J Mar Sci Technol (2002) 7:31^2 Journal of

Marine Science and Technology © SNAJ 2002

Finite-volume simulation method to predict the performance of a sailing boat

HiROMicHi AKIMOTO ' and H I D E A K I MIYATA^

' Department of Applied Matliematics and Physics, Tottori University, Minami 4-101, Koyama, Tottori 680-8552, Japan 2 Department of Environmental and Ocean Engineering, The University of Tokyo, Tokyo, Japan

Abstract A flow-simulation method was developed to predict

the performance of a sailing boat in unsteady motion on a free

surface. The method is based on the time-marching, finite-

volume method and the moving grid technique, including

consideration of the free surface and the deformation of the

under-water shape of the boat due to its arbitrary motion. The

equation of motion with six degrees of freedom is solved by

the use of the fluid-dynamic forces and moments obtained

f rom the flow simulation. The sailing conditions of the boat

are virtually realized by combining the simulations of water-

flow and the motion of the boat. The availability is demon­

strated by calculations of the steady advancing, rolling, and

maneuvering motions of International America's Cup Class

( l A C C ) sailing boats.

Key words Free surface

motion • Sailing boat

IVIoving boundary • Unsteady

Introduction

Recent advances in computational f luid dynamics (CFD) enables us to predict the performance of a ship in steady advancing motion.^ There have also been some attempts to evaluate the maneuvering abilities of a ship by CFD techniques.^.^ However, most of these motions are restricted within steady two-dimensional motion, e.g., steady circling or obhquely advancing motions.

I n the case of high-speed ships, the attitude of a ship depends on its forward speed because of the large dynamic pressure acting on its huh. Experimental stud­ies are difficult owing to the large amplitude motion of the ship. In the course of maneuvering, a ship makes

Address correspondence to: H . Akimoto (e-mail: akimoto@damp.tottori-u.ac.jp) Received: December 25, 2001 / Accepted: March 26, 2002

rol l , yaw, pitch, sway, and heave motions, which are often 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 their comphcated performance, which is different f rom that of commercial ships."» I n upwind sailing conditions, the boat cruises with a large heel angle to obtain thrust f rom the wind and a small leeway angle to generate lateral force to cancel the unwanted component of the sail force. 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 evaluated in order to solve the equation of motion of the boat (Fig. 1). I t is difficult to investigate the performance of a sailing boat in an experimental facility because of the complexities of the saihng dynamics and wind condi­tions. I n a towing tank, we can only postulate the wind action and the resuhant balance of the boat by adjusting the towing position and ballast weights on a model ship.

A method of overcoming these difficulties is to extend the CFD techniques so that the freely moving properties of the boat can be incorporated. Some studies have attempted to realize self-propelling and steady maneuvering motions by the CFD technique. Most of these have been numerical realizations of a towing test, where a ship is fixed in a computational grid system. We can remove this restriction by solving the equation of motion of the ship simultaneously with the CFD simulation, including the changes in geometry of the free surface and the body boundary.

I n this work, a flow simulation code W I S D A M - V I P is developed for the flow around a freely moving ship, in particular for the large motions of a sailing boat (Fig. 2). I t employs the finite-volume method in the framework of an O-O-type structured grid system that is fitted to both the free surface and the hull surface. The grid system is generated at each time step. Deformations of the computational domain are treated by a waterline search procedure and a moving grid method that is

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

V

Fig. 1. Coordinate system

Governing equations

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

df JK(() h(t]

dt ^' H'V '

(1)

(2)

where V(t) and S(t) denote the volume and surface area of the control volume, respectively, u is the fluid veloc­ity vector, and v is the moving velocity of the surface of the control volume. dS is the product of the infinitesimal area element d^ and the outward normal vector // on the surface of the control volume. Using the eddy viscosity model for Reynolds stress, the stress tensor T is expressed as

-{u - v)ii -PI + Re

• + v. (3)

Models of appendages

^ rudder) force Equation of motion with

6 degrees of freedom

force CFD simulation ofthe hull

"WISDAM-Vir

ction of the shin 'I

( Control j

Fig. 2. Block diagram of the performance prediction system

similar to that of Rosenfeld and fCwak.*" The fiuid-dynamic forces obtained in the CFD simulation are introduced into the equation of motion of the boat. Although the forces and moments acting on the sahs, keel, and rudder are given by empirical equations in this method, this system provides the basis of a virtual reahty system for aU saihng boats.

