Monika WARMOWSKA, Jan JANKOWSKI[*]

**Ship motion affected by moving liquid cargo in holds **

Liquid cargo moving in holds (sloshing) of a ship sailing in waves affects ship motion. Sloshing forces are taken into account in the equations of ship motion. The motion of liquid cargo in ship holds is described by boundary value problems in which the essential element is the moving free surface of the liquid. This problem is solved by Boundary Element Method.

This paper presents the analysis of ship motion effected by moving molten sulphur in partly filled holds. The analysis has been carried out for different regular and irregular waves inducing ship motion coupled with sloshing.

A numerical solution of the sloshing boundary-value problem and the equations of ship motion make it possible to develop a visualization of the ship motion in waves.

## 1.Introduction

Ship motion in irregular waves is described by non-linear equations of motion written in the non-inertial reference system. Normally, the forces generated by sea waves and the inertial forces determine equations of ship motion. However, in case of liquid cargo carried by ship with partly filled holds, sloshing induces additional forces acting on the ship, which have been taken into account in these equations.

The sloshing flow of liquid cargo in holds is generated by the ship’s motion in waves. It has been experienced in ship operation that sloshing induced forces can affect ship motion, which means that these phenomena are coupled and therefore the problem describing ship motion and the problem of sloshing should be solved jointly.

The sloshing flows coupled with ship motion have been analyzed, among others, by Kim [2] and Yamaquchi & Shinkai [7]. Kim [2] applied the linear equations of ship motion and the sloshing problem (velocity and pressure) was determined using the SOLA scheme [3]. Yamaquchi & Shinkai [7] used the SOLA-SURF Scheme to solve the Navier-Stokes equations describing the liquid flow in anti-rolling tank, while ship motions are described by ordinary strip theory, [5].

Ship motion in waves and sloshing phenomena, previously described and solved independently in the Polish Register of Shipping, have been coupled. In the numerical solution of ship motion equations the sloshing problem is solved in each time step. This paper presents ship motion influenced by sloshing.

## 2.The problem of ship motion in irregular waves

Simulation of ship motion in waves is based on a numerical solution of non-linear equations of motion. The hydrodynamic forces and moments defining the equations are determined in each time step. The accuracy of the simulation depends on the accuracy of calculating hydrodynamic wave forces and moments.

The use of general fluid equations of motion to determine the hydrodynamic forces in the simulation procedure is impractical as these are complicated and have to be solved in each time step. Therefore, the simulation is based on simplified models, which give sufficient accuracy in practical application. It is assumed that the hydrodynamic forces acting on the ship moving in irregular waves can be split into: Froude-Krylov diffraction and radiation forces and forces caused by water on deck, sloshing forces, rudder forces and non linear damping.

The Froude-Krylov forces are obtained by integrating over the actual wet ship surface the pressure caused by irregular waves undisturbed by the presence of the ship.

The diffraction forces (caused by the presence of the ship diffracting the waves) are determined as a superposition of diffraction forces caused by the harmonic components of the irregular wave. It is assumed that the ship diffracting the waves is in its mean position. This is possible under the assumption that the diffraction phenomenon is described by a linear hydrodynamic problem.

The radiation forces are determined by added masses for infinite frequency and by the so-called memory functions (given in the form of convolution), which take into account the disturbance of water, caused by preceding ship movements, affecting the moving ship in the time instant in which the simulation is calculated.

The equations of ship motion in irregular waves are written in the non-inertial reference system. The system Q is fixed to the ship in the centre of its mass and the equations of ship motion assume the following form [1]:

m[vQ(t)+Q(t)vQ(t)]=FW(t)+FD(t)+FR(t)+FS(t)-FS(t=0)+FA(t)+mD-1G,

L(t) +(t) L(t)=MQW(t)+ MQD(t)+ MQR(t)+ MQS(t)MQS(t=0)+ MQA(t),

vQ(t)=RUQ(t) + Q(t)RUQ(t),(1)

