6dof Rigid Motion Cfd

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J Mar Sci Technol (2002) 7:31–42

Finite-volume simulation method to predict the performance of a sailing boatH iromichi A kimoto 1 and H ideaki M iyata 2

1 Department of Applied Mathematics and Physics, Tottori University, Minami 4-101, Koyama, Tottori 680-8552, Japan2 Department of Environmental and Ocean Engineering, The University of Tokyo, Tokyo, Japan

roll, yaw, pitch, sway, and heave motions, which areoften neglected in the case of a low-speed ship.

Predicting the performance of a sailing boat is an

interesting topic in ship hydrodynamics owing to theircomplicated performance, which is different from thatof commercial ships. 4 In upwind sailing conditions, theboat cruises with a large heel angle to obtain thrust fromthe wind and a small leeway angle to generate lateralforce to cancel the unwanted component of the sailforce. The attitude of the boat depends on the hydro-and aerodynamic forces and moments acting on its hull,sails, keel, and rudder. These forces must be evaluatedin order to solve the equation of motion of the boat(Fig. 1). It is difcult to investigate the performance of asailing boat in an experimental facility because of thecomplexities of the sailing dynamics and wind condi-tions. In a towing tank, we can only postulate the windaction and the resultant balance of the boat by adjustingthe towing position and ballast weights on a model ship.

A method of overcoming these difculties is toextend the CFD techniques so that the freely movingproperties of the boat can be incorporated. Somestudies have attempted to realize self-propelling andsteady maneuvering motions by the CFD technique.Most of these have been numerical realizations of atowing test, where a ship is xed in a computational gridsystem. We can remove this restriction by solving theequation of motion of the ship simultaneously with the

CFD simulation, including the changes in geometry of the free surface and the body boundary.In this work, a ow simulation code WISDAM-VII 5 is

developed for the ow around a freely moving ship, inparticular for the large motions of a sailing boat (Fig. 2).It employs the nite-volume method in the frameworkof an O-O-type structured grid system that is tted toboth the free surface and the hull surface. The gridsystem is generated at each time step. Deformations of the computational domain are treated by a waterlinesearch procedure and a moving grid method that is

Abstract A ow-simulation method was developed to predictthe performance of a sailing boat in unsteady motion on a freesurface. The method is based on the time-marching, nite-volume method and the moving grid technique, includingconsideration of the free surface and the deformation of theunder-water shape of the boat due to its arbitrary motion. Theequation of motion with six degrees of freedom is solved bythe use of the uid-dynamic forces and moments obtainedfrom the ow simulation. The sailing conditions of the boatare virtually realized by combining the simulations of water-ow and the motion of the boat. The availability is demon-strated by calculations of the steady advancing, rolling, andmaneuvering motions of International America’s Cup Class(IACC) sailing boats.

Key words Free surface · Moving boundary · Unsteadymotion · Sailing boat

Introduction

Recent advances in computational uid dynamics(CFD) enables us to predict the performance of a shipin steady advancing motion. 1 There have also beensome attempts to evaluate the maneuvering abilities of a ship by CFD techniques. 2,3 However, most of thesemotions are restricted within steady two-dimensionalmotion, e.g., steady circling or obliquely advancingmotions.

In the case of high-speed ships, the attitude of a shipdepends on its forward speed because of the largedynamic pressure acting on its hull. Experimental stud-ies are difcult owing to the large amplitude motion of the ship. In the course of maneuvering, a ship makes

Address correspondence to: H. Akimoto(e-mail: [email protected])Received: December 25, 2001 / Accepted: March 26, 2002



32 H. Akimoto and H. Miyata: Predicting sailing boat performance

similar to that of Rosenfeld and Kwak. 6 The uid-dynamic forces obtained in the CFD simulation areintroduced into the equation of motion of the boat.Although the forces and moments acting on the sails,keel, and rudder are given by empirical equations inthis method, this system provides the basis of a virtualreality system for all sailing boats.

Fluid-dynamics simulation

The CFD code WISDAM-VII was developed to simu-late the incompressible viscous water ow around a shipin arbitrary motion. 5 Assuming that the interactions of the ow between the hull and other parts of the boat aresmall, the CFD simulation is performed for the hull onlyin this method. Thus, the target of the system is thehydrodynamic evaluation of hull con gurations underthe in uences of all the lifting surfaces.

