Fluid Mechanics for Sailing Vessel Design

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

FLUID MECHANICS FOR SAILING VESSEL DESIGN  .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r    f   1    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t    l   n  o   n   P    A  y    b

 Jerome  Jerome H. Milgram Massachusetts Institute of Technology, echnology, Cambridge, Massachusetts 02139; e-mail: [email protected] KEY WORDS: WORDS:

sailing vessels, ships, numerical numerical hydrodynamics, model testing, numerical simulation, velocity prediction, hull design

ABSTRACT The design of sailing vessels is an ancient art that places an ever-increasing reliance on modern engineering sciences. Fluid mechanics shares the forefront of  this reliance along with structural mechanics. This review focuses on the application cation of fluid mechanics mechanics in modern modern sailing sailing vessel design. design. It is now common practice to predict sailing performance with what are called velocity prediction computer programs. The validity of the predictions is crucially dependent on accurate curate modeling modeling of the air and water forces on the vessel. This article article reviews reviews existing methods of force modeling that include theory, experimentation, and numerica numericall fluid mechanics mechanics and aerodyn aerodynamics amics.. The accuracy accuracy and reliability reliability of  the numerical methods are considered on the basis of experimental results and full-scale performance in areas for which the information is available.

1.

INTR INTROD ODUC UCTI TION ON

The The last last sail sailin ing g vess vessel el arti articl clee to appe appear ar in the the Annual Review of Fluid Mechanics was was the the exce excelle llent nt and and thoro thorough ugh arti articl clee by Lars Larsso son n (1990) (1990).. Beca Becaus usee cons consid ider erab able le lite litera ratur turee both both befo before re and and sinc sincee has has appe appear ared ed on the the subje subject ct,, this this arti articl clee includ includes es only that information the author believes to be the most interesting, important, and timely. timely. Portions Portions of this review review are similar, similar, if not identical, identical, to earlier earlier work  (Milgram 1996); however, this article is more concise and also includes new information from more recently published literature as well as from important older articles. During the past seven years, the author participated in the design of several International America’s Cup Class (IACC) racing yachts. Because of the high 613 0066-4189/98/0115-0613$08.00

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614

MILGRAM

level of racing competition in the use of these vessels, all available capabilities must must be brough broughtt to bear bear on thei theirr desi design. gn. This This incl include udess cons conside idera rabl blee expe experi rime menntal and numerical numerical fluid mechanics. mechanics. Much of what is contained contained in this article was developed for that activity.

2.  .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r    f   1    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t    l   n  o   n   P    A  y    b

EVALU EVALUA ATION OF DESIGN DESIGNS S AND AND DESIGN DESIGN IDEAS

A sailing vessel is a complex interconnected system, and most design changes influe influenc ncee more more than than one one kind kind of fluid fluid force force.. For For exam example ple,, to reduc reducee the fric fricti tiona onall resistance of the hull by reducing its wetted surface while still maintaining its length requires a reduction in beam that causes a reduction in heeling stability, whic which, h, for for pres prescr crib ibed ed sail sail shap shapes es,, lead leadss to an incr increa ease se in heel heel angl angle. e. The The chan change ge in 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 advantageous or not? More importantly importantly,, how can one evaluate evaluate the effects effects if the sail shapes are simulta simultaneo neously usly changed changed to optimiz optimizee them them for the altere altered d hull? hull? Short Short of  complete, full-scale sailing experiments, answering these questions requires a numerical numerical method of predicting performance performance.. A computer program that does this is a velocity prediction program (VPP). A brief description of VPPs follows, and a consideration of the individual forces a VPP needs in order to work properly forms the framework for much of the remain remainder der of this articl article. e. Howe Howeve verr, there there is a vast amount amount of literaliterature ture about about VPPs. VPPs. The interest interested ed reader reader is referr referred ed to Kerwi Kerwin n (1975), (1975), where where the basis of the first fundamentally sound VPP is described, and to Larsson (1990), Milgram (1993), and Van Oossanen (1993) for additional information and various perspectives on these programs.

