Scientific Methods in Yacht Design

Annu. Rev. Fluid Mech. 1990. 22: 349-85 TECHNISCHE UNIVERSITE1 Copyright (CD 1990 by Annual Reviews Inc. All rights reserved Laboratorium voor Scheepahydrornechanica Archief Makelweg 2, 2628 CD Delft 1OL. 015 - man- Fax: 015- 781 eaS SCIENTIFIC METHODS IN YACHT DESIGN

Lars Larsson SSPA Maritime Consulting, P.O. Box 24001, S-400 22 Gothenburg, Sweden, and Chalmers University of Technology, Department of Marine Hydrodynamics, S-41296 Gothenburg, Sweden

INTRODUCTION One of the most remarkable achievements in the history of sport was the victory of the Australian yacht Australia II in the 1983 America's Cup races. The cup, brought to the US from England in 1851 after the victory of the yacht America, had since then been successfully defended 24 times, and was considered permanently bolted to its table in the New York Yacht Club. American technology and financial resources, combined with the skill and experience of some of the most well-known yachtsmen in the world, had always been an insurmountable obstacle to challengers from different countries. From a scientific point of view the victory of Australia II is of special interest, since the successful outcome of the races was, to a very large extent, the result of a strong technological effort (see van Oossanen 1985). The hull and, particularly, the keel represented radical departures from the traditional design of the 12-m America's Cup yachts. Before the breakthrough in 1983 the evolution of the 12-m class had almost ceased, and there was widespread opinion among designers that the optimum design had been reached. Furthermore, due to some notable failures in the early 1970s, confidence in yacht research, and tank testing in particular, was very low. After the 1983 races interest was boosted, not only in the Cup itself but also in yacht research and development. It had become

'Twelve meters is the rating of the yacht according to the International R-rule. The real overall length is about 20 m. Yachts of the 12-m class were used in the America's Cup between 1958 and 1987. 349 0066-4189/90/0115-0349$02.00 350 LARSSON obvious to everyone in the yacht-design community that much practical information could be obtained from the application of scientific methods, and that there was still room for improvement in the 12-m class design. In the 1987 campaign most participating syndicates spent large efforts on tank testing and numerical flow calculations. This was particularly true of the US syndicate Sail America, whose leading spokesman, Dennis Connor, had lost the Cup as skipper of Liberty in 1983. More than 30 naval architects, engineers, and scientists were engaged in the development of the new yacht, and after a very successful campaign Connor was able to bring the Cup back to the US in February 1987. What is interesting in this effort, from a scientific point of view, is that most of the work has been published (see Salvesen 1987, Chance 1987, Letcher et al. 1987a,b, Letcher & McCurdy 1987, Oliver et al. 1987, Boppe et al. 1987, Scragg et al. 1987). Not much has been reported from other US syndicates, although some informationabout the research in the America II and Heart of America campaigns has been released (see Xia & Larsson 1986, van Hem- men 1986, Larsson 1987). Experiences from testingseveral of the non-US challengers have been collected in two papers on keels by van Oossanen & Joubert (1986) and van Oossanen (1987), and some infolination on the Australian efforts is given in the papers by Cox & Whitaker (1987) and Klaka & Penrose (1987). Experiences from the design of the first fiberglass 12-m yachts, KZ 3 and 5 from New Zealand, are reported by Bowler & Honey (1987). Although the America's Cup has been the source of much recent yacht research, important developments have taken place over the years at insti- tutions not linked to the Cup. The most comprehensive collection of papers on yacht research may be found inthe transactions from several series of symposia on the subject held regularly in the US and the Netherlands. The Chesapeake Sailing Yacht Symposium is held biannually on the US East Coast, and the AIAA Symposium on the Aero/Hydronautics of Sailing is an annual event on the West Coast.Recently, another series, the Tampa Bay Sailing Yacht Symposium, was started in Florida. The Dutch Sym- posium on Developments of Interest in Yacht Architecture is held in Amsterdam biannually. Several decades of yacht research at Southampton University are reported in the excellent books by Marchaj (1979, 1982, 1986), which deal with the fundamental aspects of sailing in a very clear and concise manner. Other books on the same topic are those by Kay (1971), Hammitt (1975), and Gutelle (1984). A book on the principles of yacht design by Joubert & Larsson (1990) is about to be published. There are two different disciplines that must be mastered by the successful yacht designer: fluid mechanics and structural mechanics. Bearing SCIENTIFIC METHODS IN YACHT DESIGN 351 this in mind, it may seem surprising that the vast majority of papers on yacht research deal with only the first area. An important reason for this unbalance may be the 12-m rule, which prohibits exotic materials and specifies robust scantlings. There has been no room for advanced structural optimization. While the fluid mechanics of sailing includes both aero- and hydrodynamics, the emphasis of this review is on the latter. This is the area where most of the development has taken place in recent years. Readers interested in sailing aerodynamics are referred to Marchaj (1977, 1979), Milgram (1968, 1971a,b,c, 1972, 1978), Thrasher et al. (1979), Wiersma (1977, 1978, 1979a,b) and Register & Irey (1983). Today, hydrodynamic data for a hull can be obtained either by improved techniques for tank testing or by means of numerical methods. In either case a computer program is requiredfor predicting the performance of the yacht, given the hydrodynamic input. In fact, the program may itself generate such data from semiempirical formulas. The results will be less accurate, but a large number of alternatives may be evaluated in a very short time. This type of program, called VPP (Velocity Prediction Program), is now a most important tool for top yacht designers. In the next section the VPP theory is outlined and some examples of applications are given. Recent developments in the tank testingtechniques are described thereafter, and in the final section the results of application of computational fluid dynamics (CFD) are reported.

