Applied Ocean Research 28 (2006) 171–181www.elsevier.com/locate/apor

Development of a VoF-fractional step solver for floating body motionsimulation

Roozbeh Panahi∗, Ebrahim Jahanbakhsh, Mohammad S. Seif

Marine Laboratory, Department of Mechanical Engineering, Sharif University of Technology, P.O. Box 11365-9567, Tehran, Islamic Republic of Iran

Received 22 May 2006; accepted 22 August 2006Available online 8 January 2007

Abstract

Numerical simulation of floating or submerged body motions is presented based on a Volume of Fluid (VoF)-fractional step coupling. Solvinga scalar transport equation for volume fraction of two phases results in a single continuum with a fluid property jump at the interface. In addition,velocity and pressure fields are coupled using the fractional step method. Based on integration of stresses over a body, acting forces and momentsare calculated. Using the strategy of non-orthogonal body-attached mesh and calculation of motions in each time step result in time history ofhydrodynamic motions. To verify the accuracy of the numerical procedure in simulation of two-phase flow, sloshing and dam breaking withobstacle problems are investigated. Besides, motions simulation strategy is evaluated by using a cylinder water entry test case. To demonstrate thecapability of the simulation, barge resistance is calculated in two cases of fixed and free motion (2-DoF). All of the results are in good concordancewith experimental data. The present method can be extended for full nonlinear motion of ships in waves.c© 2006 Elsevier Ltd. All rights reserved.

Keywords: Numerical hydrodynamics; Volume of fluid; Fractional step; Ship motion

1. Introduction

Nowadays, numerical simulations are becoming a commonway for assessment of ship performance in early design stages.Although model tests using an experimental approach are stillvery useful they have their own restrictions and expenseswhich have motivated us to employ numerical simulations.Taking into account the advances in computer hardware, useof Computational Fluid Dynamics (CFD) is becoming the bestchoice in many cases.

In practice, a ship hydrodynamic problem includes turbulentviscous flow with complex free surface deformation andsometimes fluid–structure interaction. Investigation of allaspects of such a problem necessitates a robust numerical tool.One of the practical ways to study the aforementioned coupledcomplicated case is to decouple it by either completely ignoringthe less important phenomena or approximating them.

In solving such a problem, one encounters to threesubproblems which are: (a) velocity and pressure distribution,(b) free surface deformation and (c) rigid body motion.

∗ Corresponding author.E-mail address: Roozbeh Panahi@yahoo.com (R. Panahi).

0141-1187/$ - see front matter c© 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.apor.2006.08.004

Methods for solving the Navier–Stokes and continuityequations are typically categorized as pressure-correctorschemes and projection or fractional step schemes.

In pressure-corrector methods like SIMPLE [1] and PISO[2,3], a pressure correction equation is solved for severaliterations in each time step to reach a divergence free velocityfield. In contrast with such iterative methods, in fractional stepschemes a pressure or pseudo-pressure Poisson equation issolved once in each time step to enforce continuity. Therefore,using fractional step is preferable, especially in unsteadyproblems [4]. The fractional step method, which has been usedwidely over the past two decades, is introduced by Chorin[5,6], based on Hodge decomposition. In the 1980s, severalsecond-order projection methods are proposed by Goda [7],Bell et al. [8], Kim and Moin [9] and Van Kan [10]. Accuracyof such methods is discussed by Brown et al. [11], resulted inintroducing a modified scheme based on Bell et al. [8].

On the other hand, the existing approaches for handlingfluids interface are [12]: interface tracking or surface methodsand interface capturing or volume methods.

Interface tracking methods are characterized by anexplicit representation of the interface. In other words, the

172 R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181

computational grid is moved and updated in each time stepto have no flow across it while satisfying force equilibriumon a fluid at the interface [4]. The common drawback of thiscategory is the inability to handle complex geometries andoverturning waves. This problem leads to interface capturingmethods where the interface is captured as a part of the physicaldomain. Volume of Fluid (VoF) is an approach in volumemethods. One of the most interesting schemes in this categoryis to solve an additional convection equation. This results in avolume fraction which implies the availability of two phasesin each Control Volume (CV) for the whole computationaldomain. In discretisation of such a transport equation one needsto face values which must be estimated using an appropriateinterpolation scheme. This interpolation must ensure bothboundedness and availability criteria [13] to have physicalvolume fraction values. It means the value of a flow propertyin the absence of source or sink cannot be higher or lowerthan those values prescribed on the boundaries of a cell. Inaddition, the amount of flow convected over a face during atime step should be less than or equal to the amount availablein a doner cell. Simple interpolations have some problem withthe transitional area between two phases while introducingnumerical diffusion [14] or disobeying the local boundedness[15]. There are some composite schemes which switch betweentheir options according to the received signals about the flow, tohave a physical distribution of fluids in the whole domain [16,17]. CICSAM [17] (Compressive Interface Capturing Schemefor Arbitrary Meshes) which is used in this study is a promisingmethod which appropriately retains the transitional regionbetween two fluids while successfully establishes all criteria incomparison to other composite interpolations [18].