Fluid-dynamics simulation

The CFD code W I S D A M - V I I was developed to simu­late the incompressible viscous water flow around a ship in arbitrary motion.^ Assuming that the interactions of the flow between the hull and other parts of the boat are small, the CFD simulation is performed for the hull only i n this method. Thus, the target of the system is the hydrodynamic evaluation of hufl configurations under the infiuences of all the l if t ing surfaces.

where P is the kinematic pressure defined by F = p/p -z/Ftf, p is the pressure, p is the density of water, Fn (=U/^[gL) is the Froude number, and U and L are the standard speed and length of the boat, respectively. ƒ is the unit tensor. Re (=UL/v) is the Reynolds number, and V, is the kinematic "eddy viscosity. Equations 1 and 2 are solved numerically by a MAC-type time-marching algorithm. The spatial discretization of the usual convection flux uu is by the 3rd-order upwind scheme. The additional flux due to grid motion vu is treated by the method of Rosenfeld and Kwak.'' To satisfy the minimum geometric conservation, this uses the momentum in the volume of a hexahedron which is swept by the face under consideration due to the motion of the grids. The pressure and the diffusive fluxes are evaluated by 2nd- and Ist-order spatial descretiza-tion, respectively. Time discretization is Ist-order Euler-explicit.

Boundary condition for velocity

We assume that the boat is rigid. Then the velocity of the surface on the hull is determined f rom the motion of the boat and the relative position of the point of interest Xf, f rom the gravitational center.

VK =Vn+COrX ( • V b - - V G ) (4)

where .VQ is the center of gravity of the boat, VQ is the forward velocity of XQ, and COQ is the angular velocity vector of the boat with respect to XQ. TO impose the no-slip boundary condition (u = vJ, we use dummy velocity points located in the body (Fig. 3). Their velocity

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

body boundary MQ (dummy point)

Fig. 3. Body boundary condition for velocity

Fig. 4. Deformation of grids in a cross section

vectors «dummy are defined by ttte no-slip condition and liner extrapolation f rom the fluid velocity of the nearest point

ll dummy (5)

Without disturbing the no-slip condition, surface grid points slip on the hull surface according to the regenera­tion of the grids at every time step, so that excessive distortion of the grid system does not occur even when the boat makes a large rolling motion (Fig. 4).

Boundary condition of pressure

To evaluate the pressure on the moving body surface, an A L E (arbitrary Lagrangian and Eulerian) fo rm of the Navier-Stokes equation is employed on the body surface.

(6)

Imposing the no-slip condition u = v^, with Eq. 6, we obtain an equation for the pressure gradient on the moving body surface.

(VP) body dt Re

(V) 'body

I n Eq. 7, the flrst term on the right-hand side is the acceleration of the body surface determined by Eq. 4. By taking the inner product of Eq. 7 and the outward normal vector on the body surface «,,, the normal pres­sure gradient on the moving body surface is obtained in the form

dP

dt (8)

Here, the contribution of the diffusion term is neglected because it is parahel to the body surface in the boundary layer. Pressure at the dummy point is extrapolated using the normal pressure gradient dPIdn^ in Eq. 8. When the acceleration of the body is zero, this is equivalent to the usual zero-normal-gradient condition for pressure.

Free surface condition

We assume here that there is no turning over or break­ing motion of free surface waves. Then the wave height is expressed by a single-valued function z = h{x, y, t). Although this assumption is not rigorously vahd in the bow region, this is restricted to a smafl area and has a relatively small influence on the dynamics of Interna­tional America's Cup Class ( lACC) boats. The kine­matic and dynamic boundary conditions are given on the free surface. The kinematic condition is expressed as

dh

dt

I \dh / xdh (9)

where u - [uj , 112, « 3 ] is the fluid velocity vector on the free surface, and x - [x^, x^, X3] and v = [vj , V j , V3] are the position and velocity vectors, respectively, of grid points on the free surface. Wi th the assumption that the sur­face tension and viscous stress on the free surface are negligibly small, the dynamic condition is written as

P

du_

dn. 0

Fn2 Fn2

(on the free surface)

(10)

wherep„ is the atmospheric pressure on the free surface, and diildn^ is the velocity gradient in the direction nor­mal to the free surface. The quantity p^ is assumed to be a constant. We can set = 0 without loss of generality.