VELOCITY PREDICTION PROGRAMS Davidson (1936), more than 50 years ago, proposed a method for predicting the close-hauled (upwind) performance of sailing yachts from towing-tank data. Using full-scale measurements on board the yacht Gim- crack, he was able to derive a set of sail coefficients, which has been in use at many towing tanks until recently (Murdey1978). A more general evaluation procedure was, however, proposed by Herreshoff (1964), enabling predictions of all points of sailing to be made. Methods for predicting performance without access to towing-tank data appeared in the mid-1970s (see Myers 1975, Letcher 1974, 1975a, 1976. Dawson 1976, Curtiss 1977). A few years later, van Oossanen (1979) presented a method especially designed for 12-m yachts. The real breakthrough of the VPPs did not appear, however, until the early 1980s as a result of the H. Irving Pratt project at Massachusetts Institute of Tech- nology (Kerwin & Newman 1979). The purpose of the project was to improve the handicap rules for sailing yachts. A VPP was developed, which has since become the basis for the American Measurement Handicapping 352 LARSSON System (MHS) and, more recently, for the International Measurement System (IMS) (see Poor 1986, Kirkman 1987). In the development of the 12-m yachts for the 1987 America's Cup the VPPs playeda very important role, as explained by Oliver et al. (1987) and van Hemmen (1986). For the special Cup in 1988, where a catamaran competed againsta monohull, performance predictions must have been even more important, and since a new rule has now been adopted for the America's Cup, VPPs will continue to be imperative as a tool for the designer. Structure of the VPP Until recently, all velocity prediction programs have been based on the equations for static equilibrium. The essence of suchprograms is thus a method for satisfying the equilibrium equations (although usually not in all six degrees of freedom). To accomplish this, info' illation is required about the hydrodynamic, aerodynamic, and stability properties of the yacht (see Figure 1). The output of the program is the yacht perfoithance under varying wind conditions. As can be seen in Figure 1, the info' 'nation required can be obtained in a variety of ways. Static stability calculations are usually carried out by the designer, and some empirical correction for the effect of forward speed is added in the program. Accurate aerodynamic data are more difficult to find. While it would be possible to obtain such data from either experiments

Empirical a and c very data rare

Wind tunnel Aerodynamic c Numerical results model results

Stability Solution of Speed model equilibrium Heel

equalions Leeway et

Towing tank Hydrodynamic Numerical results model results

Empirical Either one of d - f data can be used

Figure 1VPPstructure. SCIENTIFIC METHODS IN YACHT DESIGN 353 or numerical calculations, this is seldom done owing to the high cost involved. Instead, standard aerodynamic coefficientsare used. The hydrodynamic input may be obtained from three different sources: towing-tank results, numerical predictions, or semiempirical relations. In the lattercase the hydrodynamic coefficients are computed by the VPP itself, andvery rapid evaluations of design alternatives may be made. Equilibrium Equations Figure 2 shows the forces acting on a sailing yacht under equilibrium conditions. In the top view (left) the horizontal components of the forces are displayed. When the hull is driven through the water, a resistance is developed. Under equilibrium conditions this has to be balanced bya driving force from the sails. Unfortunately, this cannot be accomplished without at the same time producing a side force, which in turn has to be balanced by the hydrodynamic side force. Since the turning moment under equilibrium conditions must be zero, the resulting aero- and hydrodynamic forces (in the horizontal plane) have to act along the same line. In the view from behind (right in Figure 2) the force balance ina transverse plane (at right angles to the direction of motion) is presented. It is seen that the resulting aerodynamic side force is at right angles to the mast. This is an assumption that was introduced already by Davidson (1936) and has been adopted by all sailing-yacht investigators since then. The heeling moment from the side force is balanced bya righting moment from the couple created by the buoyancy force and the weight of the yacht. Apart from the equilibrium equations, the velocity triangle (see Figure

Aerodyn 'Heel side force angle Aerodyn Leeway Dr side force angle force angle Resistance Heeling force App. wind angle Hydrodyn. side force Weight Buoyancy App. wind, Hydrodyn side force force Figure 2Forces on a sailing yacht. 354 LARSSON

Figure 3Velocity triangle.

3) plays an important role. As the figure shows, the trianglerelates the true and apparent wind velocities via the yachtspeed. The VPPs generally solve a set of five equations: ,0 Aerodynamic driving force =Hydrodynamic resistance, Aerodynamic side force = Hydrodynamic side force, is Aerodynamic heeling moment=Hydrodynamic (-static)righting' moment,, co Apparent wind velocityfrom velocity triangle, Apparent wind direction from velocity triangle. Hydrodynamic Model A typical distribution of resistance components for asailing yacht beating to windward is shown in Figure 4. It is seenthat at lower speeds the viscous

Resistance EN]

1.500

1100G

Reeking induced 500 Wave Residuary Schoenherr Upright or Viscous ITTC -57 1_

3 4 5 :6 7' Speed [knots]

Figure 4Resistance componentsofa 7.6-m cruisingyacht, SCIENTIFIC METHODS IN YACHT DESIGN 355 resistance dominates, while at higher speeds the wave resistance becomes increasingly more important. Upwind, the induced resistance amounts to about one fourth of the total resistance at low and intermediate speeds, less at high speed. A fourth component, the heeling resistance, is introduced to take the effect of heel on the viscous and wave components into account. For some yachts with very inclined ends, this fourth component may be negative. VISCOUS RESISTANCE The traditional way to compute viscous resistance in hydrodynamics is to refer to some empirical formula based on tests with flat plates. In the US, Schoenherr's formula has always been the most popular one, but according to the recommendations of the International Towing Tank Conference (ITTC), the ITTC-57 Ship Model Correlation Line should be used (Comstock 1967). This is not a pure flat-plate formula, since it includes some form effect as well.. As a matter of fact, many VPPs use this formula for the total viscous resistance. Some attempts have been made to compute individual "form factors" for each hull (van Oossanen 1979,, 1981), but the accuracy of the empirical formulas is questionable. When calculating the Reynolds number Ra required for the ITTC-57 formula, some account has to be taken of the fact that the chord lengths of appendages are much smaller than the length of the hull. Therefore :individual calculations are often made of the different parts of the underwater body (van Oossanen 1979, 1981, Larsson 1981b).. WAVE (RESIDUARY) RESISTANCEThe fact that the computed viscous resistance does not include the full form effect may not be too serious, provided that the formula for computing the rest of the upright resistance takes this into account., This component is normally referred to as the residuary resistance, and it contains essentially wave resistance but also, to some extent, the viscous form effect. Several different relations for the residuary resistance have been employed in VPPs. Van Oossanen (1979, 1981) used a semiempirical relation due to van Oortmerssen (1971),, based on tests with 93 different small-craft hull forms at the Dutch towing tank MARIN. An advantage with this formula is that it is based on wave-resistance theory, and very good predictions applying the formula for Antiope were obtained by Larsson (1979) and van Oossanen (1981). For 12-m yachts, van Oort- merssen's relation may be the best but for lighter, more beamy yachts another expression, derived on purely empirical grounds from a systematic series of sailboat tests by Gerritsma et aL (1981), should be more accurate. This formula, which contains the displacement, displacement/length ratio, prismatic coefficient,' longitudinal position of the center of buoyancy,,

The prismatic coefficient is defined as the volume displacement divided by the Maximum sectional area times the waterline length. It is thus a measure of the fullness of the' yacht.. 356 LARSSON and beam/draft ratio, was shown to give quite good correlation with measurements when applied to the 22 models of the test series.