Governing equations

The governing equations are the conservation laws of momentum and mass in control volumes which deformtime-dependently. They are expressed as

dd

d dt

V V t S t

u T S ( ) ( )Ú Ú = ◊ (1)

dd

dt

V t S t

( ) = -( )◊( )Ú u v S (2)

where V (t ) and S(t ) denote the volume and surface areaof the control volume, respectively, u is the uid veloc-ity vector, and v is the moving velocity of the surface of the control volume. d S is the product of the in nitesimalarea element d S and the outward normal vector n on thesurface of the control volume. Using the eddy viscositymodel for Reynolds stress, the stress tensor T isexpressed as

T u v u I u u= - -( ) - + +Ê Ë Á ˆ ̄ ˜ — + —( )ÈÎÍ ˘̇̊P t T 1

Ren (3)

where P is the kinematic pressure de ned by P = p /r -z/Fn 2, p is the pressure, r is the density of water,Fn (=U /÷ g

—L ) is the Froude number, and U and L are the

standard speed and length of the boat, respectively. I is the unit tensor, Re ( =UL /n ) is the Reynolds number,and n t is the kinematic eddy viscosity. Equations 1and 2 are solved numerically by a MAC-type time-marching algorithm. The spatial discretization of theusual convection ux uu is by the 3rd-order upwindscheme. The additional ux due to grid motion vu istreated by the method of Rosenfeld and Kwak. 6 Tosatisfy the minimum geometric conservation, this usesthe momentum in the volume of a hexahedron which isswept by the face under consideration due to the motionof the grids. The pressure and the diffusive uxes areevaluated by 2nd- and 1st-order spatial descretiza-tion, respectively. Time discretization is 1st-orderEuler-explicit.

Boundary condition for velocity

We assume that the boat is rigid. Then the velocity of the surface vb on the hull is determined from the motionof the boat and the relative position of the point of interest x b from the gravitational center.

v v x x b G G b G= + ¥ -( )w (4)

where x G is the center of gravity of the boat, vG is theforward velocity of x G, and w G is the angular velocityvector of the boat with respect to x G. To impose the no-slip boundary condition ( u = vb), we use dummy velocitypoints located in the body (Fig. 3). Their velocity

Fig. 1. Coordinate system

Fig. 2. Block diagram of the performance prediction system



33H. Akimoto and H. Miyata: Predicting sailing boat performance

vectors udummy are de ned by the no-slip condition andliner extrapolation from the uid velocity of the nearestpoint u1.

u v udummy b= -2 1 (5)

Without disturbing the no-slip condition, surface gridpoints slip on the hull surface according to the regenera-tion of the grids at every time step, so that excessivedistortion of the grid system does not occur even whenthe boat makes a large rolling motion (Fig. 4).

Boundary condition of pressure

To evaluate the pressure on the moving body surface,an ALE (arbitrary Lagrangian and Eulerian) form of the Navier –Stokes equation is employed on the bodysurface.

∂∂ + -( )◊— = -— + —u

u v u ut

P b Re1 2 (6)

Imposing the no-slip condition u = vb with Eq. 6, weobtain an equation for the pressure gradient on themoving body surface.

—( ) = - ∂∂ + —Ê Ë Á

ˆ ¯ ˜ P

t bodyb

bodyRe

vu

1 2 (7)

In Eq. 7, the rst term on the right-hand side is theacceleration of the body surface determined by Eq. 4.By taking the inner product of Eq. 7 and the outwardnormal vector on the body surface nb, the normal pres-sure gradient on the moving body surface is obtained inthe form

∂∂ = - ◊ ∂∂

P n t b

bbn

v(8)

Here, the contribution of the diffusion term is neglectedbecause it is parallel to the body surface in the boundarylayer. Pressure at the dummy point is extrapolated usingthe normal pressure gradient ∂P /∂nb in Eq. 8. When theacceleration of the body is zero, this is equivalent to theusual zero –normal-gradient condition for pressure.