Model of turbulence

A hybrid turbulence model is employed to evaluate the kinematic eddy viscosity v,. This is a combination of the

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

Baldwin-Lomax algebraic turbulence modeP (BL) and the Smagorinsky eddy viscosity* in the subgrid-scale turbulence model (SGS). The B L model is applied on the fore part of the hull where the boundary layer is thin and locally two-dimensional. However, in the rear part of the body and in its wake region, the boundary layer rapidly increases in thickness^" and the B L model tends to overestimate the eddy viscosity. Therefore, we estimate v, by the following equation:

V, =

0

ySGS

(x<Xpp)

(xpp < X < X[^,jjp,)

( -^-MID < X < X ^ p )

( X A P < X )

(11)

where v?^ and vp^ are kinematic eddy viscosities obtained f rom the B L and SGS model, respectively.' Xpp, X M I D , and x^p denote the x-coordinates of the fore end point, mid-ship position, and after end point of the wetted surface of the hull, respectively. The parameter P is selected as ^ = ^s{x)/s(x^io), where ^(x) is the sec­tion area that is the longitudinal distribution of the huh volume beneath the still water plane i n the upright position.! ^ is 1 at the midship, decreases in the aft direction, and becomes 0 at . Ï A P -

I n the original B L model, smah calculated vf^ flows are treated as laminar. However, we assumed that the turbulent boundary layer starts f r o m Xpp even at a rela­tively low Reynolds number condition. This is because our target is the predicition of forces on the fuh-scale boat.

ySGs jg determined by

C , A { l - e x p ( - y V 2 5 ) }

where y+ is the dimensionless wall distance, A is the minimum grid spacing of the local control volume, and \Sij\ is the magnitude of the strain-rate tensor. The Samgorinsky constant is 0.5 in this calculation. The estimate of the eddy viscocity changes gradually f rom the B L model to the SGS model in the aft half of the hull. I n the wake region after x^p, the eddy viscocity is determined by the SGS model only.

Motion of the boat

where m is the mass of the boat, and the right-hand-side terms are forces acting on the main huU, sails, rudder, and keel, respectively. These forces include hydrody­namic or aerodynamic forces and gravity. The last term, fdamp. is an artificial damping force. This is added to reduce unnecessary transient oscillations of the boat when the target is a steady-state solution. The conserva­tion law for the angular momentum has a similar form.

dt rudder -'^darap (13)

where ho is the angular momentum vector of the boat with respect to its center of gravity. The right-hand-side terms are the moment vectors of all parts and the artifi­cial damping moment.

A ship has natural damping mechanisms due to the viscous effect and the dynamic forces of its appendages. Since their magnitude of damping is usually small, occa­sional transient oscillating motions of the boat waste considerable CPU time. Therefore, F^^^^ and M^^^p are added to reduce the CPU time for unnecessary transient motions. These are set at zero if the unsteady motion of the boat must be simulated rigorously.

i huu is expressed as

Js P-

Fn2 ^5- -' DdS-

Js

/77,, Fn^

(14)

where dS means the stirface integration on the wetted surface of the hull, / J Ï ^ ^ , , is the mass of the hull , D is the viscous stress tensor, and is a unit vector orienting vertically upward. The right-hand-side terms are the pressure force, viscous force, and gravitational force, respectively. The hydrodynamic and gravitational mo­ments on the hull are express as

^ h u u = - j i ^ - ^ | { ( - ^ - - - V G ) x n } d S

+ j ^ ( x - . v , ) x ( z > . « ) d ^ - ^ ( . v , , , - x , ) x e

(15)

where ( X - . V Q ) is the position vector originating f romxo, and Xh„u is the center of gravity of the main huU. I t should be noted that the buoyancy of the hull is in­cluded in the surface pressure integration that includes the gravity potential. The effect of added mass around the hull is also included in this integration.

