Fluid Mechanics for Sailing Vessel Design

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November 11, 1997 10:39 Annual Reviews AR049-20

Annu. Rev. Fluid Mech. 1998. 30:613–53Copyright c© 1998 by Annual Reviews Inc. All rights reserved

FLUID MECHANICS FOR SAILINGVESSEL DESIGN

Jerome H. MilgramMassachusetts Institute of Technology, Cambridge, Massachusetts 02139;e-mail: [email protected]

KEY WORDS: sailing vessels, ships, numerical hydrodynamics, model testing, numericalsimulation, velocity prediction, hull design

ABSTRACT

The design of sailing vessels is an ancient art that places an ever-increasing re-liance on modern engineering sciences. Fluid mechanics shares the forefront ofthis reliance along with structural mechanics. This review focuses on the appli-cation of fluid mechanics in modern sailing vessel design. It is now commonpractice to predict sailing performance with what are called velocity predictioncomputer programs. The validity of the predictions is crucially dependent on ac-curate modeling of the air and water forces on the vessel. This article reviewsexisting methods of force modeling that include theory, experimentation, andnumerical fluid mechanics and aerodynamics. The accuracy and reliability ofthe numerical methods are considered on the basis of experimental results andfull-scale performance in areas for which the information is available.

1. INTRODUCTION

The last sailing vessel article to appear in theAnnual Review of Fluid Mechanicswas the excellent and thorough article by Larsson (1990). Because considerableliterature both before and since has appeared on the subject, this article includesonly that information the author believes to be the most interesting, important,and timely. Portions of this review are similar, if not identical, to earlier work(Milgram 1996); however, this article is more concise and also includes newinformation from more recently published literature as well as from importantolder articles.

During the past seven years, the author participated in the design of severalInternational America’s Cup Class (IACC) racing yachts. Because of the high

6130066-4189/98/0115-0613$08.00

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level of racing competition in the use of these vessels, all available capabilitiesmust be brought to bear on their design. This includes considerable experimen-tal and numerical fluid mechanics. Much of what is contained in this articlewas developed for that activity.

2. EVALUATION OF DESIGNS AND DESIGN IDEAS

A sailing vessel is a complex interconnected system, and most design changesinfluence more than one kind of fluid force. For example, to reduce the frictionalresistance of the hull by reducing its wetted surface while still maintaining itslength requires a reduction in beam that causes a reduction in heeling stability,which, for prescribed sail shapes, leads to an increase in heel angle. The changein heel angle not only changes the hull shape, it also changes the sail forces.How does one determine whether the sum of all these effects is advantageousor not? More importantly, how can one evaluate the effects if the sail shapesare simultaneously changed to optimize them for the altered hull? Short ofcomplete, full-scale sailing experiments, answering these questions requires anumerical method of predicting performance. A computer program that doesthis is a velocity prediction program (VPP).

A brief description of VPPs follows, and a consideration of the individualforces a VPP needs in order to work properly forms the framework for muchof the remainder of this article. However, there is a vast amount of litera-ture about VPPs. The interested reader is referred to Kerwin (1975), wherethe basis of the first fundamentally sound VPP is described, and to Larsson(1990), Milgram (1993), and Van Oossanen (1993) for additional informationand various perspectives on these programs.

2.1 Fundamental Principles for a VelocityPrediction Program

The primary purpose of a VPP is to predict the boat speed for any prescribedwind conditions and sailing angle (βT) between the wind direction and the courseof the boat. In a computational model this is achieved by balancing counteract-ing aerodynamic and hydromechanic forces and moments. The course of thevessel differs from the heading of its centerline by the yaw (leeway) angle,λ.

Figure 1 shows the aerodynamic and hydromechanic force and moment com-ponents in the deck plane, which is perpendicular to the center plane of thevessel. Those involved in the VPP force and moment balance are:Fa f , theaerodynamic forward force in the course direction;Fah, the aerodynamic heelforce, which is perpendicular to the forward force and parallel to the deck plane;Mah, the aerodynamic heeling moment, whose vector is along the centerline

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SAILING VESSEL DESIGN 615

Figure 1 Forces and moments in the deck plane.

of the yacht;Fwr , the resistance of the yacht in the direction opposite to thecourse direction;Fwh, the hydromechanical heel force component, which isperpendicular to the course and parallel to the deck plane (Fwh is exclusive ofcomponents of that part of the buoyancy force that balances the weight of theyacht); andMwh, the righting moment of the water on the yacht, whose vectoris in the direction of the yacht centerline and which includes both hydrostaticand hydrodynamic components.

For any equilibrium sailing condition there are three balance equations in-volving these forces and moments:

Fwr (Vb, φ, λ) = Fa f (Vb, φ, λ),

Mwh(Vb, φ, λ) = Mah(Vb, φ, λ),

Fwh(Vb, φ, λ) = Fah(Vb, φ, λ),

(1)

whereVb is the boat speed in the direction of the course,φ is the heel angle,andλ is the leeway (yaw) angle.

For prescribed values of the wind speed and the sailing angleβT, all six termsin Equation 1 depend on the boat speed, the heel angle, and the leeway angle.Figure 2 shows a block diagram of the VPP model, which solves the threeequations for these three unknowns.

Numerical solution of the balance Equations 1 is the most straightforward partof a VPP. Conversely, modeling all the forces involved is an approximate andimperfect science. Hydrodynamics, as we do it in terms of theory, experiment,

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Figure 2 A block diagram of the velocity prediction program (VPP). This emphasizes the factthat solving the force balance equations is the minor and more ordinary part of the process. Themodeling of the forces is necessarily imperfect and requires most of the effort in developing afaithful VPP.

and numerical computation, makes its greatest contribution to this field bypredicting forces and teaching us how to model them.

In addition to the basic force models, two additional VPP features, whichinvolve feedback, are shown in Figure 2: the sail shape optimizer, which ad-justs the sail shapes and their associated aerodynamic characteristics to maxi-mize the boat speed for the prescribed wind conditions; and the direction opti-mizer, which adjusts the angleβT between the course and the wind to maximize“speed made good,” VMG= |Vb cosβT |, when the desired course is upwind ordownwind.

2.2 Use of a Velocity Prediction ProgramFigure 3 shows the effect on time that a 1% change in total resistance has onan International America’s Cup Class (IACC) yacht sailing a course 17.2 kmupwind and 17.2 km downwind. A 1% change in resistance corresponds to achange in race course sailing time of 24–68 s, depending on wind speed. Thesetimes relate to substantial margins of victory or defeat. When the tactical ad-vantages of the faster boat are considered as well, the influences of the speeddifferences are even greater.

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Figure 3 Time differentials, in sailing a 34.3-km course, that result from a 1% change in resistance.

The time differentials shown in Figure 3 correspond to about 0.3% differencesin the average speed. Even smaller speed differences can be meaningful forracing vessels, so differences of very small magnitude need to be considered inmethods of evaluating candidate designs.

It is not possible to predict absolute boat speeds for a prescribed design towithin 0.3% or less of the actual sailing speed. However, this extreme accuracyis not required on an absolute basis, only on a relative basis, and can be achieved,to a greater or lesser degree depending on the design area under consideration,if the technology is pushed to its limit for some experimental and numericalmethods.

3. DECOMPOSITION OF THE FORCECOMPONENTS

A vessel under sail with non-zero heel and yaw angles involves water flows thatare asymmetric port and starboard and that are influenced by sea waves andunsteady ship motions. It also involves a complicated air flow over the sails,

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hull, and waves. In the face of these complications, to effectively apply thepresent state of the art in fluid mechanics to analyze the flows and to supportthe design process, it is necessary to make simplifying approximations.

3.1 Hydrodynamic ResistanceThe essential goal in modeling hydrodynamic resistance is determination ofthe functionFwr (Vb, φ, λ), or alternativelyFwr (Vb, φ, Fwh), for any prescribedhull form in prescribed sea conditions. A useful approach, described in detailby Milgram & Frimm (1993), is to use an additive resistance model of thefollowing form:

Fwr = Dh f + Dr + Da f + Dhi + Dw − Td, (2)

whereFwr is the total hydrodynamic resistance (drag);Dh f is the frictionaldrag of the hull;Dr is upright residuary resistance of the entire vessel;Da f isthe friction and interference drag of the appendages;Dhi is the drag resultingfrom heel and yaw (leeway) or, equivalently, from heel and heel force produc-tion; Dw is the resistance resulting from sea waves (added resistance); andTd

is the mean dynamic thrust resulting from interactions of appendages with theunsteady flow, which is due to vessel seakeeping motions and sea wave orbitalvelocities. Nondimensional force coefficients,C, are obtained by dividing cor-responding forces by12ρwV2

b Sh, whereρw is the density of the water andSh isthe wetted surface.