Free surface condition

We assume here that there is no turning over or break-ing motion of free surface waves. Then the wave heightis expressed by a single-valued function z = h( x, y, t ).Although this assumption is not rigorously valid in thebow region, this is restricted to a small area and has arelatively small in uence on the dynamics of Interna-tional America ’s Cup Class (IACC) boats. The kine-matic and dynamic boundary conditions are given onthe free surface. The kinematic condition is expressedas

∂∂ = = - -( ) ∂∂ - -( ) ∂∂ht

v u u v h x

u v h x3 3 1 1

12 2

2(9)

where u = [u1, u 2, u3] is the uid velocity vector on thefree surface, and x = [ x1, x2, x3] and v = [v1, v2, v3] are theposition and velocity vectors, respectively, of grid pointson the free surface. With the assumption that the sur-face tension and viscous stress on the free surface arenegligibly small, the dynamic condition is written as

P p x x

n= - = - ∂

∂ =a2 2

sFn Fnr 3 3 ,

u0 (10)

(on the free surface)

where p a is the atmospheric pressure on the free surface,and ∂u/∂n s is the velocity gradient in the direction nor-mal to the free surface. The quantity p a is assumed to bea constant. We can set p a = 0 without loss of generality.

Model of turbulence

A hybrid turbulence model is employed to evaluate thekinematic eddy viscosity n t . This is a combination of the

Fig. 3. Body boundary condition for velocity

Fig. 4. Deformation of grids in a cross section




Baldwin –Lomax algebraic turbulence model 7 (BL) andthe Smagorinsky eddy viscosity 8 in the subgrid-scaleturbulence model (SGS). The BL model is applied onthe fore part of the hull where the boundary layer is thinand locally two-dimensional. However, in the rear partof the body and in its wake region, the boundary layerrapidly increases in thickness 10 and the BL model tends

to overestimate the eddy viscosity. Therefore, weestimate n t by the following equation:

n n

bn b n

n

t t

t t

t

x x

x x x

x x x

x x

=

<( )< <( )

+ -( ) < <( )<( )

Ï

Ì

ÔÔÔ

Ó

ÔÔÔ

0

1

FP

BLFP MID

BL SGSMID AP

SGSAP

(11)

where n t BL and n t SGS are kinematic eddy viscositiesobtained from the BL and SGS model, respectively. 1

xFP , xMID , and xAP denote the x-coordinates of the foreend point, mid-ship position, and after end point of thewetted surface of the hull, respectively. The parameterb is selected as b = ÷ s

—( x)/ s

—( x —

MID ), where s( x) is the sec-tion area that is the longitudinal distribution of the hullvolume beneath the still water plane in the uprightposition. 1 b is 1 at the midship, decreases in the aftdirection, and becomes 0 at x AP .

In the original BL model, small calculated n t BL owsare treated as laminar. However, we assumed that theturbulent boundary layer starts from x FP even at a rela-tively low Reynolds number condition. This is becauseour target is the predicition of forces on the full-scale

boat.n t SGS is determined by

n t ij C y SSGS = - -( ){ }[ ]+s expD 1 25

2

where y+ is the dimensionless wall distance, D is theminimum grid spacing of the local control volume,and |Sij | is the magnitude of the strain-rate tensor. TheSamgorinsky constant C s is 0.5 in this calculation. Theestimate of the eddy viscocity changes gradually fromthe BL model to the SGS model in the aft half of thehull. In the wake region after x AP , the eddy viscocity isdetermined by the SGS model only.

Motion of the boat

Equation of motion of the boat

The equation for the transverse motion of the boat iswritten as

mt

dd

Ghull sail rudder keel damp

v F F F F F = + + + + (12)

where m is the mass of the boat, and the right-hand-sideterms are forces acting on the main hull, sails, rudder,and keel, respectively. These forces include hydrody-namic or aerodynamic forces and gravity. The last term, F damp , is an arti cial damping force. This is added toreduce unnecessary transient oscillations of the boatwhen the target is a steady-state solution. The conserva-

tion law for the angular momentum has a similar form.dd

Ghull sail rudder keel damp

hM M M M M

t = + + + + (13)

where hG is the angular momentum vector of the boatwith respect to its center of gravity. The right-hand-sideterms are the moment vectors of all parts and the arti -cial damping moment.

A ship has natural damping mechanisms due to theviscous effect and the dynamic forces of its appendages.Since their magnitude of damping is usually small, occa-sional transient oscillating motions of the boat waste

considerable CPU time. Therefore, F damp and M damp areadded to reduce the CPU time for unnecessary transientmotions. These are set at zero if the unsteady motion of the boat must be simulated rigorously.