INDUCED RESISTANCE The induced resistance may be computed either from slender-body theory or lifting-line theory (see, e.g., Newman 1977). The former approach was chosen in the MIT work (Kerwin 1976). In this approximation the induced resistance is independent of the detailed shape of the underwater part of the yacht, being inversely proportional to the draft squared. A correction for the displacement effect of the hull on the trailing vortex system was, however, applied by Kerwin. More detailed studies of the effect of varying the shape of the appendages can be made if the lifting-line theory is adopted, as proposedby van Oossanen (1979, 1981). The induced resistance is then computed indi- vidually for the hull, keel, and rudder. Since in the lifting-line theory the induced resistance is inversely proportional to the aspect ratio and an "induced drag factor" (Abbott & von Doenhoff 1949), dependent on the load distribution, the planform of the keel/rudder comes into play. In aerodynamics the influence on the induced drag factor by s-weepback and taper is known (see, e.g., Hoerner 1965), but for a sailing yachtthe influence is more complex owing to the interaction between the vortex field and the free surface. A correction for taper was introduced by van Oossanen (1981). The aspect ratio is geometrically defined as the span divided by the mean chord, or, equivalently, the span squared divided by the area. For a lifting surface attached at right angles to an infinite plane, the effective aspect ratio is twice the geometric one, and this assumption is usually adopted for the hull as well as for the keel and rudder. The hull is thus considered reflected in the free water surface, which is a good approximation at low and moderate Froude numbers but less accurate when significant waves are being generated (see Slooff 1984). In the most modern VPPs (Oliver et al. 1987, Larsson 1989), the effect of the keel wings is taken into account. After their introduction on Australia II in the 1983 races, such wings are now always used on 12-m yachts. The wings are mounted on the tip of the keel, thereby interfering with the overflow from the pressure to the suction side. The strength of the vortex left behind the keel is then reduced, as is the induced resistance. Larsson's relations were derived by results from systematic numerical predictions by Letcher et al. (1987b), where the size and planform of the wings were varied, and the same data, combined with extensive towing tank information, were used by Oliver et al. HEELING RESISTANCE When the yacht heels,the effective waterline length as well as the wetted surfacechange. Both effects may be accurately SCIENTIFIC METHODS IN YACHT DESIGN 357 predicted in the effective CAD systems now available to yachtdesigners, but this is not usually done. While very approximate methods areused for the length variation, the change in wetted surface is notconsidered at all.

SIDE FORCE Obviously, the same theories used for computing theinduced resistance can also be employed for the side force(hydrodynamic lift). Nomoto & Tatano (1979) showed that by combiningslender-body theory for the hull with lifting-line theory for the appendages, an accuratecal- culation of the side force and the center of effortcould be made. In this method the side force of the hull and rudder wasobtained by extending them through the hull to the waterline, Theinteraction between the hull and the appendages is then approximately taken into account.A more exact calculation of this interaction wassuggested by van 'Oossanen (1981). When the VPP is used for evaluating experimental data orresults from numerical flow calculations, the residuary resistance isusually represented by a spline function through the data, while theinduced and heeling resistance, as well as the side force, are 'computedusing semiempirical formulas, where the constants are determined by regressionanalysis (see Gerritsma et al,. 1981, Letcher 1975b). Aerodynamic Model The most difficult part of the VPP theory is to find agood aerodynamic model. For many years, Davidson's (1936) sail coefficients werethe only ones available, and they havebeen used until recently for predicting upwind performance from towing-tank tests. A simplemodel for all wind directions was proposed by Herreshoff (1964), and Myers (1975)suggested a model based onwind-tunnel data (Marchaj, 1982) and lifting-line calculations (Milgram 1971c). To obtain a more reliablemodel, full-scale data were recorded on two ocean racers, Standfast and BayBea, in the mid-1970s. The results were reported by Kerwin et al. (1974)and Gerritsma et al. (1,975), and the model proposed wasused in the original VPP developed in the Pratt project at MIT (Kerwin 1976). A rather serious disadvantage of the proposed model wasthat it provided no means for taking the actual shape of the sail planinto account. The model was, valid for yachts with a rig not toodifferent from the masthead rig of the ocean racing yachts on which the measurementshad been made. For very different rigs, as on a 12-myacht, the model could not be used without corrections. Apossible improvement, suggested by Kerwin (1976), was to make use of the sail modelof the International Offshore Rule. (IOR) for ocean racing. By meansof this model, the actual rig could be converted to an equivalent Standfast typeof rig, for which the measured coefficients could be used. 358 LARSSON The most widely accepted aerodynamic model today is the one presented by Hazen (1980). As in all other models, the forces are presented as functions of the apparent wind angle. There are thus no means of trimming the sails in different ways; they are always assumed to be set at their optimum. Hazen compiled data from various sources, including the Bay BeaStandfast results, and was able to obtain coefficients for lift and parasitic (viscous) drag for each individual sail. A weighted average based on the area of each sail may then be used for their combined effect. The induced drag is computed according to lifting-line theory by taking 1.1 times the height of the rig as the effective span. The most serious disadvantage of Hazen's model is that the interaction between different sails is not included, but it has nevertheless become accepted as the best choice available. The effect of heel is taken into account in two principally different ways. The original Bay BeaStandfast data (Kerwin 1976) indicated a linear force reduction with heel angle. Kerwin (1976), on the other hand, suggested that the effect of heel could be considered in the velocity triangle. In a coordinate system heeling with the yacht, the apparent wind angle will depend on the heel angle, and a fixed set of coefficients that are only dependent on the apparent wind angle can be used. It should be noted that due to the atmospheric boundary layer, the wind velocity has to be computed at the actual height of the center of effort of the sails. Thus, the true wind velocity also depends on the heel angle.

