offshore Department in Shanghai Central Office of BV for their supporting.

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