2.1

Fundamental Principles for a Velocity Prediction Program

The primary purpose of a VPP is to predict the boat speed for any prescribed wind wind condi conditi tions ons and and sail sailin ing g angl anglee (βT ) betw betwee een n the the wind wind direc directi tion on and and the cour course se of the boat. In a computational model this is achieved by balancing counteracting aerodynamic aerodynamic and hydromechanic hydromechanic forces forces and moments. moments. The course of the vessel differs from the heading of its centerline by the yaw (leeway) angle, λ. Figur Figuree 1 show showss the the aero aerodyn dynam amic ic and and hydrom hydromec echa hanic nic forc forcee and and mome moment nt comcomponents in the deck plane, which is perpendicular to the center plane of the vessel. Those involved in the VPP force and moment balance are: F a f  , the aerodynamic forward force in the course direction; F ah ah , the aerodynamic heel forc force, e, whic which h is perp perpen endi dicu cula larr to the the forw forwar ard d forc forcee and and para parall llel el to the the deck deck plan plane; e;

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SAILING VESSEL DESIGN

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r    f   1    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t    l   n  o   n   P    A  y    b

Figure 1

615

Forces and moments in the deck plane.

of the yacht; F wr , the resistance of the yacht in the direction opposite to the course direction; direction; F wh , the hydromechanical heel force component, which is perpendicular to the course and parallel to the deck plane ( F wh is exclusive of  components of that part of the buoyancy force that balances the weight of the yacht); and M wh , the righting moment of the water on the yacht, whose vector is in the direction of the yacht centerline and which includes both hydrostatic and hydrodynamic components. For any equilibrium sailing condition there are three balance equations involving these forces and moments: F wr ( V b , φ , λ ) = F a f  ( V b ,φ,λ),  M wh ( V b , φ , λ ) = M ah ah ( V b ,φ,λ),

(1)

F wh ( V b , φ , λ ) = F ah ah ( V b ,φ,λ),

where V b is the boat speed in the direction of the course, φ is the heel angle, and λ is the leeway (yaw) angle. For prescribed prescribed values values of the wind speed speed and and the sailing sailing angle angle βT , all six terms in 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 three equations for these three unknowns. Nume Numeri rica call solu solutio tion n of the bala balanc ncee Equa Equati tions ons 1 is the the most most stra straigh ightf tfor orwa ward rd part part of a VPP. Conversely, modeling all the forces involved is an approximate and

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616

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r    f   1    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t    l   n  o   n   P    A  y    b

MILGRAM

A block diagram of the velocity prediction program (VPP). This emphasizes the fact that solving solving the force balance balance equations equations is the minor and more ordinary part of the process. process. The modeling of the forces is necessarily imperfect and requires most of the effort in developing a faithful VPP. Figure 2

and numerical computation, makes its greatest contribution to this field by predicting predicting forces and teaching us how to model them. In addition to the basic force models, two additional VPP features, which involv involvee feedback, are shown in Figure 2: the sail shape optimizer, optimizer, which ad justs the sail shapes and their associated aerodynamic characteristics to maximize the boat speed for the prescribed prescribed wind conditions; conditions; and the direction direction optimizer, which adjusts the angle βT  between the course and the wind to maximize “spe “speed ed made made good, good,”” VMG VMG = |V b cos βT |, when hen the the desi desire red d cours oursee is upwi upwind nd or downwind.

2.2

Use of a Velocity Prediction Program

Figure 3 shows the effect on time that a 1% change in total resistance has on an International America’s Cup Class (IACC) yacht sailing a course 17.2 km upwind and 17.2 km downwind. A 1% change in resistance corresponds to a change in race course sailing time of 24–68 s, depending on wind speed. These times relate to substantial margins of victory or defeat. When the tactical ad-

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SAILING VESSEL DESIGN

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r    f   1    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t    l   n  o   n   P    A  y    b

Figur Figuree 3

617

Timediffere imedifferenti ntial als, s, in saili sailing ng a 34.3-k 34.3-km m course course,, that that result result from from a 1% change change in resist resistanc ance. e.

The The time time diff differ eren enti tial alss show shown n in Figur Figuree 3 corr corres espo pond nd to abou aboutt 0.3% 0.3% diff differ eren ence cess in the average average speed. Even smaller smaller speed differen differences ces can be meaningful meaningful for racing vessels, so differences of very small magnitude need to be considered in methods of evaluating candidate designs. It is not possible to predict absolute boat speeds for a prescribed design to within 0.3% or less of the actual sailing speed. However, this extreme accuracy is not not requ requir ired ed on an abso absolu lute te basi basis, s, only only on a rela relati tiv ve basi basis, s, and and can can be achi achieeved, ved, 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 numerical methods.