Coupling of two aforementioned subproblems results insimulation of the interfacial flow. This subject is recentlydeveloped by many researchers especially based on volumemethods [19,20] and used in simulation of breaking waves[21,22], green water [23,24], sloshing [25] and wave–structureinteraction [26].

Interfacial flow simulation results in velocity and pressuredistribution which yield tangential and normal stresses,respectively. Integration from such stresses over the bodyresults in forces and moments acting on it. Here, the motionsof the rigid body in the 6-DoF are determined by solving theequations of variation of linear and angular momentum. Thisis the last subproblem which gives a time history of bodymotions in one or two phases, e.g. submarine maneuvering orship seakeeping. The important point is that, although potentialtheory methods [27] are capable of predicting motions withlower run time they are not suitable where viscous effects orbreaking waves play an important role. For several practicallyimportant cases large errors can be introduced by the potentialtheory assumptions.

The motion of a floating body is a direct consequence of theflow-induced forces acting on it while at the same time theseforces are a function of the body movement itself. Therefore,the prediction of flow-induced body motion in viscous fluidis a challenging task and requires a coupled solution of fluidflow and body motion. In the recent two decades, with the

change in computer hardware, motion simulation is the subjectof much hydrodynamics research. It is developed from therestricted motions such as trim or sinkage by Miyata [28],Hochbaum [29], Alessandrini [30] and Kinoshita [31] to theevaluation of 6-DoF motions by Miyake [32], Azcueta [33],Vogt [34] and Xing-Kaeding [35].

In this paper an integrated finite volume procedure forsimulation of three-dimensional interfacial flow interactionwith rigid body motion is presented. The governing equationsare reviewed in Section 2.1. Discretisation of the pressureintegral in Navier–Stokes equations is of great importanceespecially when there are two phases with high density ratio,e.g. water and air. Discretisation of the governing equationsis presented in Section 2.2 besides the introduction of anew pressure integral discretisation. Especial care must betaken in motion simulation about the Reference System (RS)selection and its effect in governing equations which are appliedeverywhere needed.

Computer software is developed based on the presentedmathematical formulation. The interfacial flow part of thesoftware is verified using sloshing and dam breaking withobstacle problems. The strategy of motion simulation isassessed with simulation of a cylinder water entry problem.Finally, barge resistance is calculated in fixed and free motion(2-DoF) cases. Comparisons of all test cases show the abilityof the proposed procedure to be extended as a numericalhydrodynamic tank for full nonlinear ship motion simulation.

2. Numerical method

2.1. Governing equations

As mentioned earlier, complex deformation of the interfacein hydrodynamic problems leads to the use of volume methodsin free surface modeling. Here a scalar transport equation whichis extracted from the continuity equation [36] (Eq. (1)) is solvedto determine the volume fraction of each phase (i.e. air andwater) in all computational cells:

∂α

∂t+

⇀

∇ ·(α⇀u ) = 0. (1)

This results in α distribution as:

α =

1 for cells inside fluid 10 for cells inside fluid 20 < α0 < 1 for transitional area.

(2)

Here, an effective fluid with variable physical properties isintroduced:

ρeff = αρ1 + (1 − α)ρ2

νeff = αν1 + (1 − α)ν2(3)

where subscripts 1 and 2 represent two phases, e.g. water andair.

Such an effective fluid is used in solving the fluid main gov-erning equations. These are momentum and continuity equa-tions which are as follows:

∂ui

∂t+ u j

∂ui

∂x j= −

1ρ

∂ P∂xi

+ ν∂2ui

∂x j∂x j+ gi (4)

R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181 173

Fig. 1. Motion simulation strategy and reference system.