F hull is expressed as

F S D S e hull 2hull

2Fnd d

Fn= - -Ê

Ë Á ˆ

¯ ˜ - ◊ -Ú Ú P z m

S S 3 (14)

where Ú S d S means the surface integration on the wettedsurface of the hull, m hull is the mass of the hull, D is theviscous stress tensor, and e 3 is a unit vector orientingvertically upward. The right-hand-side terms are the

pressure force, viscous force, and gravitational force,respectively. The hydrodynamic and gravitational mo-ments on the hull are express as

M x x n

x x D n x x e

hull 2 G

Ghull

2 hull G

Fnd

dFn

= - -Ê Ë Á

ˆ ¯ ˜ -( ) ¥{ }

+ -( ) ¥ ◊( ) - -( ) ¥

Ú

Ú

P z

S

S m

S

S 3

(15)

where ( x - x G) is the position vector originating from x G,and x hull is the center of gravity of the main hull. It

should be noted that the buoyancy of the hull is in-cluded in the surface pressure integration that includesthe gravity potential. The effect of added mass aroundthe hull is also included in this integration.

Correction of the viscous force

It is not practical to perform a simulation of a full-scaleboat with our limited CPU power because the Reynoldsnumber of such a boat is about 10 8. Therefore, we haveto estimate the motion of a full-size boat from the com-




putational results with a lower Reynolds number. Forthis purpose, only the frictional component of forcecalculated by CFD is extrapolated to that of the full-sizeboat as

f f ship

ship

CFD CFD= C

C f

f (16)

where f ship is the estimated frictional force of the full-scale ship, and f CFD is the computed frictional force, i.e.,integrated tangential stress on the wetted surface of thehull. C f ship and C f CFD are the frictional force coef cientsof a at plate at the Reynolds number of the full-scaleship and the CFD simulation, respectively. Althoughthe range of Reynolds numbers in the CFD simulationsis from 10 5 to 10 6, the calculated pressure force can beused in the equation of motion of the full-scale boat,with a small error only, because the scale effect is be-lieved not to be large for this component.

Models of appendages

The uid-dynamic forces acting on sails, rudder, andkeel are evaluated by a wing theory and empiricalequations.

For example, we describe here the model of the rud-der. The advancing velocity vector of the rudder vrudder isexpressed as

v v x x rudder G G rudder G= + ¥ -( )w (17)

where x rudder is the position vector of the rudder. Thenthe uid velocity relative to the rudder is

v u va rudder= - (18)

A local coordinate system is used to decompose thehydrodynamic force into the drag and lift components(Fig. 5). The new coordinate system is chosen as

X v v Y n X Z X Y r a a r r r r r r= = ¥ = ¥, , (19)

where n r is the unit normal vector of the rudder. Theangle of attack a is

a p = - ◊( )-2 1cos X nr r (20)

where a is positive when the lift force is in the Z r direc-tion. The hydrodynamic force on the rudder is

F Z X vrudder L r D r a r= +( )C C S12

2r (21)

where C L and C D are the lift and drag coef cients, re-spectively, and Sr is the area of the rudder. There areinteractions among the appendages and the hullthroughout the ow eld. Because of the complexitiesof these interactions, we incorporate here only the in-duced velocity of the forward-mounted keel. The uiddynamic coef cients in Eq. 21 are determined by thefollowing empirical equations 5:

C C

C C

LL L

keel

keel 3Ddd

C= -Ê Ë Á

ˆ ¯ ˜ a

a l 1

C C C C C

D L D0L2

rudder= +( ) +1 2 pl

here l rudder and l keel are aspect ratios of the rudder andkeel, respectively, and C Lkeel is the lift coef cient of thekeel. d C L/da , C D0, and C 3D are coef cients of the lift-curve slope, the base drag, and the three-dimensionalcorrection factor, respectively. C 1 and C 2 are factorsof the interaction between the keel and the rudder.These coef cients are obtained from experiments with amodel rudder. The forces acting on the sails and thekeel are evaluated in a similar manner.