3.

DECOMP DECOMPOSI OSITIO TION N OF THE FORCE FORCE COMPONENTS

A vess vessel el unde underr sail sail with with nonnon-ze zero ro heel heel and and ya angl angles es invo involv lves es wate waterr flows flows that that

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618

MILGRAM

hull, hull, and waves. waves. In the face of these these complica complication tions, s, to effec effecti tive vely ly apply apply the present state of the art in fluid mechanics to analyze the flows and to support the design process, it is necessary to make simplifying approximations.

3.1

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r    f   1    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t    l   n  o   n   P    A  y    b

Hydrodynamic Hydrodynamic Resistance

The essential goal in modeling hydrodynamic resistance is determination of  the function F wr (V b , φ , λ), or alternatively F wr ( V b , φ , F wh ), for any prescribed hull form in prescribed sea conditions. A useful approach, described in detail by Milgram & Frimm (1993), is to use an additive resistance model of the following form: F wr  = Dh f  + Dr  + Da f  + Dhi + Dw − T d  d ,

(2)

where F wr  is the total hydrodynamic resistance (drag); Dh f  is the frictional drag of the hull; Dr  is upright residuary resistance of the entire vessel; Da f  is the friction and interference drag of the appendages; Dhi is the drag resulting from heel and yaw (leeway) or, equivalently, from heel and heel force production; Dw is the resistance resulting resulting from sea waves waves (added resistance resistance); ); and T d  d  is the mean dynamic thrust resulting from interactions interactions of appendages appendages with the unsteady flow, which is due to vessel seakeeping motions and sea wave orbital velocities. Nondimensional force coefficients, C , are obtained by dividing corresponding forces by 12 ρw V b2 S h , where ρ w is the density of the water and S h is the wetted surface. Figure 4 shows the fraction of resistance contributed by each component, exclusive exclusive of  o f  T d  d , versus wind speed from VPP computations for an IACC yacht sailing upwind using tank test data and measured sea spectra in San Diego, Califor California nia,, as input. input. Figure Figure 5 shows shows the fracti fractions ons for sailin sailing g downw downwind ind.. Tack Tacking ing angles for optimum speed-made-good are used both upwind and downwind. Although the general features exhibited in Figures 4 and 5 are common to a broad range of vessel types, precise values of resistance components depend on vessel vessel type and sailing conditions. The hull friction is always the largest component for upwind sailing, and in light winds it is larges largestt for downwi downwind nd sailing. sailing. For higher higher wind speeds speeds in downwind sailing, the residuary resistance becomes the largest because of the high boat speeds in these conditions. conditions. The effects effects of resistance resistance resulting resulting from heel, side force, and added resistance are negligible in downwind sailing and are neglected in Figure 5. For upwind sailing, in the stronger winds, all other resistance components are similar in magnitude, with the exception that the appendage friction becomes less consequential as the wind speed increases. Strictly speaking, the terms in Equation 2 are not independent of each other.

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SAILING VESSEL DESIGN

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t

Figure 4

619

Fractions of total resistance for each component for upwind sailing.

from the others, the interactions interactions between terms are not accounted for. for. HowHowever, if scale model tests are conducted, the interactions are captured if the model scale is not too small; they get mixed into the various terms in the decomposition. composition. For example, example, if the upright residuary residuary resistance resistance is defined as the measured resistance minus the presumed frictional resistance, the sum of these two resistances resistances automatically automatically includes the interactions interactions.. The work of Kirkman & Pedrick (1974) suggests that scale model waterline lengths need to be 5 m or

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 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t

620

MILGRAM

Figure 5

Fractions of total resistance for each component for downwind sailing.

βa , as

F a f  = L a sin βa − Da cos βa F ah ah = L a cos βa + Da sin βa .

and

(3) (4)

Similarly, the aerodynamic heeling moment, M ah determined ned from these these ah , is determi forces forces and the heights of their centers. centers.