REEFING, FLATTENING, AND TWISTING Most modern VPPs have options for reefing and flattening the sails. In fact, in most programs there are automatic reefing and flattening procedures for finding the optimum performance of the yacht under varying conditions. Not only is this possibility a necessary feature for realistic predictions,it also serves as a guide to the skipper of how to make optimum use of his sails. The reefing R and flattening F factors in the sail model have different effects on the sail forces. If R is proportional to some linear dimension of the sails, and the shape of the sail plan is assumed constant, the area and hence the forces will be proportional to R2. At the same time the height of the center of effort is also approximately proportional to R. F, on the other hand, is proportional to the lift of the sails, but it has no influence on the centerof effort. Since the induced drag is proportional to lift squared, decreasing the F-factor means increasing the lift/drag ratio,i.e. rotating the sail force forward. As most sailors know, it is more beneficial to flatten the sailsthan to reef them when the heeling moment gets too large. An interesting feature was introduced in the VPP used by the Sail America syndicate in the 1987 America's Cup (Oliver et al. 1987).Since 12-m yachts never reef, this parameter was replaced by a twist function, SCIENTIFIC METHODS IN YACHT DESIGN 359 by which the spanwise loading of the sails couldbe varied. In this way the actual sail trim of the 12 m in the strong winds off Fremantlecould be modeled quite well. The effect of the twist parameter wasobtained from lifting-line theory. Flow Diagram A simplified flow diagram of a typical VPP isshown in Figure 5.. To obtain the equilibrium solution, an iterative method isrequired. Thus, the speed and heel angle of the yacht are first assumed. Thisyields the apparent wind speed and direction, which are used to find the sail forces.The heel equation is then solved to find a new heel angle, and new sailforces can be calculated. Upon convergence of the heel iteration, all resistance components canbe computed and the corresponding speed is obtained. Thisis not usually close enough to the assumed one, so several iterations arerequired. Know- ing the speed and heel angle one can compute explicitlythe leeway angle. The calculations are repeated for varying truewind speeds and directions,, and the results can be presented in the form of a polarplot (see Figure 6). The polar plot gives the speed of the yacht versusthe true wind angle. Each curve represents a certain wind speed. Points ofspecial interest are the upper- and lowermost points of the curves, wherethe yacht has its maximum speed upwind and downwind, respectively. Thecorresponding angles can be used by the helmsman to optimize upwind anddownwind performance. Sailing in Waves A weak point of most VPPs is the prediction ofthe performance in waves. In many programs the wave effect is neglected, at leastexplicitly. It may be argued that since the aerodynamic modelis based, to some extent, on results from actual sailing conditions, some effect ofthe waves may be buried in the sail coefficients. However, when obtainingthe data, rough sea conditions were avoided.It would have been too difficult to include the complex effects of a seaway in the simplesail coefficients_ Waves create motions in all degrees of freedom. Additional meanforces are generated both on thehull and the sails. Furthermore, the sails influence the motions considerably,thereby rendering standard prediction techniques useless in certain degrees of freedom. Therefore, acomplete model for the wave effects is out of reach at present, but inthe approach chosen 'by the author (Larsson 1981b, 1989) and some otherinvestigators (Klaka & Penrose 1987, Oliver et aL 1987), the most importanteffect is taken into account., Thus, the added resistance in wavesis calculated using a strip theory (see, e.g.., Frank & Salvesen 1970), and this componentis added to the still-water resistance., 360 LARSSON

START

True wind velocity given

True wind direction given

Boat speed guessed

App. wind vet. dir from wind triangle

Heel angle guessed

Aerodyn forces from sail model

Heel angle from heel equation I

1 Figure 5 VPP flow diagram (schematic). Heel cony?

speed LBoatfrom X- equation

Speed cony.?

Leeway angle from '(-equation

Yes More 'wind directlions?

Yes More wiad velocities?

STOP SCIENTIFIC METHODS IN YACHT DESIGN 361

WIND DIRECTION

BOAT SPEED 30° [knots' 10I LW*SPEED MADE 6000

60°

irgs' 90° ri441, 1111/ 1200 t6M/s OM s 10 SPEED MADE GOOD MIN 12

Figure6Polar plot (example).

To investigate whether or not strip theory is appropriate for motions and added resistance of yachtlike hulls, Gerritsma & Moeyes (1973), Klaka & Penrose (1987), and Gerritsma & Keuning, (1988) carried out tests in waves and compared them with calculations. In general, fairly good correspondence was noted. For heavy hulls a considerable reduction in added resistance with heel angle was found, indicating that the strip-theory calculations should be carried out for varying heel. This turned out to be unnecessary, however, for lighter, more beamy hulls. The effect of leeway was found to be negligible. Skinner (1982) investigated numerically the influence of the aerodynamic damping of the sails on the pitch motion and the added resistance. A reduction of the latter of about 14% due to damping for a 12-m yachtwas obtained. He also carried out a simple calculation for the effect of pitching on the driving force from the sail and found it to be negligible. While Skinner's investigation involved important simplifications, it still lends some support to the approach used in the VPPs. 362 LARSSON

urgdX)

Pitch Yaw

Sway `11), Heave '(Z) Figure 7Yacht motions (definitions),

New Developments of the VPP

DYNAMIC VI)13 A new type of VPP has recently been developed at SSPA Maritime Consulting in Sweden. The purpose of the newdevelopment has been to be able to predict the dynamics of maneuvers and otherunsteady conditions apart from wave motions. Such effects are of greatimportance in match racing and should be taken into account wheninvestigating the performance of, for example, America's Cup yachts. - The mathematical model is based on the one presented byRutgersson & Ottosson (1987) and Kallstrom & Ottosson (1983) forship-maneuvering studies. Four dynamic equations are solved (cf, Figure 7,where the ter- minology is defined): a Surge acceleration fromtotal surge force, Sway acceleration from total sway force, Roll acceleration from total roll Moment, Yaw acceleration from total yaw moment., These nonlinear ordinary differential equations in time areintegrated using the SIMNON computer program for nonlinear systems(Astrom 1982). On the SSPA Masscomp 5400 microcomputer, updating canbe achieved up to 10 times per second, soreal-time simulations can be made. This possibility is utilized in the SSPA sailing simulator, wherematch racing between two yachts can be practiced. The simulator, seenin Figure 8, consists of two identical setups, each with two screensfor displaying the yacht in a horizontal and a vertical view. On thelatter screen, which has a zooming facility, thecompeting yacht and the marks on the course are displayed too. All instruments used on a real yacht areshown in the horizontal view. A steering wheel and controls for sailsand trim tab' are included.

A trim tab is a keel flap.. SCIENTIFIC METHODS IN YACHT DESIGN 363

Figure 8The SSPA sailing simulator (two identical setups).