Figure 4 shows the fraction of resistance contributed by each component,exclusive ofTd, versus wind speed from VPP computations for an IACC yachtsailing upwind using tank test data and measured sea spectra in San Diego,California, as input. Figure 5 shows the fractions for sailing downwind. Tackingangles for optimum speed-made-good are used both upwind and downwind.Although the general features exhibited in Figures 4 and 5 are common to abroad range of vessel types, precise values of resistance components dependon vessel type and sailing conditions.

The hull friction is always the largest component for upwind sailing, andin light winds it is largest for downwind sailing. For higher wind speeds indownwind sailing, the residuary resistance becomes the largest because of thehigh boat speeds in these conditions. The effects of resistance resulting fromheel, side force, and added resistance are negligible in downwind sailing andare neglected in Figure 5. For upwind sailing, in the stronger winds, all otherresistance components are similar in magnitude, with the exception that theappendage friction becomes less consequential as the wind speed increases.

Strictly speaking, the terms in Equation 2 are not independent of each other.For example, the viscosity, which is mainly responsible for friction drag,causes boundary layers whose displacement thickness influences the residuarywavemaking resistance. If each term in Equation 2 is estimated independently

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Figure 4 Fractions of total resistance for each component for upwind sailing.

from the others, the interactions between terms are not accounted for. How-ever, if scale model tests are conducted, the interactions are captured if themodel scale is not too small; they get mixed into the various terms in the de-composition. For example, if the upright residuary resistance is defined as themeasured resistance minus the presumed frictional resistance, the sum of thesetwo resistances automatically includes the interactions. The work of Kirkman& Pedrick (1974) suggests that scale model waterline lengths need to be 5 m ormore for a reasonable assurance of reliable results in the experimental processand expansion of its data to full scale.

Once a set of resistance components that includes these interactions is deter-mined by model testing, it is often useful to estimate numerically the differencein a resistance component between two designs. Although this may not capturedifferences in the interactions, it often provides a good approximation to thedifferences that are the most significant.

3.2 Aerodynamic ForcesThe forward and heeling aerodynamic forces,Fa f andFah, are given in terms ofthe aerodynamic lift and drag forces,La andDa, and the apparent wind angle,

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Figure 5 Fractions of total resistance for each component for downwind sailing.

βa, as

Fa f = La sinβa − Da cosβa and (3)

Fah = La cosβa + Da sinβa. (4)

Similarly, the aerodynamic heeling moment,Mah, is determined from theseforces and the heights of their centers.

On a moving vessel, the apparent wind speed and angle depend on height,since the components due to true wind speed depend on height and the compo-nent due to vessel speed is independent of height. The reference wind speed,Va, and angle,βa, are commonly taken as the speed and angle at a height of10 m above the water.

The lift force,La, can be accurately determined from lifting surface theory,as described in Section 6 for upwind and close-reach sailing, where the local sailcambers and incidence angles are modest. For offwind sailing, flow separationis almost always large enough to materially influence the lift, so experimentaldata are required to construct a mathematical model for it.

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The aerodynamic drag force includes the induced drag of the sails as wellas the frictional and parasitic drag on the sails, mast, rigging, and hull. Forwindward and close-reach sailing, the induced drag data can come from thesame computational implementation of lifting surface theory that provides thelift. However, all the drag for offwind sailing and the friction and parasitic dragfor upwind sailing must come from experiments or empirical estimates.

Aerodynamic lift and drag coefficients,CLa andCDa , are

CLa ≡La

12ρaV2

a Saand CDa ≡

Da12ρaV2

a Sa, (5)

whereρa is the air density andSa is the actual sail area.Each sailing condition has a different lift and drag coefficient for optimal

performance. The usual modeling approach is to determine a maximal-allowedlift coefficient as a function of apparent wind angle,CLmax(βa). For each appar-ent wind angle and operating lift coefficient, which can be any positive valueless than or equal toCLmax(βa), there is an associated drag coefficient. The VPPchooses the amount of sail area to set, up to a maximal-allowed amount, foroptimal performance. To complete the specification, the drag coefficient needsto be modeled as a function ofCL andβa.

The author has had success in modeling the drag coefficient as

CD(CL , βa) = CDo(βa)+ C2LCi (βa)+ C2

LCDp(βa), (6)

whereCDo(βa) includes the friction drag of the sails and the profile drag coef-ficient of the hull, mast, and rigging;Ci (βa) is a coefficient of induced drag;andCDp is a coefficient of lift-dependent profile drag.

The wind tunnel data of Campbell (1997) indicates that using an exponentgreater than 2 on the lift coefficient in the last term in Equation 6 improvesthe description of the sail forces in his experiments. Euerle & Greeley (1993)developed procedures for modeling sail forces for differing vertical distributionsof lift by alteringCi in ways that can be well approximated theoretically.

Two approaches can be taken for estimatingCDo(βa) andCDp(βa). One isto add estimates of the drags from existing data. As examples, rigging dragcan be based on published drag coefficients for cylinders, and sail parasiticdrag can be based on section data (Milgram 1971). The other approach is tomeasure the drag and subtract the induced drag computed from lifting surfacetheory for the sail shapes in use to obtain the other drag components. Thishas been done by Milgram (1993) and by Masuyama & Fukasawa (1997) withinstrumented sailing vessels. Each of these vessels has all parts of the sails andrig affixed to a rigid frame that is connected to the hull through a six-componentcomputer-interfaced force and moment dynamometer as described by Herman(1988). Further information on combining data from a sailing dynamometer

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with numerical results to develop an upwind VPP sail force model, and forusing the sailing dynamometer data alone for developing an offwind VPP sailforce model, is provided by Peters (1992) and by Milgram et al (1993).

4. TOWING TANK TESTING

A thorough review of towing tank testing of model scale sailing vessels isprovided collectively in the works of Larsson (1990), Van Oossanen (1993),and Milgram (1993). Here, the process is outlined and some special problemsare described.

When data are obtained by model testing, the frictional terms,Dh f andDa f ,are subject to Reynolds scaling, whereas the other terms,Dr , Dhi , Dw, andTd,are subject to Froude scaling. The upright quantities, hull friction, appendagefriction, and residuary resistance are determined in the same way in ordinaryresistance tests of vessels that are not powered by sails. Appendage friction isestimated on the basis of appendage geometry, and the hull friction coefficientis taken as

Ch f (Re) = (1+ k)C f (Re), (7)

whereReis the Reynolds number based on length,k is the form factor evaluatedfrom the tank data by the method of Prohaska (1966), andC f (Re) is the “flatplate” frictional resistance. The difference between the measured resistanceand the estimated frictional resistance is taken as the residuary resistance,Dr .

In addition to straight-ahead tests with the vessel upright, a sailing vesselmodel needs to be tested with non-zero heel and yaw (leeway) angles withboth resistance and side force measured. This greatly increases the numberof tank runs required for a sailing vessel as compared to an engine-propelledvessel. About 135 test combinations of speed, heel, and leeway are typicallyrequired to fully quantify the hydrodynamic forces on a sailing vessel. Thisnumber of test runs is based on determination and use of a single form factor,k, obtained from a large number of low-speed upright (zero heel and zero yaw)runs and a single value of wetted surface and length for expanding the modeltank data to full scale.

In principle, a different form factor applies to each different heel angle, anddifferent values of wetted surface and length apply to each combination of heeland speed. The additional measurements and the additional test runs for deter-mination of variable form factors, lengths, and wetted surfaces would greatlyincrease the complexity and cost of a model test program. Mantzaris (1992) andMilgram (1993) showed that for 31% scale tank models with waterline lengthsof about 6 m, the effect of a variable form factor in the analysis for the time foran IACC yacht to sail a 47.3-km race course is about 4 s; the effect of including

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the speed-dependent wetted surface on the sailing time is between 7 and 10 s,depending on the wind speed. These effects increase as the model size is madesmaller because in the tank data analysis, errors in the frictional resistance ap-pear in the residual resistance, which is expanded to full scale differently thanthe frictional resistance.