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SAILING VESSEL DESIGN

621

The aerodynamic drag force includes the induced drag of the sails as well as the frictiona frictionall and parasit parasitic ic drag on the sails, sails, mast, rigging, rigging, and hull. For windward and close-reach sailing, the induced drag data can come from the same computational implementation of lifting surface theory that provides the lift. However, all the drag for offwind sailing and the friction and parasitic drag for upwind sailing must come from experiments or empirical estimates. Aerodynamic lift and drag coefficients, C  L  L a and C  D  Da , are  .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t

C  L  L a ≡

La 1 ρ V 2 S  2 a a a

and

C  D  Da ≡

Da 1 ρ V 2 S  2 a a a

,

(5)

where ρ a is the air density and S a is the actual sail area. Each sailing condition has a different lift and drag coefficient for optimal perf perform orman ance ce.. The usua usuall mode modelin ling g appr approa oach ch is to dete determ rmine ine a maxim maximal al-a -allo llowe wed d lift coefficient as a function of apparent wind angle, C  L  L ma x (βa ). For each apparent wind angle and operating lift coefficient, which can be any positive value less than or or equal equal to C  L there is an assoc associat iated ed drag drag coef coeffici ficient ent.. The VPP  L ma x (βa ), there chooses the amount of sail area to set, up to a maximal-allowed amount, for optimal performance. To complete the specification, the drag coefficient needs to be modeled as a function of  C  L  L and βa . The author has had success in modeling modeling the drag coefficient coefficient as 2 2 C  D  D (C  L  L , βa ) = C  D  Do (βa ) + C  L C i (βa ) + C  L C  D  D p (βa ),

(6)

where C  D  Do (βa ) includes the friction drag of the sails and the profile drag coefficient of the hull, mast, and rigging; C i (βa ) is a coefficient of induced drag; and C  D  D p is a coefficient of lift-dependent profile drag. The wind tunnel data of Campbell (1997) indicates that using an exponent greater than 2 on the lift coefficient in the last term in Equation 6 improves the description of the sail forces in his experiments. Euerle & Greeley (1993) devel develope oped d procedur procedures es for modeling modeling sail sail forces forces for differ differing ing verti vertical cal distri distribut butions ions of lift by altering C i in ways that can be well approximated theoretically. Two approaches can be taken for estimating C  D  Do (βa ) and C  D  D p (βa ). One is

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622

MILGRAM

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

4.

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t

TOWIN TOWING G TANK TESTIN TESTING G

A thorough review of towing tank testing of model scale sailing vessels is provided collectively in the works of Larsson (1990), Van Oossanen (1993), and Milgram (1993). (1993). Here, Here, the process process is outlined outlined and some special special problems are described. When data are obtained by model testing, the frictional terms, Dh f  and Da f  , are subject to Reynolds scaling, whereas the other terms, Dr , Dhi , Dw , and T d  d , are subject to Froude scaling. The upright quantities, hull friction, appendage friction, and residuary resistance are determined in the same way in ordinary resistance tests of vessels that are not powered by sails. Appendage friction is estimated on the basis of appendage geometry, and the hull friction coefficient is taken as C h f  ( Re) = (1 + k )C  f  f  ( Re),

(7)

where Re is the the Reyn Reynol olds ds numb number er base based d on leng length th,, k is the form form fact factor or evalua valuate ted d from the tank data by the method of Prohaska (1966), and C  f  f  ( Re) is the “flat plate” plate” frictional resistance. resistance. The difference difference between between the measured measured resistance resistance and the estimated frictional resistance is taken as the residuary resistance, Dr . In addition to straight-ahead tests with the vessel upright, a sailing vessel model needs to be tested with non-zero heel and yaw (leeway) angles with both resistance and side force measured. This greatly increases the number of tank runs required for a sailing vessel as compared to an engine-propelled vessel. About 135 test combinations of speed, heel, and leeway are typically required to fully quantify the hydrodynamic forces on a sailing vessel. This number 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)

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SAILING VESSEL DESIGN

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w  .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :   n    0  e    3  .   C    8  .    b    9   i    9   B    1  .      h   o   c   e   n   a    l    i    M    d   M    i   u   i    l    d    F  o  .   c   v   i   e   n    R  c   e  .   i   u   t