∂ui

∂xi= 0. (5)

Velocity and pressure distribution around a body causes it tomove in 6-DoF. These movements can be estimated by solvingthe linear and angular momentum equations:∑ ⇀

F = m⇀a (6)∑ ⇀

MG = IG⇀α +

⇀ω ×IG

⇀ω . (7)

Finally, it must be mentioned that a body-attached meshwhich follows the time history of body motions is used for rigidbody motion simulation. In addition, all of the fluid governingequations are written for a moving Control Volume (CV) inNewtonian RS (Fig. 1).

2.2. Discretisation

The finite volume discretisation of volume fraction transportequation (Eq. (1)) is based on the integration over the CV andthe time step:∫ t+δt

t

(∫V

∂α

∂tdV)

dt +

∫ t+δt

t

(∫V

⇀

∇ ·(α⇀u )dV

)dt = 0. (8)

The first term in Eq. (8) is a common integral form andapplying the Gauss theorem on the second term results in:

(αt+δt− αt )

V1t

+12

(n∑

f =1

αt+δtf F t+δt

f −rel +

n∑f =1

αtf F t

f −rel

)= 0 (9)

where α f is the face volume fraction. F f −rel = (⇀

U f −⇀

U m)·⇀

A f

is the relative volumetric flux at the CV face.⇀

U f is calculatedin an especial manner and will be discussed later in the solution

algorithm (Section 2.3.1). In addition,⇀

U m is the only effect ofmoving CV in the governing equations.

The time integral of the second term is discretised usingthe Crank–Nicholson scheme. Assuming a linear and smallvariation of F f −rel in the small time step results in using themost recent value of it. Therefore, rearranging Eq. (9) leads to:

αt+δt V1t

+

n∑f =1

12αt+δt

f F f −rel = SαP (10)

where 1t is the time step and the source term SαP in Eq. (10)is written as follows:

Fig. 2. Flow direction (arrow) determines doner, acceptor and upwind cells, foreach CV face.

SαP = αt V1t

−

n∑f =1

12αt

f F f −rel (11)

α f in Eqs. (10) and (11) must be approximated based onCVs’ center values. It has been mentioned earlier that simpleinterpolations result in non-physical volume fraction values.This leads to using a high order composite interpolation.

Most composite interpolations typically switch betweentwo high and low order interpolations in order to use theiradvantages and minimize their disadvantages. Here, the maindistinctions are in how and when they switch between theseschemes according to flow information.

CICSAM (Compressive Interface Capturing Scheme forArbitrary Meshes) uses CBC [37] (Convection BoundednessCriteria) (α fCBC) and UQ [14] (ULTIMATE QUICKEST)(α fUQ) interpolations by introducing a weighting factor γ f(Eq. (12)) which takes into account the slope of the freesurface relative to the direction of motion. CBC is the mostcompressive scheme which stipulates robust local bounds onα f : nevertheless it does not actually preserve the shape ofinterface. Here UQ is used for its ability in preserving theinterface shape. Based on NVD [14] (Normalized VariableDiagram) normal face value is obtained as follows:

α f = γ f α fCBC + (1 − γ f )α fUQ . (12)

NVD definitions are shown in Fig. 2 where doner, acceptorand upwind cells (subscripts D, A and U respectively) aredefined according to flow direction for each CV face:

α f =α f − αU

αA − αU. (13)

Using the definition of Eq. (13) in Eq. (12) results inestimation of α f .

α f contains all information regarding the fluid distributionin the doner, acceptor and upwind cells as well as the interfaceorientation relative to flow direction. To avoid the non-physicalα, a correction step is added to the volume fraction calculationprocedure and used in the developed software, details of whichcan be found in Ubbink and Issa [17].

By integration of the Navier–Stokes equations over a movingCV it becomes as Eq. (14) in Newtonian RS:

ddt

∫V

⇀u dV +

∫A

⇀u (

⇀u Rel ·

⇀n ) · dA

=

∫A

ν⇀

∇⇀u ·

⇀n dA −

1ρ

∫V

⇀

∇ PdV +

∫V

⇀g dV . (14)

174 R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181

Fig. 3. PLI for CV’s face pressure calculation.

For the fluid velocity ui the diffusion term (the first termin the rhs of Eq. (14)) is discretised using over-relaxedinterpolation [38]:∫

Aν

⇀

∇ ui ·⇀n dA =

∑faces

ν f⇀

A f · (⇀

∇ ui ) f . (15)

In order to discretise the convection term (the second termin the lhs of Eq. (14)), one needs fluid velocity at CV face ui− fas well as the volumetric flux which appeared in Eq. (16):∫

Aui (

⇀u Rel ·

⇀n )dA =

∑faces

ui− f F f −Rel (16)

where ui− f is calculated using Gamma interpolation scheme[38].