A conclusion is that for model waterline lengths of at least 5 m and for scalefactors on the order of 1:3, it is not necessary to use heel-dependent form fac-tors. The errors in predicted speed and sailing times caused by estimating thefrictional resistance from a constant length and wetted surface for the cases ex-amined by Mantzaris and Milgram are similar to those from force measurementerrors on the order of 0.5%, which is the best currently achievable (Parsons &Pallard 1997). Thus, we would need to be concerned if two very differentvessel types were being compared. However, for similar vessel types, theseerrors would be similar for candidate designs and can be neglected in com-parisons. The ability to obtain satisfactory full-scale performance-predictioncomparisons from tank data with estimated frictional resistance based on singlevalues of length, wetted surface, and form factor for each design when the modelis large enough is of major importance in making model-scale tank testing apractical endeavor.

In conducting tank tests of vessels to be used for racing, accuracy and re-peatability are of paramount importance. Section 2.2 describes the sensitivityof racing performance to small changes in resistance. Since total accuracyis impossible, a reasonable approach is to strive to limit measurement errorsor lack of repeatability to 1% or less, and to take special measures when onehas to choose between designs whose predicted performances differ by lesseramounts. For example, the scale model of each design can be tank tested atfour separate times and the results then averaged together. This reduces theerroneous data variability by a factor of 2.

One way to minimize some of the errors in tank data is to use speed, heelangle, and heel force as independent variables instead of speed, heel angle, andleeway angle. The primary influence of leeway angle on hull drag is the draginduced by the heel force associated with the leeway. Therefore, if there is a verysmall error in leeway measurement or an unmeasured cross-tank current duringpart of a data acquisition run in a tank test, erroneous effects are minimizedbecause the correct relationship between the induced drag and the heel forcestill occurs. Details of this approach are provided by Milgram (1993).

5. INDIVIDUAL COMPONENTS OF RESISTANCE

This section focuses on the components of resistance individually, with em-phasis on numerical methods for estimating some of them. The most reliable

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ways of obtaining several of the estimates are thorough and carefully done ex-periments. Existing numerical methods cannot provide speed predictions thatare precise on an absolute basis, but they can provide differences in some ofthe resistance components for differing designs and thereby aid in the designprocess.

5.1 Hull FrictionAlthough hull friction is often the largest of the resistance components, it is theone that is least amenable to numerical hydrodynamics. Recently, a number ofinvestigators applied computational Reynolds-averaged Navier Stokes (RANS)methods to the viscous resistance of ships and boats (cf Larsson et al 1989,Farmer et al 1995, Miyata 1996). None of these references provided a compar-ison between computation and experiment for hull friction drag. Larsson et al(1989) gave a comparison for pressure coefficient and wall friction velocity atseveral locations on a large commercial ship hull form with an error betweentheory and experiment for the wall friction of as much as 30% in the aft partof the ship where the boundary layer is thick. In addition to uncertainty aboutaccuracy, one problem with using RANS codes to compare forces between dif-fering designs is that they require an extreme amount of computer time. Thisissue is discussed at some length by Farmer et al (1995), who describe ongoingresearch to deal with it.

Currently, none of the computational methods for hull friction and its inter-action with free surface effects are adequate to provide support to the designprocess. We can expect improvements in RANS codes for free surface flowson ships and boats as time goes on. A future alternative to RANS codes forfaster computation of viscous free surface flows may be outer inviscid solutionsstrongly coupled with inner integral boundary layer equations, as is describedfor appendages in Section 5.2. It would be helpful to have data from broadsystematic experimental studies of the relationship between sailing vessel hullparameters and the friction drag coefficient, but these have not been done, atleast not to my knowledge.

Without either a robust numerical tool or an experimental database for fric-tion drag and its interaction influence on the other drag components, tow-ing tank experiments with the friction drag estimated by Prohaska’s methodis state of the art. This cannot estimate the friction drag precisely, since ityields a coefficient that depends only on the Reynolds number, whereas thereare Froude number effects on the friction drag. Thus, the error in the esti-mate gets mixed into the other resistance components, which are all subject toFroude scaling, except for the appendage friction. Since the scale correctionsdiminish with model size, large models give more reliable results than smallones.

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5.2 Computation of Viscous Drag on AppendagesAppendages on a fin-keeled sailing vessel include the keel fin and the rudder,and possibly a ballast bulb and winglets on the keel or rudder, or both (foran example showing such appendages, see Figure 14 later in the article). Inprinciple, the viscous drag on appendages could be computed either with aRANS method or with a method that strongly couples an outer inviscid flowwith inner boundary layer equations.

With the existing state of the art, the drag forces provided by RANS codeshave not had the accuracy needed for the design of racing vessels. Develop-ment of outer inviscid solutions strongly coupled with inner integral boundarylayer equations is in the formative stage for three dimensional flows and showsconsiderable promise for the future. On the other hand, the strongly coupledmethod for two-dimensional (2D) flows is very advanced and shows excellentagreement with experiments. It can be used in support of design, because therudder and keel fin and optional winglets of a high-performance sailing vesselare high-aspect–ratio lifting surfaces, so their friction drag can be estimatedfrom 2D section analysis.

A review of the coupled method in two and three dimensions is given here.The flow away from the immediate vicinity of the lifting surface is largely in-viscid, but viscous effects are important in the boundary layers. For many yearsresearchers tried to iterate between the inviscid and boundary layer solutions.The idea was to compute an inviscid flow, use its pressure gradients in solvingthe integral boundary layer equations, solve again for the inviscid flow with theairfoil thickened by the displacement thickness of the boundary layer, includingthe viscous wake, solve the boundary layer equations again, etc. This approachfailed when the boundary layer thickened rapidly, such as often occurs close tothe trailing edge.

The strongly coupled method, which solves for the outer and boundary layerflows simultaneously, is much more robust. Advanced 2D developments havebeen achieved by Drela & Giles (1987) and Drela (1989). Recently, Nishida(1996) and Milewski (1997) developed the methodology and associated com-puter codes for three-dimensional (3D) flows and applied them to some simplecases. For purposes of explanation, with a focus on incompressible water flow,the outer inviscid flow is described by its velocity potential,φ, continued to thebody surface in the boundary integral equation form used by Milewski (1997).The perturbation velocity potential isφ, to which the potential of the free stream,φ inf , must be added to obtain the total velocity potential,

2(`−2)πφ =∫

sb

(G∂φ

∂n− φ ∂G

∂n

)ds−

∫sw

1φw∂G

∂nds+

∫(sb+sw)

σGds, (8)

where is 2 for 2D flow or 3 for 3D flow,G is the Rankine source Green Function

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(logr for 2D or 1/r for 3D, with r the distance between field and source points),n is the normal into the lifting surface,dsis the differential element of length orarea (2- or 3D),sb is the path or surface (2- or 3D) around the airfoil,sw is thepath or surface (2- or 3D) on the center of the wake,1φw is the jump in potentialacross the wake from top to bottom (constant along each wake streamline), andσ

is a fictitious transpiration source strength distribution on the lifting surface andwake that has to be determined so as to make the outer flow the same as the realboundary layer would cause. The solution to Equation 8 with the last term re-moved, and subject to the usual Neumann boundary condition and the Kutta con-dition, is called the inviscid potential,φ inv. The total velocity potential is then

8 = φ inf + φ inv +∫(sb+sw)

σGvds, (9)

whereGv is the sum ofG and a body-shape-specific dipole distribution on thesurface, chosen such that the normal derivative ofGv is zero except where thesource and field points coincide.

TWO-DIMENSIONAL FLOW The surface velocity, which corresponds to the tan-gential velocity at the outer edge of the boundary layer, is calledUe and isobtained as the derivative of the total potential with respect to the tangentialcoordinate,s.