623

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 made smaller because in the tank data analysis, errors in the frictional resistance appear in the residual resistance, resistance, which is expanded expanded to full scale differently differently than the frictional resistance. A conclusion is that for model waterline lengths of at least 5 m and for scale factors on the order of 1:3, it is not necessary to use heel-dependent form factors. The errors in predicted speed and sailing times caused by estimating the frictional resistance from a constant length and wetted surface for the cases examin amined ed by Mantz Mantzar aris is and and Milg Milgra ram m are are simi simila larr to those those from from forc forcee meas measur urem emen entt errors on the order of 0.5%, which is the best currently achievable (Parsons & Palla Pallard rd 1997). Thus, Thus, we would would need need to be concer concerned ned if two very differe different nt vessel vessel types were being compared. Howev However er,, for similar vessel vessel types, these errors would be similar for candidate designs and can be neglected in comparisons. parisons. The ability to obtain satisfactor satisfactory y full-scale full-scale performanceperformance-predic prediction tion comp compar aris ison onss from from tank tank data data with with esti estima mate ted d fric fricti tiona onall resi resist stan ance ce base based d on sing single le value valuess of lengt length, h, wett wetted ed surfa surface ce,, and and form form fact factor or for each each desi design gn when when the the mode modell is large enough is of major importance in making model-scale tank testing a practical endeavor. endeavor. In conducting tank tests of vessels to be used for racing, accuracy and repeatability peatability are of paramount paramount importance. importance. Section Section 2.2 describes the sensitivity sensitivity of racing racing perform performanc ancee to small small change changess in resist resistanc ance. e. Since Since total total accura accuracy cy is impossible, a reasonable approach is to strive to limit measurement errors or lack of repeatability to 1% or less, and to take special measures when one has to choose between designs whose predicted performances differ by lesser amounts amounts.. For exampl example, e, the scale scale model of each design design can be tank tested tested at four separate separate times and the results then averaged averaged together together.. This reduces reduces the erroneous erroneous data variabili variability ty by a factor of 2. One way to minimize some of the errors in tank data is to use speed, heel angle, and heel force as independent variables instead of speed, heel angle, and leeway leeway angle. The primary influence influence of leeway leeway angle on hull drag is the drag

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624

MILGRAM

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

5.1  .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :    0  n

Hull Friction

Although hull friction is often the largest of the resistance components, it is the one that is least amenable to numerical hydrodynamics. Recently, a number of  inves investiga tigators tors applie applied d computa computatio tional nal Reynol Reynoldsds-av avera eraged ged Navie Navierr Stokes Stokes (RANS) (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 comparison between between computation and experiment experiment for hull friction drag. Larsson Larsson et al (1989) gave a comparison for pressure coefficient and wall friction velocity at several locations on a large commercial ship hull form with an error between theory and experiment for the wall friction of as much as 30% in the aft part of the ship where the boundary layer is thick. In addition to uncertainty about accuracy, one problem with using RANS codes to compare forces between differing designs designs is that they require an extreme amount amount of computer time. This issue is discussed at some length by Farmer et al (1995), who describe ongoing research research to deal with it. Currently Currently,, none of the computational computational methods for hull friction and its interaction with free surface effects are adequate to provide support to the design W ct improvements improvements in RANS codes for free surface surface flows

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SAILING VESSEL DESIGN

5.2

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :    0  n

625

Computation of Viscous Drag on Appendages

Appendages 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 (for an example example showing showing such appendage appendages, s, see Figure Figure 14 later in the article) article).. In principle, the viscous drag on appendages could be computed either with a RANS method or with a method that strongly couples an outer inviscid flow with inner boundary layer equations. With the existing state of the art, the drag forces provided by RANS codes have have not had the accuracy accuracy needed for the design of racing vessels. vessels. DevelopDevelopment of outer inviscid solutions strongly coupled with inner integral boundary layer equations is in the formative stage for three dimensional flows and shows considerable considerable promise promise for the future. future. On the other hand, the strongly coupled coupled method for two-dimensional (2D) flows is very advanced and shows excellent agreement with experiments. It can be used in support of design, because the rudder and keel fin and optional winglets winglets of a high-performance high-performance sailing vessel vessel are high-aspect–ratio lifting surfaces, so their friction drag can be estimated from 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 invisc viscid id,, but but visco viscous us effe effect ctss are are impor importa tant nt in the bounda boundary ry laye layers rs.. For For many many year yearss researchers tried to iterate between the inviscid and boundary layer solutions. The idea was to compute an inviscid flow, use its pressure gradients in solving the integral boundary layer equations, solve again for the inviscid flow with the