It must be mentioned that the Crank–Nicholson scheme isused for time discretisation of diffusion and convection termsin Eq. (14).

The pressure term (second term in the rhs of Eq. (14)) isdiscretised as:∫

AP

⇀n dA =

∑face

P f⇀

A f . (17)

Using the common Linear Interpolations (LI) for calcula-tion of face pressure P f results in severe oscillations in velocityfield. This is of great importance especially when there are twofluids with high density ratio, e.g. water and air. Here a Piece-wise Linear Interpolation (PLI) is introduced as shown in Fig. 3.It is based on a constraint for lines L A f and L B f which connectpressure values at CVs’ center PA and PB to P f as follows:

Slope of L A f

Slope of L B f=

ρA

ρB(18)

where ρA and ρB are the densities of CVs A and B, respectively.Therefore P f can be estimated by using the pressure value atCVs’ center PA and PB by Eq. (19):

P f = PAκ + PB(1 − κ) (19)

where the weighting factor κ is calculated as follows:

κ =ρBδB

ρAδA + ρBδB. (20)

In this equation δA and δB are distances from face center fto CVs’ center A and B, respectively.In order to show the effect of such an interpolation, thehydrostatic pressure distribution along the near free surface

Fig. 4. Hydrostatic pressure distribution at near free surface CVs’ center.

Fig. 5. Pressure at CV’s centers and faces.

Fig. 6. Effect of interpolation on pressure integral term.

cells (Fig. 4(a)) is plotted in Fig. 4(b). In Fig. 4(a) points andsolid lines stand for CVs’ centers and faces, respectively.

Fig. 5 shows the pressure distribution at CVs’ facescalculated by using LI and PLI. It is obvious from Fig. 5 thatusing LI results in non-physical pressure at CV’s face, but PLIcalculates the CV’s face pressure exactly which is zero for theface at x = 0 m. This difference affects the pressure integral ofeach CV consequently, as shown in Fig. 6. Although it seems

R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181 175

Fig. 7. Solution algorithm.

that such a difference by using LI in comparison to PLI can beneglected, dividing the pressure integral term by CV’s density(Eq. (6)) exaggerates this error. Especially in the case of largedensity ratio of two phases and for the CV next to the freesurface in the light fluid (here at x = 0.5 m) it is very important.Therefore using PLI restrains severe oscillations of the velocityfield by better estimation of P f .

2.3. Solution algorithm

As mentioned before, one encounters three problemsthrough the simulation of a hydrodynamics motion. These partswhich are marked with dashed lines in Fig. 7 are solved in aloop to reach the time history of motions.

2.3.1. Velocity and pressure distributionIn solving the Navier–Stokes and continuity equations one

needs physical properties – density and viscosity distribution –

in the computational domain. Solving the volume fractiontransport equation (Eqs. (10) and (11)) using the CICSAMinterpolation results in effective fluid properties using Eq. (3).For calculation of velocity and pressure fields, the fractionalstep method of Kim and Choi [39] is applied as shown in Fig. 8.

The first step is to solve the momentum equation for thefirst intermediate velocity ui with the lagged pressure gradient(Eq. (21)) and then to calculate the second intermediate velocityu∗

i by Eq. (25). Using Eq. (26) results in new pressure fieldPn+1. Especial care must be taken into account for calculationof the rhs integral of Eq. (26) in boundary faces. In other

words, the fluid velocity at face⇀

Un

f , calculated in the previoustime step, must be used at the boundary face rather than the

intermediate one⇀

U∗

f . Also the intermediate CV’s face velocityis calculated using the linear interpolation. Eqs. (27) and (28)

Fig. 8. Fractional step method of Kim et al. [39].

176 R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181

are used in calculation of fluid velocity at CV’s center andCV’s face, respectively. It must be noted that in the colocatedarrangement used in the current algorithm, face velocitiesmust be calculated separately including the effect of pressuregradient to overcome the checkerboard pressure and Eq. (28)is used for such a purpose. In Eq. (28), LI() is the linearinterpolation of fluid velocity at CV’s center [40].