Not only does the dynamic VPP require acompletely new solution procedure, but also many new empirical relations areneeded. Since the yaw equation is considered, arudder model including stall must be provided, and stall effects of the keel at low speeds mustbe taken into account as well. The aerodynamic modelincludes the interference between the two yachts, and there is a sheeting model,which takes into account the movement of the center of effortfor different sheeting angles. Obviously, added masses for hull, keel, andrudder must be computed for all motions considered. The sailing simulator has been in operationsince the summer of 1988, but no extensive report on the basic theory has yetbeen published. A brief description is given by Larsson (1989). The dynamicVPP has now replaced the static one (Larsson 1981b) for allperformance predictions at SSPA, including the ones based on towing-tank dataand numerical flow predictions. When the dynamic program is used as aVPP, an autopilot steers the yacht, and a prescribed number of tacks isautomatically performed. As in the manually operated simulator, awind spectrum with variations in speed and direction is included. 364 LARSSON

EXTENDED USE OF THE VPP Extensive use of a VPP is reported by Oliver et al. (1987) in the development of the America's Cup contender Stars and Stripes. With knowledge of the wind statistics for Fremantle, the probability of winning a race against a known opponent could be calculated. In fact, in one approach, quasi-steady time-domain studieswere carried out where the two yachts were raced around the fullcourse, using the wind history from statistical data. A matrix of win/loss probabilities could then be obtained by systematically varying the sizes of the two yachts. Table 1 gives one example, where Stars and Stripes as well as the opponent were varied in size, the waterline lengths being changed in steps of 1ft from 46 ft to 49 ft. Obviously, when the sizes are equal (identical yachts), the probability must be 50%. It is seen that the best chance of winning is when Stars and Stripes is 48 ft and the opponent 46 ft. In order to make the most rational decision regarding the hull size, game theory was applied. Thus, match racing was considered a two-person, zero-sum game. For the probabilities of Table 1 (the "game matrix"), this theory leads to a "single strategy" with 48 ft as the best option. Choosing this length there is no way the opponent can get a higher probability of

-winning. Other game matrices may lead to a "mixed strategy," where a weighted random selection between the choices must be made. It is also possible to compute the result of a series of match races where the fleet of competing yachts has an assumed composition of lengths.

TOWING-TANK TESTING

The first reported extensive tank test of a sailing yacht was a disaster. The British designer G. L. Watson spent nine months tank testing his America's Cup challenger Shamrock II, only to be beaten in three straight races by the Herreshoff-designed defender Columbia in 1901. Watching the losing

Table 1 Percentage win probabilities for Stars and Stripes (Perth, January)

Opponent

Waterline (ft) 46 47 48 49

46 50 45 41 44 Stars 47 55 50 47 47 and 48 59 53 50 52 Stripes 49 56 53 48 50 SCIENTIFIC METHODS IN YACHT DESIGN 365 battle, Watson made his now classic remark: "I wish Herreshoff hada towing tank" (Burgess 1935). While this may not be a fair verdicton the towing tank, it is quite clear that the early tests with sailing-yacht modelswere not entirely reliable. It was not until Davidson (1936) pointed out the importance of measuring not only the resistance but also the side force thatmore accurate predictions could be made from the tank. Thereare, however, still a number of pitfalls in the sailboat test technique, which inmany respects is considerably more complex than regular ship model testing. Testing Techniques As pointed out many years ago by Davidson (1936), thereare two principally different techniques that can be used when investigatinga sailing- yacht model. The apparently most straightforwardway is to tow the yacht at the center of effort of the sails and let it attain its equilibrium condition under heel. By measuring speed, heel, and leeWay fora given sail force, the merits of the design can be evaluated. This method was not, however, adopted by Davidson, who favored the other technique, in which the hull is fixed in all degrees of freedomexcept heave and pitch, and where a matrix of speeds, heels, and leeways is covered in each test. By applying suitable interpolation routines and using a VPP type of approach, the equilibrium of the yacht under different conditions can be determined.. This semicaptive techniquewas adopted by other towing .tanks and is still the most popularone.

THE SEVIICAPTIVE TECHNIQUE While most tanks have stuck to the .original. idea of releasing only heave and pitch in the semicaptive techniques (Kirk- man 1974, 1979, Murdey et al. 1987, DeBord 1987, Takarada & Obokata 1987, Gonzalez 1987, Campbell & 'Claughton 1987),an alternative approach is to release also the roll motion andmeasure the heel angle during the run.. By applying a weight sliding transversely,a static moment can be applied to the hull to vary the heel angle. This technique has been used particularly for large models to avoidan excessive roll moment in the supporting mechanism due to the hydrodynamic force (see Herreshoff & Newman 1967, Kirkman 1974, 1979).. Thereare, however, many other ways of avoiding this problem. In Figure 9 the equipment used at SSPA for testing large models is presented. The towing force is appliedto a vertical mast, approximately at the center of effort of the sails in the upright condition. The horizontal component of the heeling force is measuredat the same position.. Fore and aft, the hull is guided by vertical posts, giving the model the freedom to heave and pitch. Side-force transducers.are fitted between the hull and the vertical posts, which do not exertany longitudinal 366 LARSSON

Yaw and Pitch free Hinge

Y force X force

Universal joint Carriage

Yforce

universal joint Y force

Figure 9SSPA's yacht dynamometer (principle).

forces on the hull. In this design the lever arms for heeland yaw moments are very large, and the setupis quite stiff, enabling accurate setting of the angles. When the heel angle is changed, the foot ofthe mast, pivoting around a bolt at deck level, is moved sideward. Sincethe side forces are applied horizontally in this equipment, a weight has tobe put on the hull to account for the vertical componentof the sail force. No correction is applied to account for the slightly too largepitching moment from the mast force when the model isheeling.

THE FREE-SAILING TECHNIQUEDavidson's free-sailing idea was first developed by Allan et al. (1957). A force was applied atthe center of effort of the sails, and the hull was free to attainequilibrium heel and leeway angles corresponding to the towing speed.Care was taken to apply the towing force at right angles to the mast. Unstable asthis setup may seem, steady conditions were obtained in most cases.Some restraining moment (which was measured) at the top of the mast washowever, occasionally needed. Equipment, in principle very similar tothat used by Allan et al., was developed at the National Research Council(NRC), Canada (see Murdey 1978, Murdey et al. 1987). The majordifference is that in the NRC tests, the rudder angle is used as anindependent variable. By varying the rudder angle, the heel is changed systematically.The main reason for including the rudder is that in practice it is veryimportant for balancing the yacht.. More realistic conditions can then beobtained in the tank., SCIENTIFIC METHODS IN YACHT DESIGN 367 Some instability was reported by Murdey (1978), and to avoid this problem a slightly different approach was chosen at MARIN, the Nether- lands, where the most modern free-sailing equipment has been built (see Gommers & van Oossanen 1984, van Oossanen 1985). The equipment is seen in Figure 10. In this system the yaw angle is fixed. The towing force is applied at a vertical position corresponding to the center of effort of the sails. When the hull starts moving, a side force is developed. This causes the model to heel and a yawing torque to be developed. By using servo- motors, the mast structure moves longitudinally until the torque is elim- inated, and the tow point moves axially along the mast until there is no axial component left of the tow force. Equilibrium conditions are obtained after 15-20 s. The major advantage of the free-sailing systems is that fewer test points are required. Rather than one needing to interpolate in the speed-heel- leeway matrix, each test condition represents a realistic combination of the three variables. According to Murdey et al. (1987), experiences from the NRC system indicate that about half as many test points are required as compared with the conventional semicaptive systems. On the other hand, predictions of prototype perfoi mance can be made only for the stability tested. In the semicaptive approach, stability is taken into account in the VPP prediction and can thus be varied arbitrarily. Another disadvantage of the free-sailing tests is that the cost of producing a model is