Ue(s) = ∂(φ inf + φ inv)

∂s+∫(s′b+s′w)

d M

ds′∂Gv(s, s′)

∂s′ds′, (10)

where the mass defect,M, the transpiration source strength,σ , and the boundarylayer displacement thickness,δ∗ = ∫ (1−u/Ue)dη (η is the coordinate normalto the surface), are related by

σ = d M

dsand M = Ueδ

∗. (11)

The objective is to find a solution forM(s) and the other boundary layerparameters that satisfy the integral boundary layer equations

dθ

ds+ (2+ H)

θ

Ue

dUe

ds− C f

2= 0 and (12)

θd H∗

ds+ H∗(1− H)

θ

Ue

dUe

ds− 2CD + H∗

C f

2= 0, (13)

whereθ ≡ ∫ (u/Ue)(1− u/Ue)dη is the momentum thickness,H ≡ δ∗/θis the shape parameter,C f ≡ 2τwall/[ρ(Ue)2] is the skin friction coefficient,θ∗ ≡ ∫ (u/Ue){[1− (u/Ue)2]dη} is the kinetic energy thickness,H∗ ≡ θ∗/θis the kinetic energy shape parameter,CD ≡

∫τ(du/dη)dη/[ρ(Ue)3] is the

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dissipation coefficient, andτ is the shear stress andu is the local velocity in theboundary layer.

Drela & Giles (1987) give the semi-empirical equations for all the aboveboundary layer parameters in terms ofθ , M, andUe for laminar flow, whichis then entirely specified by these parameters and the simultaneous solutionof Equations 10, 12, and 13. The user must either specify the location of thetransition point from a laminar to a turbulent boundary layer or use a semi-empirical relation to estimate where natural transition occurs. One commonmethod is based on an estimate of the ratio of the amplitude of the most unstableTollmien-Schlichting wave at the transition point to its value at the first locationof growth with the ratio expressed asen. The value ofn at transition has beencorrelated withTf , the ratio of root mean square–free stream turbulence speedto mean speed, by Mack (1977) as

n = −(8.43+ 2.4 logTf ). (14)

Drela & Giles (1987) give a semi-empirical function,f1, for the rate of changeof n along the chord

dn

ds− f1(H, θ) = 0. (15)

For turbulent boundary layers, they provide semi-empirical relations for allof the boundary layer parameters in terms ofθ , M, Ue, and the coefficientof maximum shear stress in the boundary layer,Cτ , for which they provide asemi-empirical differential equation in the form of

d logCτ

ds− f2(cτ , H, H∗, θ) = 0. (16)

Therefore, where the boundary layer is turbulent, the flow is entirely specifiedby four parameters and the simultaneous solution of Equations 10, 12, 13, and16. The solution is obtained by discretizing the airfoil surface and Equations10, 12, 13, and 16 and solving them by Newton’s method using the inviscidsolution for a starting point.

Figure 6 demonstrates the capabilities of the method. M Drela (privatecommunication) has provided experimental results and those from XFOIL, hisprogram that implements the method, for an LA203A section exhibiting somecomplicated flow features whose experimental characteristics were measuredby Liebeck & Camacho (1985). Drela set the angle of attack in the compu-tations to the one that achieved the same lift coefficient as was achieved inthe measurements. A laminar separation bubble occurs atx/c ≈ 0.62, withturbulent reattachment thereafter, as can be seen in both the suction side pres-sure distribution and in the upper boundary layer streamline. Near the trailing

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Figure 6 Calculated and measured pressure coefficients on an airfoil section. Theupper partofthe figure shows the pressure coefficients with the inviscid calculation (dashed line) and the coupledboundary layer calculation (solid line). The lower partof the figure shows the airfoil section andthe streamlines at the outer edge of the boundary layer.

edge, there is very substantial boundary layer thickening on the suction sideand reduced thickness on the pressure side. This is shown by deviations fromthe inviscid pressures in both the experimental and computed results.

The remarkably good comparison between theory and experiment is due toseveral factors. Equations 10, 12, and 13 are fundamentally correct. However,they are not all that matters. The solution is crucially dependent on the variousrelations between boundary layer parameters (cf. Drela & Giles 1987) andthese are semi-empirical. The reason for their accuracy is that vast amounts ofexperimental data, gathered over many years, have been used in generating thesemi-empirical relations.

We do have XFOIL computations and experimental data for the forces onfoil sections used for sailing vessel appendages. Figure 7 shows drag coeffi-cient versus lift coefficient for an airfoil section as calculated by this methodusing the program XFOIL (Drela 1989) and as measured in a water tunnel (seeKerwin 1994). For the calculation,n was set to 3.5, which corresponds to thetunnel turbulence level of 0.7%. Experimental forces were based on velocitiesaround a large rectangle surrounding the section, measured with a laser dopplervelocimeter and applying the Bernoulli equation outside the wake and momen-tum conservation principles to the flow (Kinnas 1991). Agreement betweennumerical prediction and experiment is excellent.

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Figure 7 Calculated (line) and measured (circles) section drag versus lift coefficients.

Figure 8 (left) shows the airfoil section characteristics calculated by thismethod for two section shapes that could be used for the keel fin of a sailingvessel. One section has a thickness fraction of 0.13, and the other has athickness fraction of 0.17. The calculation is done forn = 9, which cor-responds to an incident stream with negligible turbulence. The figure indicatesthat for lift coefficients in excess of 0.33, the thicker section has less fric-tion drag as a result of more laminar flow. Design experience runs counterto this result. A boat goes faster with a keel thickness fraction of 0.13 thanwith a thickness fraction of 0.17, even though typical keel lift coefficientsexceed 0.33 for windward sailing. The reason for the disparity between thecomputed results and sailing experience is most probably that the combina-tion of free stream turbulence, particulate matter, and small gas bubbles inocean water near the surface cause transition near the leading edge on thesuction side—no matter what the section shape (cf Lauchle & Gurney 1984).Figure 8 (right) shows the airfoil characteristics with suction side transitionset at a maximum downstream location of 5% of the chord. For this forcedtransition condition, the thinner section has less drag up to a lift coefficientof 0.88. This is consistent with design experience, as operating lift coeffi-cients are designed somewhat lower than 0.88. The combined effects of freestream turbulence and particulate matter in the flow is an area in need of furtherstudy.

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Figure 8 Characteristics for two airfoil sections: (left) a clean, low-turbulence inflow; (right)forced transition at 5% of the chord.

THREE DIMENSIONAL FLOW Whereas Nishida (1996) coupled the boundarylayer equations with outer-flow full-potential compressible flow equations for3D, Milewski (1997) coupled the boundary layer equations to inviscid incom-pressible flow panel methods, as described above, to 3D problems.

The 3D form of Equation 10 is

EUe = iue+ kwe = EU inv +∫(a′b+a′w)

σ∇a′Gv(Ea, Ea′)da′, (17)

wherexandzare local coordinates in planes parallel to the surface in the chord-wise and cross-stream directions, respectively,Ea = i x + kz, anda signifiessurface area on the 3D airfoil or its wake.ab refers to the object surface andawrefers to the wake surface. The coordinatey is normal to the surface and takesthe role ofη used in the 2D development.

The literature for the 3D method gives the integral boundary layer equationsin a slightly different form than the literature for the 2D method: The 3Dliterature uses shear stress and dissipation rather than their coefficients. Thethree integral boundary layer equations for the 3D case that follow for chordwisemomentum (18), cross-stream momentum (19), and energy (20) in the form

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given by Mughal (1992) are used by both Milewski and Nishida.

∂

∂x

(q2

eθxx)+ ∂

∂z

(q2

eθxz)+ qeδ

∗x

∂ue

∂x+ qeδ

∗z

∂ue

∂z= τxw

ρ, (18)

∂

∂x

(q2

eθzx)+ ∂

∂z

(q2

eθzz)+ qeδ

∗x

∂we

∂x+ qeδ

∗z

∂we

∂z= τzw

ρ, (19)

and

∂

∂x

(q3

eθ∗x

)+ ∂

∂z

(q3

eθ∗z

) = 2D

ρ, (20)

whereqe = | EUe|, τxw andτzw are the two components of shear stress at the sur-face, andD is the energy dissipation per unit area. For the 3D case, there are fourmomentum thicknesses,θxx, θxz, θzx, andθzz; two displacement thicknesses,δ∗x and δ∗z ; and two kinetic energy thicknesses,θ∗x and θ∗z . Integral expres-sions for these thicknesses in terms of the velocity components through theboundary layer, and for the dissipation,D, in terms of shear stresses and strainrates in the layer, are given by Mughal (1992), Nishida (1996), and Milewski(1997).