2.3.2. Motions simulationSolving the Navier–Stokes and continuity equations for an

effective fluid result in flow forces⇀

Fflow and moments⇀

MGflowwhich are calculated by integrating the pressure field andviscous stresses over the body as Eqs. (29) and (30):

⇀

F =⇀

F ext +⇀

W +⇀

Fflow

=⇀

F ext + m⇀g +

n∑j=1

(−Pj⇀n j +

⇀τ j )A j (29)

⇀

MG =⇀

MG−ext +⇀

MGflow

=⇀

MG−ext +

n∑j=1

(⇀r j −

⇀r G) × (−Pj

⇀n j +

⇀τ j )A j (30)

where⇀

F is the total force in the Newtonian RS and⇀

MG is

the total moment around the mass center.⇀

F ext and⇀

MG−extcan be any external forces and moments acting on the bodyin the Newtonian RS. One can use them to simulate propeller

or rudder, towing, sailing, etc.⇀

W = m⇀g is the body weight

force which has only one component along the Z -direction inNewtonian RS. Subscript j stands for each CV’s face definingthe body surface. Also in Eq. (30)

⇀r j is the position vector of

the CV’s face center for all faces defining the body and⇀r G is

the position vector of the body mass center. The important pointis that, according to the selected RS, all vectors must be in theNewtonian RS.

Solving Eqs. (6) and (7) with calculated forces and momentsbased on Crank–Nicholson time discretisation results in bodymotions which are applied on the body-attached mesh in eachtime step. Such a procedure makes ready the physical geometryfor the next time step.

3. Numerical results

In order to assess the feasibility, computer software isdeveloped according to the above mentioned procedure. This issummarized in some conclusions about the accuracy, efficiencyand robustness of the two parts which are two phase flow andhydrodynamic motion simulation.

3.1. Sloshing problem

To test the interfacial flow solver, sloshing of a liquidwave with low amplitude under the influence of gravity isinvestigated. The considered situation is shown in Fig. 9 whichis the same as used by Raad et al. [41]. Initially the quiescentfluid has an average depth of 0.05 m, and its surface is defined

Fig. 9. Initial geometry of the sloshing problem.

by y(x) = 0.05 + 0.005 cos(πx). The domain is discretisedwith 160 cells in the horizontal direction and 104 cells inthe vertical direction. Besides, slip and zero-gradient boundaryconditions are used for velocity and pressure at all boundaries,respectively. In this case, the fluid begins to slosh solely underthe influence of constant gravitational field. The theoreticalperiod of sloshing of the first mode is [41]:

P = 2π√

gk tanh(kh) = 0.3739 s (31)

where k is the wave number and h the average fluid length.Initially the whole system is at rest. After a quarter of a

period the potential energy of the system has been transferred tokinetic energy and the velocities reach their maximum. After ahalf period, all the kinetic energy has been transferred back intopotential energy with the velocity almost back to zero (Fig. 10).Also, Fig. 11 shows plots of the position of the interface at theleft boundary against time. The frequency corresponds with thetheoretical one, so do the amplitudes of even periods.

3.2. Dam breaking with obstacle

A more interesting version of dam breaking occurs whena small obstacle is placed in the way of the water front [42]as shown in Fig. 12. Here, the computational grid of 45 000CVs is used as shown in Fig. 13. The grid must be fineenough, especially around the obstacle, to capture the physics.In addition, no-slip and zero-gradient boundary conditionsare used for velocity and pressure, respectively. In this case,considering the flow of both water and air is important becauseair is trapped in water; due to its much lower density, trappedair is subjected to a large buoyancy force and tends to riseup. This is obvious from Fig. 14 which shows the shape of awater column at three time instants. Comparison of numericaland experimental photos of Fig. 14 shows a good concordanceespecially in capturing the jet of water around the obstaclewhich develops to the bottom of container after t = 0.4 s.

3.3. Water entry of a horizontal circular cylinder

To evaluate the rigid body motion coupling with fluid flow,water entry of a circular cylinder is studied. The neutrally-buoyant circular cylinder and the computational grid of 18 900

R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181 177

Fig. 10. Plots of the wave position for the first period of the sloshing.

Fig. 11. Position of the interface at the left boundary.

CVs is considered as shown in Fig. 15. The cylinder is releasedfrom a position just above the still water level. It intersects thewater surface with a downward velocity of 4 m/s. Here, no-slip boundary condition at cylinder wall, zero value at downboundary and zero gradient at other boundaries are applied onvelocity. Also, the zero-gradient condition is used for pressureat whole boundaries.