Verticalcarriage attached to main carriage

Y force X force Universal joint Yaw angle (pitch, heel free)

Most movable

Figure 10MARIN's yacht dynamometer (principle). 368 LARSSON increased, since a cast lead keel is required to give the correct stability. In fact, Murdey et al. (1987), after more than 10 years of experience of the free-sailing system, conclude that the semicaptive procedure combined with a state-of-the-art VPP is the optimum combination. An implemen- tation of such a system at NRC is planned for the future. Model Size and Turbulence Stimulation An issue that has been discussed extensively in the past 15 years is the required size of the models. For almost 40 years after Davidson's pio- neering work, models at 1:13 scale were used in the Davidson Laboratory at Stevens Institute of Technology, and eight successful America's Cup defenders were developed in that way. The disappointing performance of the two 12-m yachts Mariner and Valiant in the early 1970s, however, cast doubt on the usefulness of the towing-tank technique itself, and the failures were attributed by many to the misinterpretationof the tank data due to scale effects.' An extensive test program had already been launched by the panel H-13 (sailing yachts) of the Society of Naval Architects and Marine Engineers, and a standard test case had been defined (Society of Naval Architects and Marine Engineers 1971). Different sizes of this hull, the Antiope, were tested at many different organizations in the late 1960s and early 1970s (Herreshoff & Newman 1967. Kirkman 1974). In a landmark paper, based on these and a number of othergeosim tests from different laboratories, Kirkman & Pedrick (1974) were able to draw important conclusions regarding the required size of the models. These conclusions and recommendations were later reinforced in a paper by Kirkman (1979). In essence, the results were as indicated in Figure 11. Here the correlation between the model and its prototype, or (in some cases) a larger model, is given as a function of the waterline length of the (small) model. Two important conclusions can be immediately drawn: The scatter is very large for the small scales, with many errors in the 10-20% range; and a considerable reduction in error is obtained at larger scales. To obtain the results of Figure 11 the small-model data were extrapo- lated to full scale or largest model scale using the standard procedure in yacht testing. The upright resistance is assumed to consist of two parts: the viscous resistance and the residuary resistance (cf. Figure 4). For the same Froude5 number, the latter component is proportional to the

'Brown & Savitsky (1987) have recently shown that the Mariner predictions were correct. It was predicted to be considerably slower than the competitor Courageous, which had been tested in the same tank, but this information could not be released to Mariner's designerfor proprietary reasons. 'The Froude number F, is defined as the speed divided by the square root of the waterline length multiplied by the acceleration of gravity. SCIENTIFIC METHODS IN YACHT DESIGN 369

Correlation error inpercent

20 Resistance,Fn = 0.20

So -10 Approximate bounds -20

2 3 4 5 6 7

20 Side force

-10

-20

1 2 3 4 5 6 7 Model waterline lengthinmeters Figure 11Correlation between results at model and full scale (or a large model scale). From Kirkman & Pedrick (1974).

displacement, while the former is scaled Using a standard foimula such as Schoenherr's (used by Kirkman) or the ITTC-57 correlation. The resistance components due to heel and leeway, as well as theside force, are scaled as the residuary resistance. It should be noted that more accurate scaling_ procedures, with individual extrapolation of the appendage drag and with the fotin-factor approach (see Lindgren & Dyne 1980), are used at some towing tanks. There may be several reasons for the low accuracy of the small-scale data in Figure 11. Some of them are inevitable. For instance, if the stern is very full, different separation patterns will appear in the different scales, with a large influence on the drag as a consequence. Even if separation does not appear, the relative boundary-layer thickness on a 1:10 scale model is twice as large as on the prototype. The interaction between the boundary layer and the stern waves may thus be different. The most likely sources of error, however, are the influenceof laminar flow and the parasitic drag of turbulence stimulators. Two of the author's students (Liden & Sallin 1975) carried out an 370 LARSSON

Figure 12Location of the start of transition (neutral stability point) at F = 0.12. Cal- culations are for Antiope at three different scales. From Liden & Sallin (1975). interesting calculation of the boundary layer around Antiope at different Reynolds numbers; the frictional resistance was obtained by integrating the skin-friction coefficient along the hull. In Figure 12 the location ofthe neutral stability point, computed according to Granville (1953), is shown for different scales at a constant Froude number. This point shouldbe somewhat upstream of transition, but measurements (Xia et al. 1985) have shown that, in practice, it is a relatively good representation of the start of the transition region. It is seen in the figure that at 1: 3 scale the neutral stability point has moved from a location near the bow at full scale to almost amidships. This change in the extent of the laminar region is accompanied by a drop in friction, as can be seen in Figure 13.6 It is also clear in the figure that the measured low-speed data get extremelyscattered below the critical Reynolds number. The calculations by Liden & Sallin (1975) were made withoutturbulence stimulation, as were the measurements in Figure 13. However,Kirkman (1979) reported that even for a hull with stimulators, peculiar trends ofthe viscous resistance were observed in the critical-Reynolds-number range, indicating that the stimulators were not fully effective. The most common type of stimulator in yacht investigations is a rowof studs, usually 3 mm in diameter and 1-2.5 mm high, with apitch of 10 25 mm. The studs are applied either on the hull or theappendages or both. Frequently, several rows are put on the hull at differentstations, while on the appendages the studs may be applied either close tothe leading edge or at the 25% chordline (sometimes even further aft; see Campbell & Claughton 1987). The positioning on the appendages isparticularly tricky, since the size of the stimulator is considerable for the smallermodels, as compared, for instance, with the leading-edge radiusof a keel or rudder. An unintended change in shape of the foil section maythus be caused by studs placed too close to the leading edge. Furthermore,since the use of

'Coefficients for skin friction (CO, wave resistance (Cw), lift (CD, and drag CD) inthis paper are nondimensionalized using thewetted surface and the yacht speed. SCIENTIFIC METHODS IN YACHT DESIGN 371