In developing the numerical method, Nishida and Milewski both rotate the(x, z, y) local coordinate system to an (x1, x2, y) system withx1 parallel tothe flow at the outer edge of the boundary layer andx2 perpendicular to thisstreamline. The integral boundary layer equations—18, 19, and 20—are easierto express and intuitively understand in the (x, z, y) system, but the discretizedsolution is carried out more naturally in the (x1, x2, y) system. Both Nishida(1996) and Milewski (1997) provide expressions for the displacement, mo-mentum, and energy thickness in this system. The mass defects, displacementthicknesses, and transpiration source strength, which is required in the outerflow Equation 17, are related by

M1 = ρqeδ∗1, M2 = ρqeδ

∗2, and σ = ∂M1

∂x1+ ∂M2

∂x2. (21)

This is part of the strong coupling between the integral boundary layer equationsand the outer flow equation.

The method provides the tangential friction drag and the normal pressuredrag individually. Figure 9 shows these drag coefficients (based on surfacearea) versus Reynolds number, as calculated by Milewski (1997), for a formthat might be used as the ballast bulb on a sailing vessel, with forced tran-sition at 5% of the bulb length. The form is an NACA 0020 section rotatedto form a body of revolution. Figure 10 shows near-surface streamline di-rections on a rectangular wing of aspect ratio 5 at an angle of attack of 9.5

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Figure 9 Drag coefficient, based on bulb area, for an axisymmetric ballast bulb.

degrees and a Reynolds number of 2× 106. The spanwise flow at the tip,outward on the pressure side and inward on the suction side, is captured in thecomputation.

Although the 3D method has been used only on bulb and wing forms thusfar, the thoughts for the future involve hull-keel-bulb and fuselage-wing-tailcombinations. If this can be done, the next step would be to include the effectsof the free surface.

5.3 Residuary ResistanceThe residuary resistance determined in a towing tank experiment is the sum ofthe wavemaking resistance, interactions between resistance components that arenot explicitly modeled, and errors in the presumed frictional resistance. Becauseof the Reynolds number scaling of the frictional resistance, the influence of theseerrors on predicted full-scale resistance is minimized by using large models foracquiring towing tank data. For example, if the full-size vessel is 20 m long andthe frictional resistance is in error by 4% because of an error in estimated formfactor, the full-scale predicted resistance is in error by about 0.5% if a one-third–scale model is used, but it is about 1.5% if a one-tenth–scale model is used.

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Figure 10 Flow directions of wall streamlines on a rectangular wing from a three-dimensionalcoupled integral boundary layer equation and outer potential flow computation. Angle of attack is9.45 degrees, and Reynolds number 2.1× 105.

NUMERICAL METHODS A possible use of numerical hydrodynamics for min-imizing residuary resistance is computation of the wavemaking resistance fordiffering hull geometries. The state of the art of these computations has im-proved greatly in recent years, but comparisons between experiments and thecomputations demonstrate uncertainties that lead one to turn to experimentswhenever possible. The existing computational difficulties appear to lie both inthe wave resistance computation itself and in the inability to accurately predictresistance component interactions.

Computer-based solutions to the RANS equations with free surface bound-ary conditions can, in principle, capture wavemaking and viscous flow inter-actions. The application of this method to hulls with sailing vessel forms,whose overhangs influence the flow, is just beginning. Farmer et al (1995)provide RANS code results for ship wave elevations and forces, and Miyata(1996) shows RANS-computed wave elevations and hull pressures. How-ever, comparisons between computed and measured forces, which are thecrucial quantities for design comparisons, are not generally available. Forcecomparisons between experiments and the more mature area of numerical

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determination of wave resistance for inviscid flow do exist and are reviewedhere.

The first numerical procedure giving results that were generally accurateenough to be considered for use in detailed design of real ships and boats is thatof Dawson (1977). Whereas all the prior methods linearized the mathematicalproblem about the flat free surface with a uniform stream, Dawson linearizedthe ship wave problem about the double-body flow, which corresponds to thesubmerged portion of the ship beneath a rigid free surface. This basis flowcontains many of the influences of the flow around the displacement form ofthe ship. Only the surface waves are left out, and these are approximated as alinear perturbation on the double-body flow.

Extensions and applications of Dawson’s method to sailing vessel hullsinclude those of Larsson (1987), Rosen et al (1993), Nakos et al (1993),Sclavounos (1995), Raven (1994), and Rosen & Laiosa (1997).

Comparisons of the wave drag from Dawson-type codes with experimentalresiduary resistance show differences on the order of 30% over the sailing speedrange. If this error were the same for differing designs, the results could stillbe used on a comparative basis for design. However, the analysis by Milgram(1996) of the computed and experimental results for two IACC yacht designsgiven by Rosen et al (1993) shows the error differs by up to 20% for the twodesigns at some speeds.

Larsson (1987) recognized that much of the difficulty associated with pre-diction of the wave resistance of sailing vessel hulls by Dawson-type codesinvolved lack of consideration of the portion of the hull above the still wa-terline. He demonstrated this by showing that the error was much larger fora 12-m–class yacht hull form with long and flat overhangs than it was for a5.5-m–class yacht with steeper and shorter overhangs.

Two approaches have been used to improve wave resistance calculations onvessels with overhangs: the artificial lengthening of the vessel, developed bySclavounos (1995); and a nonlinear method that treats the actual wetted portionof the vessel while it is moving forward, developed independently by Raven(1994) and by Rosen & Laiosa (1997). The approach of Sclavounos solves theinitial value problem in which the vessel is brought to speed from rest, andthe time domain computation is continued until the wave resistance becomesnearly steady. In this process, the surface elevation is computed at each timestep. Because of the shallow slope overhang profiles of the vessels under con-sideration, the wetted length changes throughout the computation. An artificethat is used is to stretch the vessel longitudinally during the computation sothe still-water length matches the computed wetted length, followed by recom-puting the double-body basis flow. Another artifice used by Sclavounos is toalter the boundary condition on some of the near-centerline free surface pan-els that border on the separation line of the stern underbody profile. Instead

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of using the usual kinematic free surface boundary condition on these pan-els, the condition of tangent separation was used inasmuch as this is what isobserved on real vessels. In spite of these special features in the numericalmethods, they show considerable overprediction of the wave resistance at highspeed.

Raven (1994) has taken the procedure one step further by calculating thewave resistance for the nonlinear problem through a set of iterations where alinear boundary integral equation (panel) method is used at each iteration, butit is linearized about the solution for the previous iteration. This is continuedto convergence, and the actual wetted shape ends up being used. Figure 11shows the percentage difference between computation and experiment for anoffshore racer-cruiser yacht with a still waterline length of 21.5 m based onthe data given by Raven. Both linear (Dawson-based) and nonlinear resultsare given. If one discounts the low Froude number result, for which the muchsmaller wave resistance is difficult to separate from the frictional resistancein the experimental data, the nonlinear theory is better. This is consistentwith a comparison made by Larsson et al (1989) between linear and nonlinearcomputations for large powered ships.

Figure 11 Percentage error in the wavemaking resistance of an IACC racer-cruiser yacht computedby Raven (1994).

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Rosen & Laiosa (1997) developed a nonlinear method equivalent to themethod of Raven (1994) in principle, although details of how the boundaryvalue problems are solved are different. They show experimental residuaryresistances and both linear (Dawson-type) and nonlinear (actual wetted hull)computations of the wave resistance for two IACC yacht designs, which differconsiderably in their bow overhangs. One is designated as a destroyer bow andthe other as a spoon bow. The experiments show less residuary resistance forthe spoon bow, and the linear code shows less wavemaking resistance for thedestroyer bow. However, the nonlinear code gives the opposite result and ranksthe boats for design purposes in the same order as do the experiments. Figure 12

Figure 12 Percentage error in the wavemaking resistance computed by Rosen & Laiosa (1997):(a) the linear computation results; (b) the nonlinear computation.