Equation of motion ofthe boat

The equation for the transverse motion of the boat is

written as

dVr ' ^ = -PhuU + -^sail + -^rudder + -^keel + -fdamp

dt (12)

Correction ofthe viscous force

I t is not practical to perform a simulation of a full-scale boat with our limited CPU power because the Reynolds number of such a boat is about 101 Therefore, we have to estimate the motion of a full-size boat f r o m the com-

H . Akimoto and H . Miyata: Predicting sailing boat performance 35

putational results with a lower Reynolds number. For this purpose, only the frictional component of force calculated by CFD is extrapolated to that of the full-size boat as

^ship

" / C F E ' (JCFD • ' C F D (16)

where f^^^^ is the estimated frictional force of the fu l l -scale ship, and/cpD is the computed frictional force, i.e., integrated tangential stress on the wetted surface of the hull. CfP and Cp™ are the frictional force coefficients of a flat plate at the Reynolds number of the full-scale ship and the CFD simulation, respectively. Although the range of Reynolds numbers in the CFD simulations is f r o m 10^ to lO'', the calculated pressure force can be used in the equation of motion of the fufl-scale boat, with a small error only, because the scale effect is be­lieved not to be large for this component.

Models of appendages

The fluid-dynamic forces acting on sails, rudder, and keel are evaluated by a wing theory and empirical equations.

For example, we describe here the model of the rud­der. The advancing velocity vector of the rudder v ,„jd„ is expressed as

"rudder = ^ G + < Ö G >< (^'rudder (17)

where x\^Mer is the position vector of the rudder. Then the fluid velocity relative to the rudder is

V , =11 (18)

A local coordinate system is used to decompose the hydrodynamic force into the drag and l i f t components (Fig. 5). The new coordinate system is chosen as

K=n,xX,, Z^=X^xY^ (19)

where is the unit normal vector of the rudder. The angle of attack a is

Yr Z

l ' a = rudder

rudder

Fig. 5. Hydrodynamic forces acting on the rudder

/ 2 - COS" K - « r ) (20)

where a is positive when the l i f t force is in the Z, direc­tion. The hydrodynanuc force on the rudder is

•\c^z^+c^x;)~p (21)

where and are the l i f t and drag coefflcients, re­spectively, and is the area of the rudder. There are interactions among the appendages and the hufl throughout the flow fleld. Because of the complexities of these interactions, we incorporate here only the in­duced velochy of the forward-mounted keel. The fluid dynamic coefflcients in Eq. 21 are determined by the foflowing empirical equations^:

dC,

da

1^ r'keel ^

a- /C,

Cp ,= ( l + C,CjCoo + ;j;^rudder

here A™''"" and A'' ' are aspect ratios of the rudder and keel, respectively, and Qf"' is the l i f t coefflcient of the keel. ACJAa, C^^, and Cj^ are coefficients of the l i f t -curve slope, the base drag, and the three-dimensional correction factor, respectively. and are factors of the interaction between the keel and the rudder. These coefficients are obtained f rom experiments with a model rudder. The forces acting on the sails and the keel 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 the appendages have not yet been ful ly considered.

As shown in Eq. 21, the treatments of wing-like ap­pendages are steady-state approximations. Because the chord length c of these appendages is relatively small in relation to the boat length L , the reduced frequency of the boat motion with a wing, (UIL)l(Ulc), is about 0.02¬0.04. This means that a quasi-steady approximation which includes the added mass of the wings is not re­quired in calm water conditions. The added mass around the sail and the bulb is not explicitly included in the present simulation, because a rough estimate of their effect was smaller than the uncertainty of the total inertial moment of the boat.

A more precise estimation of the inertia and the quasi-steady approximation of the wings wif l be re­quired when the predicted performance of a boat in waves is considered.