Although the effect of added mass around the hull is

included in the CFD part, the added mass of some of theappendages have not yet been fully considered.As shown in Eq. 21, the treatments of wing-like ap-

pendages are steady-state approximations. Because thechord length c of these appendages is relatively small inrelation to the boat length L , the reduced frequency of the boat motion with a wing, ( U /L )/(U /c), is about 0.02 –0.04. This means that a quasi-steady approximationwhich includes the added mass of the wings is not re-quired in calm water conditions. The added massaround the sail and the bulb is not explicitly included inthe present simulation, because a rough estimate of their effect was smaller than the uncertainty of the totalinertial moment of the boat.

A more precise estimation of the inertia and thequasi-steady approximation of the wings will be re-quired when the predicted performance of a boat inwaves is considered.

Grid generation

Because of the arbitrary motion of the boat and thedeformation of the free surface, the wetted part of theFig. 5. Hydrodynamic forces acting on the rudder




hull changes its shape in a time-dependent manner. Tocope with the hull con guration of a sailing boat in largechanges of attitude, an O –O-type boundary- tted struc-tured grid system was selected, as shown in Fig. 6. Thisspherical grid system is regenerated at each time stepafter the moving boundaries of the free surface and thehull have settled, so that the distortion of the grid re-mains small even for a large-amplitude motion.

Surface modeling of the hull

For ease in handling the hull con guration, a cylindricalcoordinate system was de ned, as shown in Fig. 7. Theposition vector on the hull surface X = [ X , Y , Z ] inthe hull- xed coordinate system is expressed by twoparameters of X s and q s:

X X Y r X Z r X = = ( ) = - ( )s s s s s s s, , cos , , sinq q q q (22)

where X s is the longitudinal position on the ctitioushull axis, q s is the angle from the center plane to thepoint, and r ( X s, q s) is the distance from the axis to thesurface point. The function r ( X s, q s) is designed to calcu-

late r from X s and q s using second-order interpolationfrom the given data points of the hull ’s shape. The vec-tor X is converted to the point of the ground- xed coor-dinate system x according to the position and attitude of the boat. For concise notation, we express x as x = r ( X s,q s) ∫ E ( X ( X s, q s)), where E ( X ) means the conversionfrom the body- xed coordinate to the ground- xed co-ordinate system using transverse and rotationaltransformations.

The con guration of the hull is given in the rowof position vectors generated by a CAD application.These are loaded and then converted to the cylindricalcoordinate system in CFD code. This implementationconceals the discreteness of the geometrical data in thefunction r ( X s, q s). It then become possible to handle thecomplicated geometric changes in the boundaries inthe simulation.

Search procedure for the waterline

In the rst step of the grid generation, the position of the waterline, i.e., the intersection of the free surfaceand the hull surface, is determined. Because these twosurfaces deform owing to the wave motion and thechange in the boat ’s attitude, there is no easy way todetermine the waterline.

The determination of the waterline is based on abisection search on the hull surface. If there are twopoints on the hull surface where one is above the freesurface and the other is under the free surface, the

Fig. 6. a O –O-type grid system, and b a close-up around thehull

Fig. 7. Local cylindrical coordinates representing the hull ’ssurface




position of the waterline between them is determined asdescribed below (Fig. 8).

1. Set initial points x dry = r ( X dry, q dry) and x wet = r ( X wet,q wet), where X dry, q dry and X wet, q wet are parametricexpressions of given dry and wet points.

2. Calculate the midpoint x m between x dry and x wet on thehull surface, where x m = [ xm, ym, zm] = r ( X m, q m), as

X X X m dry wet m dry wet 2= +( ) = +( )2, q q q

3. Replace x wet or x dry with x m according to the local freesurface.

If then replace withm m m wet mz h x y< ( ), , x x

If then replace withm m m dry mz h x y> ( ), , x x

where z = h( x, y) is the local free surface plane.Procedures 2 and 3 are repeated until wet and dry

points converge into a single point. The result gives thelocal waterline position.

Search procedure for the fore and the aft end points

In the case of a sailing boat, the fore end point x FP andthe aft end point x AP of the wetted surface move a lotowing to the special hull form con guration. They alsomove time-dependently owing to the motion of the boatand the free surface. We must determine these twopoints at the rst step of the grid generation becausethey are pole points of the O –O grids.

x FP is determined by the repeated use of the waterlinesearch (Fig. 9).