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626

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :    0  n

MILGRAM

(log r  for2Dor1/r for for 3D, 3D, with with r the the dist distan ance ce betw betwee een n field field and and sour source ce poin points ts), ), n is the normal into the lifting surface, ds is the differential element of length or area (2- or 3D), sb is the path or surface (2- or 3D) around the airfoil, sw is the path path or surf surfac acee (2(2- or 3D) 3D) on the the cent center er of the the wake wake,,  φ w is the the jump jump in pote potent ntia iall acro across ss the the wake wake from from top top to botto bottom m (cons (consta tant nt alon along g each each wake wake stre stream amli line ne), ), and and σ  is a fictit fictitio ious us tran transp spir irat ation ion sourc sourcee stre strengt ngth h distr distrib ibuti ution on on the the lift lifting ing surf surfac acee and and wake that has to be determined so as to make the outer flow the same as the real boundary layer would cause. The solution to Equation 8 with the last term remove moved, d, and and subje subject ct to the the usua usuall Neum Neuman ann n bounda boundary ry condi conditi tion on and and the the Kutta Kuttaco conndition, is called the inviscid potential, φ inv. The total velocity potential is then  = φ inf  + φ inv +

 

(9)

σ Gv d s ,

(sb +sw )

where Gv is the sum of  G and a body-shape-spe body-shape-specific cific dipole distribution distribution on the surface, chosen such that the normal derivative of  Gv is zero except where the source and field points coincide. The surface velocity, which corresponds to the tangential velocity at the outer edge of the boundary layer, is called U e and is obtained as the derivative of the total potential with respect to the tangential coordinate, s. TWO-DIMENSIONAL FLOW

e

U  (s ) =

∂ (φ inf  + φ inv) ∂s

+

  

(sb

d M  ∂ Gv (s , s  ) +s 

w)

ds

∂s

d s,

(10)

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SAILING VESSEL DESIGN

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :    0  n

627

dissipation coefficient, and τ  is the shear stress and u is the local velocity in the boundary layer. Drela & Giles (1987) give the semi-empirical equations for all the above boundary layer parameters in terms of  θ , M , and U e for laminar flow, which is then entirely specified by these parameters and the simultaneous solution of Equations Equations 10, 12, and 13. The user must either either specify specify the location location of the transition point from a laminar to a turbulent boundary layer or use a semiempirical empirical relation relation to estimate estimate where natural transition transition occurs. One common meth method od is base based d on an esti estima mate te of the the rati ratio o of the the ampl amplit itud udee of the the most most unst unstab able le Tollmi ollmien en-S -Sch chli lich chti ting ng wave wave at the the tran transi siti tion on point point to its valu valuee at the first first locat locatio ion n n˜ of growth with the ratio expressed as e . The value of  n˜ at transition has been correlated with T  f  f  , the ratio of root mean square–free stream turbulence speed to mean speed, by Mack (1977) as n˜ = −(8.43 + 2.4 log T  f  f  ).

(14)

Drela & Giles (1987) give a semi-empirical function, f 1 , for the rate of change of  n˜ along the chord d n˜ ds

− f 1 ( H , θ ) = 0.

(15)

For turbulent boundary layers, layers, they provide semi-empirica semi-empiricall relations relations for all of the boundary layer parameters in terms of  θ , M , U e , and the coefficient of maximum shear stress in the boundary layer, C τ τ , for which they provide a

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628

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n   r   a  .   o    F   w .   w    1    1   w    /    3   m    2    /   o   r   1    f    1    d   e   n    d  o   a   a   o   i   r    l   e   n  n   w  g   o  e    D  g   n  .   I    3   i    5   d    6   -   e    3   l   a    1   t   r    6   :    0  n

MILGRAM

Figure 6  Calculated and measured pressure coefficients on an airfoil section. The upper part  of  dashed line line) and the figure figure shows shows the pressu pressure re coeffi coeffici cient entss with with the invis inviscid cid calcul calculat ation ion ( dashed and the the coup couple led d boundary layer calculation ( solid line). The lower part  of the figure shows the airfoil section and

the streamlines at the outer edge of the boundary layer.

edge, there is very substantial boundary layer thickening on the suction side and reduced thickness on the pressure side. This is shown by deviations from

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SAILING VESSEL DESIGN

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n  r   a  .   o    F   w  .   w    1    1   w    /    3   m

629

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630

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n  r   a  .   o    F   w  .   w    1    1   w    /    3   m

MILGRAM

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SAILING VESSEL DESIGN

631

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

and  .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n  r   a  .   o    F   w  .   w    1    1   w    /    3   m