Grid generation

Because of the arbitrary motion of the boat and the deformation of the free surface, the wetted part of the

36

Fig. 6. a O-O-type grid system, and b a close-up around the hull

hul l changes its shape in a time-dependent manner. To cope with the hull configuration of a sailing boat in large changes of attitude, an O-O-type boundary-fitted struc­tured grid system was selected, as shown in Fig. 6. This spherical grid system is regenerated at each time step after the moving boundaries of the free surface and the huh have settled, so that the distortion of the grid re­mains smah even for a large-amplitude motion.

Surface modeling of the hull

For ease in handling the hull configuration, a cylindrical coordinate system was defined, as shown in Fig. 7. The poshion vector on the hull surface X = [X, Y, Z] m the hull-fixed coordinate system is expressed by two parameters of X, and d,:

X = X^, F = r ( x „ 0 j c o s 0 „ Z = - / - ( x „ 0 j s i n 0 ,

(22)

where X, is the longitudinal position on the fictitious huh axis, 9^ is the angle f rom the center plane to the point, and r(X„ 0J is the distance f rom the axis to the surface point. The function r(X„ 6,) is designed to calcu­

l i . Akimoto and H . Miyata: Predicting sailing boat performance

Fig. 7. Local cyhndrical coordinates representing the hull's surface

late /• f rom X^ and 0, using second-order interpolation f r o m the given data points of the huU's shape. The vec­tor X is converted to the point of the ground-fixed coor­dinate system .V according to the position and attitude of the boat. For concise notation, we express x as .v = r(X„ 0J = E(X(X„ 0,)), where E(X) means the conversion f rom the body-fixed coordinate to the ground-fixed co­ordinate system using transverse and rotational transformations.

The configuration of the hull is given in the row of position vectors generated by a C A D apphcation. These are loaded and then converted to the cyhndrical coordinate system in CFD code. This implementation conceals the discreteness of the geometrical data in the function r(X^, 0^). I t then become possible to handle the complicated geometric changes in the boundaries in the simulation.

Search procedure for the waterline

In the first step of the grid generation, the position of the waterhne, i.e., the intersection of the free surface and the huh surface, is determined. Because these two surfaces deform owing to the wave motion and the change in the boat's attitude, there is no easy way to determine the waterline.

The determination of the waterline is based on a bisection search on the hull surface. I f there are two points on the hull surface where one is above the free surface and the other is under the free surface, the

H . Akimoto and H . Miyata: Predicting sailing boat performance 37

Xdry(O)

Water plane

Xm(0), Xdry(l) Xdry(2)

Xmd), Xwet(2)

Xwet(O),

Fig. 8. Bisection search for the waterline

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

1. Set initial points x^.^ = liX^,,, 0 , ) and .v„,, = /-(X,,,,, Ö„et), where Xi^y, 0 , and X„,t, 0„e. are parametric expressions of given dry and wet points.

2. Calculate the midpoint x^ between x^^ and .v ,j,, on the hull surface, where .v„ = [x^, y^, z^] = r(X^, e j , as

X^ = (X,,^ + X , „ J / 2 , 0„ = K + 0 „ , ) / 2

3. Replace a;,et or x^^ with .v^ according to the local free surface.

I f Zm < M' m.ym). thcu rcplacc x,,^, with.v^

I f z^ > /ï(.r„, ,y^), then replace x^^^ with

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

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

Search procedure for the fore and the aft end points

In the case of a sailing boat, the fore end point App and the aft end point x^p of the wetted surface move a lot owing to the special huh form configuration. They also move time-dependently owing to the motion of the boat and the free surface. We must determine these two points at the first step of the grid generation because they are pole points of the O-O grids.

.Vpp is determined by the repeated use of the waterhne search (Fig. 9).

1. Set the inifial section X = Z j , i.e., on the downstream side of .Vpp

2. Search the waterline positions of both the starboard side .Vp and the port side .Vp of the section.

3. Obtain the midpoint Xy^ between . V L and % on the huh's surface.

X L ( 0 )

Xs=Xs{2)

Xs=Xs{l)

Zs=Zs(0)

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

4. Search the waterhne position Xpp on the forward stretching line f rom A^.

5. Replace with Xpp, and then go to point 2.

Procedures 2-5 are repeated until the three points .Vp, X R , and Xpp converge into one point. That point becomes .rpp at that moment. The search procedure for .v^p is performed in the same manner. Xpp and .v^p are con­nected by two waterlines on the port and starboard sides by the repeated use of the bisecfion search for the free surface around the hull.