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Figure 13 Total resistance drag areas for two IACC yachts predicted from towing tank model testdata. (left) Upright; (right) 20 degrees heel.

shows the percentage error in the numerical predictions of wave resistance inthe upright condition for these two hulls, based on the data given in the figuresof Rosen & Laiosa (1997). The nonlinear code brings the errors for each ofthe designs significantly closer together. The percentage error,E%, is definedas

E% ≡ 100× computed wave resistance−measured wave resistance

measured wave resistance. (22)

THE NEED FOR CONSIDERING HEEL ANGLE IN RESISTANCE PREDICTIONFigure13 (left) shows the upright resistance drag areas [resistance/(0.5ρwV2

b )] deter-mined by expanding towing tank test data to full scale for two IACC yachts,USA 9 and USA 23. For boat speeds slower than 11.5 knots, USA 9 is shownto have slightly less resistance. Yet, in actual use, USA 23 sailed a typical dayrace course about 2 min faster than USA 9 for these wind speeds. The speedadvantage of USA 23 was even more at higher wind speeds, in which the boatssailed faster.

The differences in resistance when the vessels are heeled is a principal reasonfor the relative performances of the two vessels. Figure 13 (right) shows the

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drag areas vs speed, as determined from the tank tests, for the same two designsat a heel angle of 20 degrees and zero side force. Differences in resistance upto 4% are seen and the design with superior performance is the one with lowerresistance. Of course, in actual sailing there would be non-zero side forceswhen the heel angle is 20 degrees, but the zero side force cases are shown hereto show the effects of heel most clearly. It is not at all uncommon for sailingvessels of differing performance potential to have similar upright resistances,but significantly different resistances when heeled.

5.4 Induced Drag of the Hull and Its AppendagesThe form of resistance decomposition shown in Equation 2 is chosen, in part,for convenience. The first three terms provide the calm water resistance in theupright condition, the fourth is the change due to the vessel operating at non-zero heel and leeway (yaw) angles, and the fifth adds the resistance increase dueto the presence of sea waves. The focus of this section is the use of theoreticaland numerical methods for estimating the differences in induced drag betweencandidate designs. It is a portion of the fourth term. Induced drag occursbecause the circulation around all side-force–producing portions of the hulland appendages induces changes in the flow direction everywhere and the locallift is perpendicular to the induced flow direction. The induced flow is in exactanalogy to the downwash of an airplane, which is responsible for its induceddrag.

In theory, the induced drag of a lift (heel force) producing object is verynearly proportional to the square of the lift for flows in the absence of a freesurface. However, towing tank tests show that the actual induced drag versuslift-squared function of a sailing vessel differs from the theoretical one basedon no free surface, both in its mean slope and in its linearity. Examples canbe found in Greeley & Cross-Whiter (1989). The disparity is certainly relatedto free surface effects, although it is likely that some nonlinearity is due to theleeway-dependent location of the aft stagnation zone on the heeled hull.

Completely numerical determination of induced drag for use in speed pre-diction at the required level of accuracy for racing vessels is beyond the stateof the art. It awaits development of robust numerical methods that accuratelysolve for the entire heeled and yawed vessel, including both the boundary layersand the largely inviscid outer flow. However, there is good reason to believethat presently available numerical hydrodynamics can evaluate the differencein induced drag between identical canoe-body hulls with different appendages.Although Larsson (1987) has developed a numerical method for calculating thecombined wave resistance and induced drag for a vessel moving with leewaybut no heel, the more common approach is to solve for the inviscid flow aboutthe heeled and yawed wetted portions beneath a rigid free surface. In practice,

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this is done by solving for the double-body flow in an infinite fluid. Two reasonsjustify this approach.

1. The free surface effects on the side force and induced drag are strongest on thecanoe body. With the same canoe body used in comparisons, the differencein these effects derived from appendage variations should be small.

2. Most of the hull-induced flow variations on the appendages are due to the hulldisplacement effects, which are captured in the rigid free surface problem.The ship wave flow effects on the appendages, which are not included inthe rigid free surface problem, are very nearly the same when only the ap-pendages are varied so that they have little effect on the differences betweenappendage forces.

Two effective methods are in general use for calculating the heel force (lift)and induced drag for the double body problem: the boundary integral equationmethod (BIEM), often called a panel method, and the vortex lattice method.The BIEM (cf. Katz & Plotkin 1991, Morino & Gennaretti 1992) is based onthe usual application of Green’s theorem, leading to the integral equation

−2πφ(x) =∫ ∫

Sb

(φ(x′)

∂G(x, x′)∂n′

+ G(x, x′) EV · En′)

dS′

+∫ ∫

Sw

1φ(x′)∂G(x, x′)∂n′

dS′, (23)

whereφ is the disturbance velocity potential caused by the body;Sis the surfaceof the body and its vortex (equivalently dipole) wake;EV is an onset flow velocityvector;x andx′ represent 3D locations of field and source points, respectively,and all points lie on the body or on the wake;G is the Rankine source Greenfunction, 1/|x – x′|; subscriptSb refers to an integral over the body with normalinto the body; and subscriptSw refers to an integral over the wake on which thepotential jump from side 1 to side 2 is1φ, and the normal to the wake is in thedirection from side 2 to side 1.

Discretization of the integral Equation 23 leads to the set of linear algebraicequations that is the panel method. In formulating it, the jump in velocitypotential across any vortex/dipole wake streamline has to be set equal to thejump in velocity potential at the position on the trailing edge of the body fromwhich the streamline originates (cf Lee 1987). Initially, the position of thewake sheet is unknown and needs to be determined interactively with streamlinetracing for maximum accuracy.

The vortex lattice method, as originally developed by Greeley & Cross-Whiter (1989), is a simplification of the panel method. It requires simpler

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program input data and provides much faster computation. Its origins stemfrom the lifting body panel method first developed by Hess (1972). That ap-proach was to use both surface source and surface dipole panels, as indicated inEquation 23, on lifting surfaces such as aircraft wings and tails, but to use onlysource panels on fuselages, which are boat canoe bodies and keel bulbs here.To properly model the circulation at the lifting surface roots, it was carried intothe fuselage to the centerline.

The Greeley & Cross-Whiter method represents the lifting surfaces as theircenterplane distributions of sources and vortex lattices (or, equivalently, dipolepanels) carried into connecting bodies and hull canoe bodies and keel bulbsas surface distributions of sources. Both Greeley & Cross-Whiter (1989) andRamsey (1996) have done extensive comparisons between the results of panelmethods and vortex lattice methods, with the finding that the two methods yieldvery nearly the same induced drag for a prescribed lift.

An example of the use of the vortex lattice code of Greeley & Cross-Whiterby J Kleene (private communication) is given. Figure 14 is a drawing of thesubmerged portion of an IACC yacht at a heel angle of 20 degrees. It iscustomary to adjust the fore and aft location of the rig (sails) such that the ruddercarries about 20% of the heel force exerted by the sails. This corresponds torudder angles of about 1 degree in light winds and 3 degrees in strong winds. Theparticular vessel geometry under consideration is more efficient at generatingheel (side) force on the keel fin than on the rudder because the keel is notonly deeper, it also has winglets. Use of a vortex lattice code has shown thatminimum induced drag occurs when the rudder side force is very nearly zero.For typical sailing conditions, the numerical computation predicts the induceddrag to be reduced by 8% when the rudder side force is reduced from 20% ofthe total to zero and the keel side force increased accordingly. Figure 15 showsthe time gained versus wind speed of an IACC yacht, when sailing a 34.4-kmwindward-to-leeward course, resulting from an 8% reduction in induced dragas determined by a VPP. All of these gains, except for a fraction of a second,occur on the 17.2-km windward portion of the course.

Figure 14 Submerged portion of an IACC racing yacht. It is shown at a heel angle of 20 degrees.

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Figure 15 Time gained when sailing a 34.4-km windward-to-leeward course when 20% of theside force is moved from the rudder to the keel and hull.