((t), (t), (t))T=D-1Q(t),

where m jest is the mass of the ship without liquid cargo, vQ=(vQ1,vQ2,vQ3) is the velocity of its mass centre, ΩQ = (ωQ1, ωQ2, ωQ3) is the angular velocity, L = (L1, L2, L3) is the moment of momentum, RUQ= (xUQ1, xUQ2, xUQ3) is the position vector of the ship mass centre in relation to the inertial system U, moving with uniform speed equal to the average speed of the ship, (, , ) are Euler’s angles, FW, FD and FR are the Froude – Krylow, diffraction and radiation forces, respectively, MQW, MQD, MQR are their moments in relation to the centre of mass G = (0, 0, -g)U, D is the angular matrix, and DΩ is the matrix which transforms Euler components of rotational velocity into ΩQ. The additional forces and moments such as damping forces or forces generated by the rudder blade are denoted by FA and MQA. Sloshing force and moment are denoted by FS and MQS.

The way of solving 3D hydrodynamic problems and determining the Froude – Krylow, diffraction and radiation forces appearing in the equation of motion are presented in [1]. The slosh-induced forces, generated by the moving ship, and the liquid kinematics and dynamics (free surface elevation, velocity and pressure field) are determined through solving the boundary value problem [6].

The non linear equations of motion are solved numerically (slamming procedure is applied) according to the method presented in [4].

The program developed basing on the equations presented and numerical methods applied enables to perform simulations of ship motions in waves.

## 3.The problem of fluid motion in partly filled ship hold

The study covered ship motion in beam seas of a ship with prismatic holds.Therefore, the parameters of liquid flow are the same in all parallel cross sections of the hold and can be represented by flow in one 2D cross section.

The gravitational and inertial forces acting on the liquid in the ship hold are of higher order than the forces generated by the viscosity, thus viscosity is neglected in the description of liquid motion in the ship hold. Such phenomena as flow separation of liquid in motion, occurrence of cushion when liquid strikes a hold wall, solubility of air in liquid moving in hold, dynamic flow interaction with flexible tank structure can also be ignored as they do not affect the ship motions in waves. Normally, it is assumed that the liquid is incompressible and the above assumptions imply that its density is constant. The liquid is in the potential gravitational field.

The boundary S of the domain occupied by liquid comprises the wet surface SC and the free surface SF . It is assumed that the liquid particle slips on the boundary S=SCSF, which results in the following conditions:

nu=nue , (xU2(t),xU3(t)) SC, t[0,T], (2)

where u is the liquid velocity field, **n is the normal vector, ue** is the velocity of hold transportation, and

, (xU2(t),xU3(t)) SF, t[0,T].(3)

It is assumed that the pressure p on the free surface is equal to the atmospheric pressure pa:

p=pa, (xU2(t),xU3(t)) SF, t[0,T].(4)

The initial conditions assume that for t=0 the liquid is at rest and the pressure in the liquid is equal to the hydrostatic pressure.

The above assumption implies that the liquid in the ship hold is irrotational which means that the liquid flow of absolute velocity u=(uU2, uU3) is potential:

u = ,(5)

where the gradient =. In this case the equations of liquid motion are transformed into Cauchy-Lagrange equation:

(6)

and the mass conservation equation takes the following form:

u = 0, (xU2(t),xU3(t)) (t), t[0,T], (7)

where t[0,T] is treated as a parameter.

The algorithm of solving equation (7) consists of three stages for each time step [6]:

- The free surface position determination from the equations:

, (xU2(t),xU3(t)) SF, t[0,T],(8)

and calculation of the velocity potential on the free surface SF from the equation:

(xU2(t),xU3(t)) SF, t[0,T].(9)

Solution of equation (9) determines the value 0 of potential on the free surface:

= 0, (xU2(t),xU3(t)) SF, t[0,T]. (10)

- Solution of the boundary – value problem comprising of:

· Laplace equation:

=0, (xU2(t),xU3(t)) (t), t[0,T], (11)

·boundary conditions:

, (xU2(t),xU3(t)) SC, t[0,T],(12)

(this condition results from (2))

0=, (xU2(t),xU3(t)) SF, t[0,T],(13)

(this condition results from (9)).

The velocity potential obtained from the boundary – value problem determines the velocity field (formula (5)).