1. Set the initial section X = X s, i.e., on the downstreamside of x FP.

2. Search the waterline positions of both the starboardside x L and the port side x R of the section.

3. Obtain the midpoint x M between x L and x R on thehull ’s surface.

Fig. 8. Bisection search for the waterline

Fig. 9. Search procedure for the fore end point ( FP ) by the

bisection method

4. Search the waterline position x FP on the forwardstretching line from x M.

5. Replace X s with X FP, and then go to point 2.

Procedures 2 –5 are repeated until the three points x L, x R, and x FP converge into one point. That point becomes x FP at that moment. The search procedure for x AP isperformed in the same manner. x FP and x AP are con-nected by two waterlines on the port and starboard

sides by the repeated use of the bisection search for thefree surface around the hull.

Distribution of the surface and volume grids

After the determination of the two waterlines, bodyboundary grids are distributed on the wetted surfacebetween the waterlines. Then the free surface grids aregenerated from the waterlines to the outer boundary.The height of each grid point from the still water planeis obtained from Eq. 9. The inner-volume grids are alge-braically distributed between the hull surface and theouter boundary. Whole grids are regenerated at everytime step.

Numerical results

Simulation procedure

The simulation procedure using this method is verysimilar to that of a model test in a towing tank. It in-cludes setting a boat a oat, adjusting the still waterlinewith ballast weights, setting the attitude, accelerating




the boat (by towing), and taking the measurements. Allof these are performed numerically.

In a typical acceleration procedure, all degrees of freedom except heave are xed to prevent unnecessaryoscillations of the boat. When the boat reaches a steadyforward speed, one can select suitable conditions of binding or release and the magnitudes of damping forthe test, in all six degrees of freedom. For example, inthe leeway –heel test, the angles of leeway and heel aregradually changed to the target position to be mea-sured. In this case, all degrees of freedom except heaveand pitch are xed. Arti cial damping terms are addedto reduce unnecessary oscillation. This is effective if time-averaged properties are the main concern.

Steady forward motion

The accuracy of the WISDAM-VII code was tested in asteady forward motion case. Figure 10 shows the distri-bution of the wave pro le along a typical IACC boat,JN35, without appendages and in an upright position.The measurements were performed in a towing tankat the University of Tokyo with a one-seventh scalemodel, and using a still camera. The target speed of the

full-size boat was 9.0 knots (Froude number 0.35,Reynolds number for this experiment 4 ¥ 106). Thenumber of control volumes was 45000. The results showthat the accuracy of the simulation was satisfactory.Although this simulation did not include local over-turning and breaking waves around the bow, the peakposition of the pro le showed good agreement.

Figure 11 shows a comparison of the pressure distri-butions on the wetted surface area. In this case an IACCboat, JN32, was in steady sailing motion with a heelangle of 21.3 ° and a leeway angle of 2.0 °. The pressure

was measured at 180 pressure holes located on the hullbeneath the still-water plane. The area of the contourmap measured was narrower than that of the computa-tion. This is because the probes were not set on the hullsurface above the still-water plane where elevated free

surface reaches only when the model is in forward mo-tion. However, the level of agreement was satisfactory.The asymmetrical pattern of the pressure is wellrealized in the computation.

Figure 12 shows a comparison of wave heights on acontour map. The wave height close to the hull was notmeasured in the experiment because we could not setthe wave height probes in the path of the towed model.Although the main wave patterns are captured in thissimulation, crests of small wavelength are dissipatedrapidly in the propagating process. This is because of

Fig. 10. Distribution of the wave height divided by L alongthe hull, Fraude number (Fn) = 0.34, NI is the number of control volumes in the longitudinal direction along the hull

Fig. 11. Comparison of the measured and computed surfacepressure ( C p) distribution: heel = 21.3°, leeway = 2.0°, contourinterval = 0.02




insuf cient resolution by the coarse free surface gridsand their rapid outward expansion. However, the agree-

ment of the wave pro le along the hull suggests that themain part of the ow aound the hull is qualitatively wellreproduced.