After the cylinder impacts on the water surface, the veloc-ity of the cylinder is decreased significantly due to the effectsof hydrodynamic forces. As shown in Fig. 16 for three timeinstants, water spray is thrown up at each side of the cylinderand travel straight upward until they become unstable. Fig. 17shows the time history of vertical displacement and accelerationof the cylinder. The instantaneous vertical positions of the cylin-der are compared with experimental data of Greenhow [43] andnumerical simulation of Xing-Kaeding [35]. It shows a reason-ably good agreement with experimental data in comparison tosimilar numerical studies.

Fig. 12. Illustration of dam breaking with an obstacle.

Fig. 13. Computational grid of 45 000 CVs for dam breaking problem withobstacle.

178 R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181

Fig. 14. Comparison of the numerical simulation of the present study (left) by experimental visualization of Koshizuka et al. [42] (right).

Fig. 15. Geometry and computational grid of water entry of cylinder problem.

3.4. Barge resistance

Ship resistance is usually evaluated with fixed trim anddraft, but they may change in moving conditions due tohydrodynamic effects. In order to show the importance ofmotion simulation – especially in ship resistance calculation– a barge ship is simulated in two cases of fixed and 2-DoFmotion and the numerical results are compared with theexperimental data. The barge model characteristics are shownin Fig. 18 and Table 1. Besides, the computational grid of36 000 cells is shown in Fig. 19. No-slip and zero-gradientboundary conditions are applied for velocity on wall andother boundaries, respectively. Also, zero-gradient boundarycondition is used for pressure in all boundaries. In order tominimize the reflection of flow a damping zone is consideredin the outlet boundaries [44,45].

An experimental test is done at V = 0.807 m/s in the marinelaboratory of Shrif University of Technology. It is obvious thatthe barge ship is not a streamlined body and therefore the wave

Table 1Barge ship characteristics

Characteristic Value

L 1.05 mB 0.29 mDraft 0.025 mCB 1.0Mass 7.26 kgIY Y 0.7 kg m2

K G 0.025 m

Table 2Barge ship resistance

Resistance Value (N) Error (%)

Experiment calculation 3.53 –Numerical calculation in fixed motion 2.71 23.2Numerical calculation in 2-DoF motion 3.32 5.9

making resistance component (deformation of free surface) oftotal resistance is of great importance in comparison to viscousresistance component. Fig. 20 shows the free surface deforma-tion in front of the barge which is of good concordance withexperiment. This results in appropriate prediction of total resis-tance although it could be captured better with a finer mesh.

In the case of 2-DoF motion, a barge is free to heaveand pitch and the resistance is calculated when the twoaforementioned motions are approximately constant. It must benoted that many factors included in the accuracy of resistancecalculation as well as turbulence flow and grid resolution arethe same in fixed and 2-DoF motion. Taking this into account,the error is decreased from 23.2% in fixed motion into 5.9% in2-DoF motion which is similar to test conditions as shown inTable 2.

R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181 179

Fig. 16. Free surface deformation in cylinder water-entry; (a): simulation, (b): experiment [43].

Fig. 17. Time history of the vertical motion (left) and the impact force (right) during the cylinder water entry.

Fig. 18. Barge ship geometry definitions.

In other words, since the model test is performed with afree model (heave and pitch motions were not restricted), thenumerical results in 2-DoF motion simulation is much betterthan for fixed condition. This means that in usual numericalresistance we always have errors if we cannot predict themoving condition (draft and trim) with good accuracy. Butthe present numerical method can find moving conditionsautomatically.

4. Conclusion

A basic mathematical formulation for motion simulationwith or without free surface is explained and details of

Fig. 19. Computational grid of barge ship resistance calculation.

numerical discretisation procedures are presented. The methodmay be applied for prediction of any 3D hydrodynamic motion

180 R. Panahi et al. / Applied Ocean Research 28 (2006) 171–181

Fig. 20. Free surface deformation in front of barge ship: (a): experiment, (b): simulation.

as well as ship and submarine simulation. Computer softwareis developed based on the derived method and validatedwith different numerical simulations such as sloshing andslamming. In all cases very good agreement with availabledata is achieved. Therefore the software may be used as a3D numerical hydrodynamic tank. As the first step, a bargeship in steady forward motion is simulated and its resistanceis evaluated with 2-DoF. Results show the possibility for fullynonlinear simulation of ship motions.

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