Mean curve through measured total resistance /

7 A

1,1 6 1:3 a 1:6 CF Measurements at 5 ITTC-57 different Fn's a 4 a / 3 Calculated CF a ° 2

icitog Rn 6.0 6.5 7.0

Figure 13Calculated skin friction for Antiope compared with the ITTC-57 correlation line and measurements at three scales. From Liden & Sallin (1975). laminar sections has become more common for keels, positioning of studs in front of the expected full-scale transition would be a mistake. There is no generally accepted simple procedure for checking the effec- tiveness of the stimulators. Hot-film measurements were made at South- ampton (Campbell & Claughton 1987) on a 1: 4 scale 12-m yacht, but this is hardly possible in general within the rather limited budgets available. The only realistic way seems to be to look at the behavior of the resistance curve at low speeds (Kirkman 1979) to see whether the slope is in accord- ance with the skin-friction curve. Another problem is the correction for the parasitic drag of the stimulators. Traditionally, this is obtained by changing the number of studs, under the assumptions that the flow is turbulent for both configurations and that the difference in drag is due to the studs themselves. If the comparison is made at high speed, all studs may be removed. Alternatively, the drag of the stimulator may be computed by assuming a velocity (inside the boundary layer) and a drag coefficient. Kirkman (1979) showed that this results in good agreement for large models but poorer agreement for small ones. He also showed that the determination of the stimulator drag is vital for the smaller models, since it may amount to more than 10% of the total resistance. Differences between different test objects are often of the order of a few percent, so the determination of the stimulator drag becomes critical. It should be apparent from this discussion that tests with small models require extreme caution, and that even with an experienced and careful 372, LARSSON test crew, some scale effects are unavoidable. This isthe reason why most America's Cup syndicates have turned to 1: 3 scale (corresponding to1:1.5 scale for Antiope) in recent years. For the typical yacht designerthis trend is unfortunate, since the cost of testing such large models isoften prohibitive. The designer therefore has to accept the somewhat lower accuracy of the smaller tank. APPLICATION OF COMPUTATIONAL FLUID DYNAMICS Computational fluid dynamics (CFD) has appeared recently as atool for the yacht designer. Chance & Company started an ambitious programin 1973 to apply calculation methods to the flow around a seriesof yacht hulls (see Boppe et al. 1987), but owing to the difficulties encountered,the only results obtained were from a potential flow solution at zeroheel and. leeway and under the assumption of a flat free' surface. One stepfurther was taken by Liden & Sallin(1975), who also included the calculation of the boundary layer at several Reynolds numbers. Morecomplete were the Antiope results presented by the author in 1979 (Larsson1979), where the viscous wave, and induced resistance as well as the sideforce were predicted using CFD. Different methods were used, however,for the lift and the free-surface calculations,, and thus the importantinteraction between the vortex system from the keel and the generated waves wasnot considered. A simplified method for taking this interactioninto account was used in the developmentof the Australia II keel (van Oossanen 1985), but it was not until the wing-keel boom of the1987 America's Cup that complete methods for lift-generating hulls in the presenceof a free surface Were developed (see van Beek etal. 1985, Xia & Larsson 1986, Larsson 1987, Boppe et al. 1987). However, quite interesting systematiccalculations were also carried out in the1987 campaign that neglected the free surface/ vortex interaction (Letcher et al. 1987b,Boppe et al. 1987, Scragg et al. 1987). These calculations are described below,followed by a survey of the more exact methods. A short accountis also given of recent viscous-flow calculationS. Calculations at Zero Froude Number In the Sail America technology team forthe 1987 America's Cup, two different groups were engaged in CFD development.Both used essentially . the same method for computing the induced resistanceand the side force, assuming no waves. In this zero-Froude-numberapproximation the free water surface is simply considered as a symmetryplane. The calculations were carried out usingthe. public-domain code VSAERO, developed. by SCIENTIFIC METHODS IN YACHT DESIGN 373 Maskew (1982) at NASA. This is a first-order panel method in which the flow around the body is simulated using constant distributions of sources and/or doublets on flat panels on the body surface. Thick components of a configuration, like the hull, are represented by sources proportional to the normal component of the free-stream velocity and an unknown doublet strength. Thin lifting surfaces, like the keel, are represented only by doublets, whose unknown strength is determined from the zero-normal-velocity condition on each panel. Doublet panels with a strength computed from the trailing-edge conditions are located in the wake, the position of which can be either iteratively determined by the program or specified by the user. Interestingly, the two groups ![Letcher et al. (1987b) at Science Appli- cations International Corporation (SAIC) and Boppe et al. (1987)] spent considerable efforts on validating the code but arrived at different conclusions. The most extensive validation was made at SAIC, where three simple cases, for which analytical results could be obtained, were investigated.. Comparisons were also made with measurements for three 12-m models tested earlier. The results seem to prove, convincingly, that the method for obtaining drag from a summation of the pressure forces on all panels is too inaccurate. A method for 'determining lift and drag from the trailing vortex system was therefore developed and proved, to be very robust. Boppe's group, on the other hand, carried out a systematic panel- ization study for a model-tested 12-m yacht and concluded that reasonable convergence using pressure summation had been obtained, for the densest arid used. Analyzing the results, one notes that both groups favored uniform grids. The finest 'one used by SAIC contained' 24 chordwise panels, whereas 5-4 panels were used by Boppe's group. The panel lengths in the chordwise direction were thus approximately 4% and 2%, respectively, of the chord. Small as these panel lengths may seem, they are not small enough to resolve the leading-edge suction. Recent panel-method calculations in the author's CFD group have indicated that for a proper resolution, the nose panels have to be only a fraction of the leading-edge radius. Panels smaller by 5-10 times than those reported in the VSAERO calculations are thus, required, and this is difficult to achieve with a uniform grid. Considerable stretching, with rather large panels (10%) in the middle of the profile, may be applied without loss of accuracy. Extensive parametric studies of keels with wings were carried out by both Sail America groups. It was found that the induced drag is relatively 'insensitive to the detailed shape of the wings, while the wingspan has a large impact on efficiency. Boppe et al. (1987) found that a forward mounting of the keel wings is favorable, but that this advantage is offset by the increased trim-tab efficiency with the wings aft. A comparison was also made between 374 LARSSON the traditional 12-m yacht Liberty and Australia II with wings. Lifting- surface calculations for an ideally loaded wing/winglet combination had indicated much larger reductions in drag as compared with the ones obtained for Australia II. This difference could be explained by the very unfavorable side-force distribution of the hull-keel-winglet combination. In Figure 14 the distributions are given for both yachts, and it is seen that neither one of them is close to the ideal elliptical one, although that of Australia II is the worst, with a large droop near the keel-hull juncture. This droop is associated with vortex shedding, which increases the drag. Obviously, in neither case is the hull a good side-force producer. Wave-Resistance Calculations at Zero Heel and Leeway