Forced roll motion

To show the applicability of this method to the unsteadymotions of a boat, a simulation of a forced rollingmotion was conducted. In this simulation, an IACCboat sailing at a constant speed (Froude number 0.35) inan upright position starts rolling with an angular veloc-

ity of 30 ° per nondimensional time unit. This is abouttwice as fast as in the nominal tacking motion of IACCracing boats. Only heave and roll motions are solved,and the other four modes of motion are xed for sim-plicity. The axis of roll is set at the center of gravity of the boat. Figure 13 shows the pressure and velocitydistributions in the transverse sections at two different

times. It shows the occurrence of a high-pressure regiondue to the acceleration of the hull ’s surface, and thelarge deformation of the free surface around the boat insection (a). Although comparative experiments havenot yet been performed, the results show the potentialof this method to treat the unsteady rotational motionson the free surface.

Comparison of course-changing maneuvers

This section considers an example of a typical maneu-vering motion in upwind sailing. An IACC boat (JN35)is in sailing motion at a constant speed of 9 knots with atrue wind angle of 45 ° and a leeway angle of 3 °. Initially,the angles of its rudder and trim tab are adjusted tobalance force and moment. The boat is then ordered tochange its heading by 4 °, so that it obtains a larger sailforce according to the increased true wind angle.This maneuver is performed by automatic control of therudder. The control routine of the rudder is a combina-tion of proportional and differential control methods

dd

ddp d

f t

G Gt

= +j j

(23)

where f is the rudder angle, j is the error of the headingangle, and G p and G d are gain parameters. In this case,pitch, roll, and heave motions are xed to simplifythe situation. Two cases of simulation were performedfor different gain values. In case 1, G p and G d were setat 3.75 and 3.0, respectively, and in case 2 they were1.88 and 3.0, respectively. The maneuver in case 2 wasexpected to be gentle with a smaller proportionalgain.

Figure 14 shows the pressure distributions on the hullin different times. At the beginning of the maneuver,there is a high-pressure region in the bow due to theyawing motion. Then the pressure magnitude decreases

when the angular velocity is approximately constant.The time-historical variations of the boat ’s speed in Fig.15 show that in case 2, the boat is accelerating moresmoothly and reaches a higher speed. The result impliesthat the maneuver in case 2 is superior to that in case 1in this situation because it steers the boat relativelygently. Figure 16 shows a series of pictures of themaneuvering motion.

This simulation of changes in course shows that thepresent method can be used to predict and to optimizethe performance of sailing boats. The simulation shows

Fig. 12. Wave-height contours. Positive values are shown asbold lines and negative values as thin lines . The contour inter-val is 1 ¥ 10- 3 of L , heel = 25°, leeway = 3.0°, Fn = 0.34, Re = 105

(computed) and 4.2 ¥ 106 (measured)




uid-dynamic data of the unsteady motions of a ship.

It is dif cult to obtain these data from experimentsbecause of the limitations of tank facilities and thecomplexity of boat motions. Thus, these examples showthe potential of this simulation method in the unsteadymotion of vessels on a free surface.

Conclusion

A new simulation method for the ow around a sailingboat in motion has been developed. The method is

based on the time-marching, nite-volume method and

the moving grid technique, with consideration of theunsteady motion of the free surface and the deforma-tion of the under-water geometry due to the motion.The equation of motion of the boat was solved simulta-neously with that of the ow eld around the hull usingthe uid-dynamic forces and moments obtained fromthe ow solver. The sailing condition of the boat wasvirtually realized in the simulation. The simulation re-sults given show the potential of this method to evaluateunsteady large-amplitude motions of moving vessels ona free surface.

Fig. 13. Pressure and velocity distributionsin rolling motion at two different times.Contour interval = 0.04, Re = 106. Positive(negative) values are shown by solid(dotted ) lines




Fig. 14. Pressure ( C p) distribution on the hull ’s surface inmaneuvering motion at different times. Contour interval =0.04, time interval = 0.25, Re = 105, positive/negative values areshown by bold /thin lines

Fig. 15. Changes in forward speed V s = |vG|/U in time

Fig. 16. Visualization of the maneuvering motion in 4 ° of course change, case 2. a t = 4.0 (maneuvering start); b t = 4.6(elapse time 0.6); c t = 5.2 (elapse time 1.2)




Acknowledgments. This research was partly supportedby a Grant-in-Aid for Scienti c Research of theMinistry of Education, Science and Culture and bythe AC Technical Committee of the Ship and OceanFoundation.

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