Distribution ofthe surface and volume grids

Af te r the determination of the two waterlines, body boundary grids are distributed on the wetted surface between the waterlines. Then the free surface grids are generated f r o m the waterlines to the outer boundary. The height of each grid point f r o m the still water plane is obtained f r o m Eq. 9. The inner-volume grids are alge­braically distributed between the hull surface and the outer boundary. Whole grids are regenerated at every time step.

Numerical results

Simulation procedure

The simulation procedure using this method is very similar to that of a model test in a towing tank. I t in­cludes setting a boat afloat, adjusfing the still waterline with baUast weights, setting the attitude, accelerating

38 H . Akimoto and H . Miyata: Predicting saUing boat performance

1 1

• Vi?

'Measured" ^e—i 'Nl=50. Re=1Cr6' - + - -• N M O . Re=10'G' -D--

f

_ .._

-0.6 -0.4 4}.2 0 O.Z 0.4 O.G

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

the boat (by towing), and taking the measurements. A l l of these are performed numerically.

I n a typical acceleration procedure, all degrees of freedom except heave are fixed to prevent unnecessary oscihations of the boat. When the boat reaches a steady forward speed, one can select suitable conditions of binding or release and the magnitudes of damping for the test, in all six degrees of freedom. For example, in the leeway-heel test, the angles of leeway and heel are gradually changed to the target position to be mea­sured. In this case, ah degrees of freedom except heave and pitch are fixed. Arf i f ic ia l damping terms are added to reduce unnecessary oscihation. This is effective i f time-averaged properties are the main concern.

Steady forward motion

The accuracy of the WISDAIVI-VII code was tested in a steady forward motion case. Figure 10 shows the distri­bution of the wave profile along a typical l A C C boat, JN35, without appendages and in an upright position. The measurements were performed in a towing tank at the University of Tokyo with a one-seventh scale model, and using a stih camera. The target speed of the full-size boat was 9.0 knots (Froude number 0.35, Reynolds number for this experiment 4 x 10''). The number of control volumes was 45 000. The results show that the accuracy of the simulation was satisfactory. Although this simulation did not include local over­turning and breaking waves around the bow, the peak position of the profile showed good agreement.

Figure 11 shows a comparison of the pressure distri­butions on the wetted surface area. I n this case an l A C C boat, JN32, was in steady saüing motion with a heel angle of 21.3° and a leeway angle of 2.0°. The pressure

Measrrred Computed

Fig. 11. Comparison of the measured and computed surface pressure (Cp) distribution: heel = 21.3°, leeway = 2.0°, contour interval = 0.02

was measured at 180 pressure holes located on the hull beneath the still-water plane. The area of the contour map measured was narrower than that of the computa­tion. This is because the probes were not set on the hull surface above the still-water plane where elevated free surface reaches only when the model is in forward mo­tion. However, the level of agreement was satisfactory. The asymmetrical pattern of the pressure is well realized in the computation.

Figure 12 shows a comparison of wave heights on a contour map. The wave height close to the hull was not measured in the experiment because we could not set the wave height probes in the path of the towed model. Although the main wave patterns are captured in this simulation, crests of small wavelength are dissipated rapidly in the propagating process. This is because of

H . Akimoto and H . Miyata: Predicting sailing boat performance 39

Measured

Fig. 12. Wave-height contours. Positive values are shown as bold lines and negative values as thin lines. The contour inter­val is 1 X 10 3 of heel = 25°, leeway = 3.0°, Fn = 0.34, Re = 10= (computed) and 4.2 x 10^ (measured)

•'insufficient resolution by the coarse free surface grids and their rapid outward expansion. However, the agree­ment of the wave proflle along the hull suggests that the main part of the flow aound the hull is qualitatively well reproduced.