∂ ∂ x

            qe2 θ  x x + qe2 θ  zx +

qe3 θ  x∗ +

∂

∂ z ∂

∂ z

∂

∂ z

qe2 θ  x z + qe δ x∗

∂ ue

qe2 θ  zz + qe δ x∗

∂ we

qe3 θ  z∗ =

2 D ρ

,

∂ x

∂ x

+ qe δ z∗

∂ ue

+ qe δ z∗

∂ we

∂ z

∂ z

= =

τ  xw ρ τ  zw ρ

,

(18)

,

(19)

(20)

 e |, τ  xw and τ  z w are the two components of shear stress at the surwhere qe = |U  face, face, and D is the the ener energy gy diss dissip ipat atio ion n per per unit unit area area.. For For the the 3D case case,, ther theree are are four four momentum thicknesses, θ  x x , θ  x z , θ  zx , and θ  zz ; two displacement thicknesses, δ x∗ and δ z∗ ; and two kinetic energy thicknesses, θ  x∗ and θ  z∗ . Integr Integral al expre expresssions for these thicknesses in terms of the velocity components through the

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632

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n  r   a  .   o    F   w  .   w    1    1   w    /    3   m

MILGRAM

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SAILING VESSEL DESIGN

 .   y    l   g   r   n   o  .   o   s   e   w  s   u   e   l    i   v  a   e   n   r   o    l   a   s   u  r   e   n  p   n  r   a  .   o    F   w  .   w    1    1   w    /    3   m

633

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634

MILGRAM

determination of wave resistance for inviscid flow do exist and are reviewed here. The first numerical procedure giving results that were generally accurate enough to be considered for use in detailed design of real ships and boats is that of Dawson (1977). Whereas all the prior methods linearized the mathematical problem about the flat free surface with a uniform stream, Dawson linearized the ship wave problem about the double-body flow, which corresponds to the submer submerged ged portion portion of the ship beneath beneath a rigid rigid free surface. surface. This This basis basis flow contains many of the influences of the flow around the displacement form of 

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SAILING VESSEL DESIGN

635

of using the usual kinematic free surface boundary condition on these panels, the condition of tangent separation was used inasmuch as this is what is observ observed ed on real real vessel vessels. s. In spite of these these special special features features in the numerica numericall methods, they show considerable overprediction of the wave resistance at high speed. Raven (1994) has taken the procedure one step further by calculating the wave resistance for the nonlinear problem through a set of iterations where a linear linear boundary boundary integral integral equation (panel) method is used at each iteration, iteration, but it is linearized about the solution for the previous iteration. iteration. This is continued

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636

MILGRAM

Rosen & Laiosa (1997) developed a nonlinear method equivalent to the method of Raven (1994) in principle, although details of how the boundary value value problems are solved are different. different. They show experimen experimental tal residuary residuary resistances and both linear (Dawson-type) and nonlinear (actual wetted hull) computations of the wave resistance for two IACC yacht designs, which differ considerably in their bow overhangs. One is designated as a destroyer bow and the other as a spoon bow. The experiments show less residuary resistance for the spoon bow, and the linear code shows less wavemaking resistance for the destroyer bow. However, the nonlinear code gives the opposite result and ranks

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SAILING VESSEL DESIGN

637

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638

MILGRAM

drag drag area areass vs spee speed, d, as dete determ rmin ined ed from from the the tank tank test tests, s, for for the the same same two two desi design gnss at a heel angle of 20 degrees and zero side force. Differences in resistance up to 4% are seen and the design with superior superior performance performance is the one with lower resist resistanc ance. e. Of course, course, in actual actual sailing sailing there would would be non-zero non-zero side forces forces when the heel angle is 20 degrees, but the zero side force cases are shown here to show the effects effects of heel most clearly. clearly. It is not at all uncommon for sailing sailing vessels vessels of differing differing performance performance potential potential to have have similar similar upright resistances, resistances, but significantly different resistances when heeled.

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SAILING VESSEL DESIGN

639

this this is done done by solv solvin ing g for for the the doub double le-b -bod ody y flow flow in an infin infinit itee fluid fluid.. Two reas reason onss

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640

MILGRAM

progra program m input input data data and provides provides much faster faster computati computation. on. Its origins origins stem

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SAILING VESSEL DESIGN

641

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642

MILGRAM

The numerical methods in use are of second order in wave amplitude, which

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