- Determination of the pressure field from the following Poisson equation, derived from Euler equations and boundary conditions, [6]:

,(14)

The Poisson boundary-value problem (14) has been used instead of equation (6), as Cauchy-Lagrange equation generated bigger numerical errors. The numerical methods of solving these problems are presented in [6].

## 4.Analysis of ship motion affected by the sloshing

A ship carrying molten sulphur has been chosen for analyzing ship motion affected by sloshing in beam seas. The main parameters of the ship are: length L=138m, breadth B=23m, draught T=8.01m. The ship roll natural period for this loading condition was 8.5s. The holds fulfilling ratio was 80% and the sloshing natural period for hold cross section was 4.22s. Four holds of the ship have been filled to the same level. The molten sulphur density is equal to 1800 kg/m3.

For comparison, calculations have also been carried out for solid cargo of the same mass as the molten sulphur.

The analysis has been carried out basing on the simulation of the ship motion in regular waves (determined by height Hr and period Tr) and irregular waves (determined by significant height Hs and averaged zero-up crossing period Tz).

The following regular wave: (Tr=4.22s, Hr=2m), (Tr=8.5s, Hr=4m) have been used to make the analyses. The results of simulations are presented in Fig.1 and Fig.2.

Fig.1 Time history of wave and slosh-induced roll moment and roll motion; regular wave

Fig. 1 shows the time histories of the wave induced moment, wave and sloshing induced moment, and corresponding roll motions.

Fig. 2. Time history of slosh – induced roll moment versus wave induced roll moment, regular wave

Fig. 2 shows that the phase difference between slosh-induced moments and wave-induced moment, which is about . This phase difference results in the reduction of total exciting moment and reduction of the roll motion.

In order to analyze the behavior of the ship in irregular waves the following waves: (Tz=4.22s, Hs=2m), (Tz=8.5s, Hs=4m) have been used. The results of simulation are presented in Fig.3. The freeze frame of ship motion is presented in Fig.4. Fig.3 shows the time histories of the wave-induced moment, wave and sloshing-induced moment, and corresponding roll motions.

Fig. 3 Time history of wave and slosh-induced roll moment and roll motion; irregular wave

## 5.Conclusions

The numerical analysis has been carried out to analyze of the coupling effects between the ship motion in waves and sloshing flows in the ship holds carrying molten sulphur. The computational results show that for the ship considered the interactions between the ship motion and sloshing flow decrease the roll motion in regular waves as the phase of slosh-induced roll moment is normally shifted about in relation to the wave–induced moment.

The irregular sea waves, being the superposition of an infinite number of regular waves of different amplitudes (determined by wave spectrum) and random phase shifts between them, generates irregular ship roll, and consequently irregular flow of liquid cargo in ship holds. The irregular interactions between the ship motion and sloshing flow normally reduce the ship roll motion but the influence of this interaction on ship roll is less significant than in case of regular waves.

Fig. 4 Freeze frame of ship motion in irregular waves

### References

[1]Jankowski J., Ship facing the waves, Polski Rejestr Statków, 2006 (in Polish).

[2]Kim Y., *A numerical study on sloshing flows coupled with ship motion – The anti-rolling tank problem*, Journal of Ship Research, Vol. 46, March 2002.

[3]Nichols B.D., N. C. Romero, *SOLA – a numerical solution algorithm for transient fluid flows*, Los Alamos Scientific Laboratory, California 1975.

[4]Raston A., First course in numerical analysis, PWN, Warsaw, 1975.

[5]Salvesen N., Tuck E.O., Faltinsen O., Ship motion and sea loads, Det Norske Veritas, Publication No. 75, March 1971.

[6]Warmowska M.,*Parameters of liquid motion in partially filled ship tank, including non-linear phenomena*, PhD Thesis, Gdańsk University of Technology, 2005 (in Polish).

[7]Yamagucki S., Shinkai A., *An advance adaptive control system for activated anti-rolling tank*, International Journal of Offshore and Polar Engineering, Vol. 5, 1995.

[*]Polski Rejestr Statków S.A., Al. Gen. Józefa Hallera 126, 80-416 Gdańsk