Forced roll motion

To show the applicability of this method to the unsteady motions of a boat, a simulation of a forced rofling motion was conducted. In this simulation, an l A C C boat sailing at a constant speed (Froude number 0.35) in an upright posiflon starts rohing with an angular veloc­

ity of 30° per nondimensional time unit. This is about twice as fast as i n the nominal tacking motion of l A C C racing boats. Only heave and rol l motions are solved, and the other four modes of motion are flxed for sim­plicity. The axis of roll is set at the center of gravity of the boat. Figure 13 shows the pressure and velocity distributions in the transverse sections at two different times. I t shows the occurrence of a high-pressure region due to the acceleration of the hull's surface, and the large deformation of the free surface around the boat in section (a). Although comparative experiments have not yet been performed, the results show the potential of this method to treat the unsteady rotational motions on the free surface.

Comparison of course-changing maneuvers

This section considers an example of a typical maneu­vering motion in upwind sailing. A n l A C C boat (JN35) is in sailing motion at a constant speed of 9 knots with a true wind angle of 45° and a leeway angle of 3°. Initially, the angles of its rudder and t r im tab are adjusted to balance force and moment. The boat is then ordered to change its heading by 4°, so that it obtains a larger sail force according to the increased true wind angle. This maneuver is performed by automatic control of the rudder. The control routine of the rudder is a combina­tion of proportional and differential control methods

dt •Gpcp + G,

d(p

'dt (23)

where/is the rudder angle, (pis the error of the heading angle, and Gp and G^ are gain parameters. In this case, pitch, rol l , and heave motions are flxed to simplify the situation. Two cases of simulation were performed for different gain values. I n case 1, Gp and G^ were set at 3.75 and 3.0, respecdvely, and in case 2 they were 1.88 and 3.0, respectively. The maneuver in case 2 was expected to be gentle with a smaller proportional gain.

Figure 14 shows the pressure distributions on the hull in different times. A t the beginning of the maneuver, there is a high-pressure region in the bow due to the yawing 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 more smoothly and reaches a higher speed. The result implies that the maneuver in case 2 is superior to that i n case 1 in this situation because it steers the boat relatively gently. Figure 16 shows a series of pictures of the maneuvering motion.

This simulation of changes in course shows that the present method can be used to predict and to optimize the performance of sailing boats. The simulaflon shows

40

t = 0.2

(a) (b) (c)

H . Ak imoto and H . Miyata: Predicting sailing boat performance

t = 0. 9

(a) (b) (c)

0.12 (c) pressure

TTTTV'., (b).

velocity-

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

fluid-dynamic data of the unsteady motions of a ship. I t is difflcult to obtain these data f rom experiments because of the limitations of tank facflities and the complexity of boat motions. Thus, these examples show the potential of this simulation method in the unsteady motion of vessels on a free surface.

Conclusion

A new simulation method for the flow around a sailing boat in motion has been developed. The method is

based on the time-marching, flnite-volume method and the moving grid technique, with consideration of the unsteady motion of the free surface and the deforma­tion of the under-water geometry due to the motion. The equation of motion of the boat was solved simulta­neously with that of the flow field around the hull using the fluid-dynamic forces and moments obtained f r o m the flow solver. The sailing condition of the boat was virtually realized in the simulation. The simulation re­sults given show the potential of this method to evaluate unsteady large-amplitude motions of moving vessels on a free surface.

H . Akimoto and H . Miyata: Predicting sailing boat performance 41

(a)t=0.00 i 1

(c)t=0.50

(e)t=1.00

(b)t=0.25

(d)t=0.75 11

( f ) t = l . 2 5 i l

Fig. 14. Pressure (C^) distribution on the hull's surface in maneuvering motion at different times. Contour interval = 0.04, time interval = 0.25, Re = 10^ positive/negative values are shown by bold/thin lines

1.3

1.25

velocity of the ship Casel velocity of the ship Case2

5 9

Fig. 15. Changes in forward speed F, = \Vf^\/Um time

^ ^ ^ ^ ^ ^ ^ i i i i i i i i i i

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

42

Acknowledgments. This research was partly supported by a Grant-in-Aid for Scientific Research of the Ministry of Education, Science and Culture and by the A C Technical Committee of the Ship and Ocean Foundation.

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