5.5 Added Resistance Due to Sea WavesPrediction of the added resistance of a sailing vessel due to sea waves ata level of accuracy comparable to important differences between vessels incompetition is still beyond the state of the art. Apart from the condition ofsurfing downwind, which can be viewed as temporarily negative-added resis-tance, the added resistance is largest and of primary importance for sailingto windward, which is the case considered here. Fully modeled experimentswould be difficult and costly because in addition to the requirement of model-ing the mass and its first and second moments, the tests should be conductedwith the waves coming from the oblique wind direction. It would requirea large wave basin capable of generating waves that are oblique to the tow-ing direction, with a model support and force measurement system that al-lowed unsteady motions and that obtained mean forces with errors limited toabout 1%. Such a facility probably does not exist, which explains why trust-worthy data, such as that provided by Gerritsma & Moeyes (1973), are forhead seas only. Even if the experiments could be done, an accurate simu-lation would require a mechanism to provide the pitch damping associatedwith unsteady sail aerodynamics, which Skinner (1982) showed is significant.Because of these experimental difficulties, added resistance is generally esti-mated from numerical methods, even though they ignore some of the importanteffects.

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The numerical methods in use are of second order in wave amplitude, whichis the lowest contributing order. Because of this, the numerical results mustbe treated with caution when determining the influence of added resistanceresulting from large waves, as has been concluded by Caponnetto (1993). Theresults do improve speed predictions, however, and are therefore commonlyused in VPP.

The added resistance operator,A(ω,9), is the ratio of the added resistanceto the square of the amplitude of a wave of circular frequencyω and angle9,which is 0 for stern seas and 180 degrees for head seas. In a directional seawave spectrumS(ω,9), the added resistance to lowest (second) order is

Dw =∫ π

−π

∫ ∞0

2A(ω,9)S(ω,9)dω d9. (24)

The role of numerical hydrodynamics is determination ofA(ω,9). Untilrecently, the procedure of Gerritsma & Beukelman (1972) gave the best agree-ment between computation and experiment. This procedure uses strip theoryto estimate the added resistance from the rate of energy imparted to radiatedwaves caused by relative vertical motions between the ship and the water. Partof the contribution to added resistance from sea wave diffraction by the ship isneglected in the procedure.

Recently, Sclavounos & Nakos (1993) and Sclavounos (1995) calculatedA(ω,9) by panel methods. As in their wave resistance computations men-tioned above, the linearized free surface problem about the double body ba-sis flow is determined. This is a more accurate approach than earlier proce-dures and provides a more representative treatment of the effects of diffractedwaves. Sclavounos reported (private communication) that theoretical resultswere brought into improved agreement with model scale measurements in headseas when the vessel length was stretched to the wetted length associated withthe steady flow.

Rosen & Laiosa (1997) formulated the unsteady ship motion and added-resistance problem in the form of a frequency domain panel method linearizedabout their nonlinear calm water forward speed solution. They show excellentagreement between computed and experimental values of the added resistanceoperator in head seas,A(ω, π), for one sailing vessel hull, whereas for a dif-ferent, but similar, hull the computations overpredict experimental values by asmuch as 100%. Likewise, both experimental and computed heave-frequencyresponses are provided that are obviously wrong for one vessel in long wavesand obviously correct for two other vessels. In spite of these shortcomings, theapproach of linearizing the unsteady problem about the nonlinear calm waterresult has merit and is likely to be the next step in development of numericalmethods for ship motions.

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Figure 16 Added resistances of two IACC designs in fully developed seas.

Rosen & Laiosa (1997) applied their method to both upright and heeledvessels. They show that the computed added resistance of an IACC yacht is notchanged significantly from the upright condition when the vessel is heeled 20degrees. However they report (private communication) that this is not the casewith all vessels. This is consistent with the results of Gerritsma & Keuning(1988), who found that the variation of added resistance with heel angle dependson the vessel type.

Figure 16 shows the added resistance based on added resistance operatorscomputed with the program of Sclavounos & Nakos (1993) and Equation 24for two IACC yachts. One is a base boat; the other has the waterline beamsincreased by 10% and the canoe-body depth decreased by 10% so as to maintaindisplacement. This pair of shapes is chosen for demonstration here because forfixed displacement and length, beam is the dominant parameter for variationsin numerically determined added resistance. In the figure, the added resistanceis shown as a fraction of the total resistance of the base boat sailing to windwardas determined through a VPP. For this example, the long crested oblique seasare propagating in the wind direction and specified by the Pierson-Moskowitzspectrum, which is

SP M(ω) = 0.081g2

ω5exp

[−0.74

(g

Uw ω

)4], (25)

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whereg is the acceleration due to gravity andUw is the wind speed at a height of19.5 m. As indicated in Figure 16, the added resistance in typical sea conditionsis of a magnitude such that it must be included in speed prediction procedures ifthey are to be accurate. Inclusion of the added resistance computed by the secondorder theory and applied to the wave spectrum of interest brings predicted andmeasured sailing speeds into better agreement, but it does not distinguish be-tween the performance of differing designs in waves to the extent that is observedin practice. It is believed that one reason for this lies in the effects of particularlylarge waves, which are not entirely captured by the second order theory.

5.6 Mean Forward Thrust Resultingfrom Unsteady Motions

When an appendage is operating in the unsteady flow associated with sea wavesand unsteady vessel motions, it has the possibility of generating a mean forwardthrust resulting from the unsteady relative motion of the fluid. This thrust is verysmall for vertical appendages such as fin keels and rudders. However, it can besignificant for horizontal appendages such as keel or rudder winglets, which isthe subject considered here. There is very little published information about themean thrust caused by unsteady motions of sailing vessels. The mathematicalanalysis that follows here was first derived for the 1992 America’s Cup defense,and it appears in similar form in Milgram (1996). A similar result was derivedindependently by Sclavounos & Huang (1997), who did a thrust analysis andcoupled it with a panel method for the unsteady motion to obtain predictionsfor the effect of rudder winglets on unsteady motions and added resistance inaddition to the unsteady thrust of the winglets.

As is sketched in Figure 17, the lift on a hydrofoil is perpendicular to theincident flow and the drag is parallel to it. Foils with symmetrical thickness

Figure 17 Forces on a wing section at an angle of attack.

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forms are used to simplify the presentation here, although the effect occurson nonsymmetrical foils as well. Lift occurs when the inflow has an angle ofattack,αw + θv, with respect to the nose-tail line of the foil, and a component ofthis lift force is forward whenαw 6= 0. θv is the pitch angle. The inflow angleof attack on horizontal appendages occurs when there is a relative verticalvelocity resulting from vessel motions or vertical components of the orbitalvelocities of the sea waves. As shown in the figure, there is a net forward forceif Lift × sin(|αw|) > Drag × cos(αw). This net forward force occurs ifαwis either positive or negative. As the vessel undergoes unsteady motions inwaves, the inflow angle to a horizontal appendage oscillates between upwardand downward. The lift vector direction can be mainly up or down, but it almostalways has a forward component. If the average of this forward componentexceeds the average of the aft component of the drag, positive average thrustoccurs. Otherwise, there is net drag.

The dynamic thrust is estimated here with lifting line theory, largely becausehorizontal appendages that are efficient at its generation necessarily have largeaspect ratios. For relative motions of frequencyfe with boat speedVb andappendage dimensions on the order of`, the thrust-producing effects havereduced frequencies,fe`/Vb, which are low enough for quasi-steady liftingline or lifting surface theory to be accurate.

Yaw motions of the vessel are neglected, and the dynamic thrust on generallyvertical appendages, such as a keel fin or a rudder, are not considered. Just thedynamic thrust on horizontal appendages such as a rudder wing or keel winglets,which have the greatest potential for thrust production, is analyzed. The effectsof spanwise flow are neglected.

The relative angle of incidence on the wing isαw + θv cosϕ, whereϕ is the(presumed steady) heel angle. The lift coefficient,CL , is

CL = 2πκ(αw + θv cosϕ). (26)

Ignoring a small increase due to thickness of the wing, the lift slope factor,κ

(cf. Glauert 1959, Thwaites 1960), is

κ = RA cosχ

2+ RA, (27)

whereRA is the aspect ratio andχ is the sweep angle of the wing.Using the approximation of an elliptical spanwise circulation distribution,

the induced drag coefficient,CDi , is

CDi =1

πRAC2

L . (28)

Thus, the forward thrust,T, to lowest order inαw and temporarily neglecting

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viscous effects, is

T = 1

2ρV2

b Ap[2πκαw(αw + θv cosϕ)− κ2πβ(αw + θv cosϕ)2], (29)

whereAp is the planform area of the wing andβ ≡ 4/RA.The angle of incidence, to lowest order, is

αw = w

Vbcosϕ, (30)

wherew is the relative vertical velocity at the fore and aft location of the wing. Itis the difference between the vertical component of the sea wave orbital velocityand the heave velocity of the boat at the wing location. Since the boat is bothheaving and pitching,w depends not only on time but also on the fore and aftlocation of the wing. Denoting statistical (or time) averages by overbars, takingthe average of Equation 29 and using Equation 30 provides the fundamentalequation for the dynamic thrust,Td:

Td = π

2ρV2

b Apκ cos2 ϕ

(2− κβ

V2b

w2 + 2− 2κβ

Vbwθv − κβ θ2

v

). (31)

The ship motions are approximated as being linearly dependent on the seastate, with complex transfer functions (frequency responses) from sea waveelevation to heave ofHζw(ωe) and to pitch ofHζθ (ωe). In practice, thesetransfer functions are determined numerically from strip theory (cf Salvesenet al 1970) or from a panel method (cf Nakos & Sclavounos 1990). Then, fora one-sided sea wave encounter frequency spectrumS(ωe),

w2 =∫ ∞

0|Hζw(ωe)|2S(ωe)dωe, (32)

θ2v =

∫ ∞0|Hζθ (ωe)|2S(ωe)dωe, (33)

and

wθv =∫ ∞

0Re[H∗ζw(ωe)Hζθ (ωe)

]S(ωe)dωe. (34)

Equation 31 is the fundamental equation for dynamic thrust and can be usedwhen the viscous and parasitic drag of the thrust-producing wing are accountedfor in the appendage friction drag. However, for force comparisons of variousdesigns of wings whose principal purpose is to generate dynamic thrust, or todecide whether to use such a wing at all, it is useful to consider the differenceof the dynamic thrust and the friction and parasitic drags of the wing.

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Here, an example is provided for a rudder wing located at a depth of 3 m and8 m aft of midship on an IACC yacht. The wing area is 1 m2. It has an aspectratio of 9, a taper ratio (tip chord/root chord) of 1/2, and a sweep angle of 25degrees. The wing has an NACA 0012 section, and its friction drag coefficientis approximated from XFOIL (Drela 1989) results at a Reynolds number of1.5 million as 0.008+ 0.164α2, where the relative angle of incidence,α, hasunits of radians. The angle of incidence used in the friction drag calculation isthe root mean square relative incidence angle, which is determined in terms ofthe quantities used in this section as

α2 =(

1

V2b

w2+ 2

Vbwθv + θ2

v

)cos2 ϕ. (35)

The interference drag area,AI , between the wing and the rudder is taken as thatgiven by Hoerner (1965) for strut junctions,

AI = t2

[17

(t

c

)2

− 0.05

], (36)

wheret is the thickness andc is the chord length at the root intersection.Figure 18 shows both the dynamic thrust and the difference between the

dynamic thrust and the sum of the friction and interference drags for upwindsailing in fully developed seas with spectra given by Equation 25. The angle

Figure 18 Dynamic thrust from a rudder wing for windward sailing.

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between the seas and the course of the boat is taken as 142 degrees (38 degreesfrom ahead). The pitch and heave frequency responses used to generate the datain the figure were calculated by the SWAN program of Nakos & Sclavounos(1990).

Numerical experiments show that dynamic thrust is greatly diminished byreducing the wing aspect ratio. So, it is important that the aspect ratio be aslarge as possible for a thrust-producing wing.

Actual sailing experience shows less windward advantage than Figure 18indicates. Possible reasons for this are erroneous predictions of pitch and heaveand possible larger-than-estimated drag, due to temporary flow separation inthe unsteady fluid motion.

6. COMPUTATION OF LIFT AND INDUCEDDRAG OF SAILS

Numerical methods for calculating the lift and induced drag of sails are pre-dominantly inviscid vortex lattice methods. The first was done 30 years ago(Milgram 1968a,b). The sails were represented by vortex lattices and the wakeswere flat. The lower boundary beneath the sails was approximated as a flat planeat a user-specified location below the sails. The procedure was aimed at theinverse design problem for which pressure distributions were specified and theshapes needed to attain them were computed. A number of proprietary de-velopments using a similar approach, and possibly incorporating some of theimprovements described below, were done by several researchers for commer-cial use, but details are not published.

The next significant published advancement was the work of Greeley et al(1989). They solved the analysis problem iteratively with the vortex wakes ofthe sails convected along the horizontal components of streamlines at each step.In addition, the sails and their images were heeled with respect to the imageplane. The resulting computer program has been broadly used to evaluate liftand induced drag of sail shapes.

The best use of a sail analysis program is for comparing performance ofcandidate designs. To do that, the form and friction drag of the sails must beincluded and the resulting total sail forces used in a VPP. A description and blockdiagram for such a procedure, which proved useful in sail shape development,is given by Milgram (1993).

Ramsey (1996) has taken the next step in the development of numerical hy-drodynamics of sails by including the aerodynamics of the above-water portionof the hull. The sails are represented as vortex lattices with convected wakegeometries determined by iteration. The hull is represented by source panels,as in the appendage numerical method of Section 5.4. If a sail is sealed to

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Figure 19 Geometry and panelization for example sail computations. The view shown is fromthe windward side with the boat heeled 20 degrees, with the viewpoint elevated and forward of themidship.

the deck of the hull, its root circulation is carried on a continuation of the sailthrough the deck down to the water plane. Figure 19 shows the hull, sails,and panelization on an IACC configuration for which Ramsey (1996) has donecomputations. The sailing conditions for this example are a wind speed of4.52 m/s at a height of 10 m, a boat speed of 4.51 m/s, a heel angle of 20degrees, a true wind angle of 40 degrees from ahead, and a boundary conditionof an impermeable free surface. The true wind velocity has a logarithmic profilewith a roughness height of 0.001 m for heights greater than 0.5 m, with a linearreduction below this height to a speed at the free surface that is half the speedat 0.5 m to avoid the actual steep gradient very close to the free surface. Thecamber and twist distributions are chosen to be consistent with good sails inpractice. Computations were carried out for the sails without a hull, for the sailsand the hull together with spaces between the bottoms of the sails and the deck,and for the sails and hull with the bottom of the jib sealed to the deck. Table 1shows the results for the forward and side forces. The existence of the hull

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Table 1 Sail forcesa

Hull

Force None No jib seal With jib seal

ForwardHull −14 39Jib 2345 2470 3084Main 1580 1615 1664Total 3925 4071 4787

SideHull 717 490Jib 6809 7080 8184Main 8549 8648 8678Total 15358 16445 17352aUnits are Newtons.

Figure 20 Circulation distributions on the sails.

adds significantly to the forces, particularly on the jib, which is closest to it. Thisis particularly true when the bottom of the jib is sealed to the deck. Figure 20shows the spanwise circulation distributions on the sails for each of thesecases. It demonstrates the influence on the mainsail of the jib, which is of lesserspan.

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7. CONCLUSIONS

Knowledge about the fluid mechanics of sailing vessels continues to increase,and the application of this knowledge to improving designs and design proce-dures is becoming more universal. We can see where the field is going, andwhen this vision is combined with engineering and scientific judgement, it isnot difficult to predict what some of the important developments over the nextseveral years are likely to be.

For calm-water hydrodynamics, numerical methods, which include heel, willbecome commonplace. A major step will have been achieved when viscous andwavemaking resistances, along with their interactions, are computed togetherwith good accuracy. Both RANS and coupled outer-flow/boundary layer meth-ods show promise, and only time will tell which, if either, can actually do thejob. The level of importance of separated hull vortices will be determined, andif they are important, researchers will work at ways to determine their strengthsand locations.

In the determination of unsteady vessel motions and the associated addedresistance, detailed and accurate methods for predicting the pitch and heave-damping caused by the sails could be developed very soon. Quantifying theeffects of large waves on the added resistance will probably take longer, largelybecause difficult and expensive experiments will probably be part of the process.

The effects of the boundary layers on sail aerodynamics will almost certainlybe incorporated in the numerical methods within a very few years. The degreeto which these methods can accurately deal with the wake of the mast that thenstreams across the sail attached to it remains to be seen.

Visit the Annual Reviews home pageathttp://www.AnnualReviews.org.

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