The introduction of the free surface complicates calculations considerably. Whereas on a solid boundary the only (inviscid) condition required is that the flow be tangential to the surface everywhere, a free surface must be given the freedom to deform in such a way that the pressure is constant at all points on the surface. Since no flow is allowed across this deforming boundary, the same kinematic boundary condition as on a solid wall also has to apply. There are thus two conditions to be satisfied, one kinematic and one dynamic, and the location where they should be applied is unknown a priori. In practice, this problem is always solved using some kind of perturbation technique. An obvious first guess of the location of the free surface is the undisturbed water level. In the linear methods this location is not updated, whereas in the nonlinear methods the boundary condition is applied to the free surface computed in a previous iteration. The kinematic and dynamic boundary conditions are usually combined and linearized about a basic, known flow, which is not updatedin the linear methods. Nonlinear

Lift

Australia II

Figure 14Lift distribution for Australia II and Liberty. Calculations by Boppe et al. (1987). SCIENTIFIC METHODS IN YACHT DESIGN 375 effects may be taken into account iteratively by using the previous solution as the basic flow. The choice of the (initial) basic flow may be made in two different ways. In the traditional thin-ship theory, the linearization is simply made about the undisturbed flow. Disturbances due to the hull then have to be small, which makes this theory less accurate for beamy hulls. Another possibility is to linearize about a double-model solution, obtained (as explained above) by considering the free surface to be a reflection plane. The major disturbances to the onset flow are then considered already in the basic flow, and more beamy and full hulls can be examined. A brief introduction and evaluation of the different methods may be found in Bai & McCarthy (1979), and a more detailed analysis of the theory is given by Yeung (1982). While there are several techniques for the solution of the free-surface problem, only panel methods are considered here. These are by far the most common ones, and they are the only ones that have been applied in sailing-yacht investigations. Havelock (1928) found a Green function that satisfies the free-surface boundary condition, linearized about the free-stream velocity. If this Have- lock source is employed, source strengths must be adjusted in such a way that the hull boundary condition is satisfied, while the free-surface boundary condition is satisfied automatically. One disadvantage of the Havelock source is that the potential function is relatively complicated. Therefore, methods employing standard Rankine sources have recently become increasingly popular, particularly since double-model linearized solutions can then be obtained. An interesting study using Havelock' sources, distributed on panels at the centerplane of the hull, was presented by Scragg et al., (1987). The purpose was to find the optimum section areacurve' for given speeds and waterline lengths. Taking the unknown ,constant source strengths on the panels as design variables and the wave resistance, computed from the energy flux through a control surface behind the hull, as the object function, the optimum source distribution could be obtained by the method of Lagrange multipliers. Two constraints could be applied: the displacement and the longitudinal position of the center of buoyancy. By integrating the volume flux from the sources in front of a particular station, the cross-section area of the hull at that station could be found. In Figure 15 an example is given where the optimum section area has been computed for varying speeds. (For proprietary reasons, actual speeds are not revealed.) The strategy adopted by the Sail America syndicate was first to compute

The section area curve represents the longitudinal ,distribution of the displacement of the hull._ 376 LARSSON

Area

A11414\4\ Figure 15Optimum section area curves. / High speed The center of buoyancy is located amidships. Calculations by Scragg et al..(1987)..

Stern Bow

the optimum section area curve,, thereafter giving an experienced yacht designer freedom to develop detailed hull lines based on the computed displacement distribution. Once this had been done, another numerical check was made before it was decided whether or not the hull should be model tested. This second calculation was carried out by a method slightly more exact than the thin-ship approximation used in the optimization procedure (see Scragg et al. 1987).. Rather than the source panels being located on the centerplane, they were now moved to the surface of the hull and assumed proportional to the normal component of the free-stream velocity. Several tests of this method were made against measured data, and it was found that the computed wave resistance was consistently overpredicted. On the other hand, the correct trends were apparently obtained. The hulls were always ranked in the right order by the numerical method, which could then be used for screening the proposed hull forms, before they were accepted for the relatively expensive model test program. If the Kelvin sources on the hull surface are computed from the exact zero-normal-velocity condition (Neumann condition), the Neumann-Kel- vin approximation is obtained. Calculations based on this theory were carried out by M.-S. Chang and the author (Larsson 1979) for the Antiope hull. The results, may be seen in Figure 16, where the predicted wave resistance is compared with measurements of the residuary resistance. A different method for solving free-surface potential-flow problems was developed by Dawson (1977). This method has been extensively tested and generalized in the author's CFD group (Larsson et al. 1989a), and several applications to sailing yachts have been made (see Larsson. 1987). Daw- son's method is linearized about the double-model solution, and Rankine sources are distributed on the hull as well as on. part of the free surface. Figure 1,7 shows typical panel distributions.. (The free-surface grid is for the yawed case, to be discussed in the next section.) The calculated wave resistance for Antiope is shown in Figure 16. Relatively good cor- SCIENTIFIC METHODS IN YACHT DESIGN 377

.10

6 0 0 Herreshoff & Newman (1967) measurements Larsson (1979) 4 0 calculations

Larsson (1987) A calculations 2 0 Kirkman (1974) / measurements

0.1 0.2 0.3 0.1+

Figure 16Comparison between measured and calculated wave resistance for Antiope.

WimmiMMIMINIIIIWr \1111111111111MMIIINE Strip '11M1111111111EMENNI IIIIM11111111/

Figure 17Panel distribution for Antiope. (Top) free surface; (bottom) hull. 378 LARSSON

Cw .10 Cakutationso

Measurements 0

0.25 0.30 0.35 040 0.20 Fn Figure 18Comparison between measured and calculated wave resistance for a 12-m yacht. respondence with measurements may be noted. Considerably worse results were obtained for a 12-m yacht, as seen inFigure 18. These results are consistent with the ones obtained by Scra2g et al. (1987). Computed values are much larger than measured ones,particularly at higher speeds. As a matter of fact this situation is inevitable when one uses a linear free-surface method. In this approximation the surface is always kept at its undisturbed level, which means that the large effects of the overhangs at both ends of the hull are not considered. At full speed the waterline length of a 12-m yacht increases by at least 25% owing to the generated wave system. This reduces the wave resistance considerably. It should be noted that the effect is much smaller for Antiope, which has less inclined ends. The only way that the change in waterline length can be taken into account is to make use of a nonlinear method, where thefree surface, as well as the submerged part of the hull, is allowed to change. Such a methodhas recently been developed by Kim (1989). Free-Surface Calculations Including Lift Three well-known aerodynamic potential-flow methods withlift have been generalized to include the free surface. Van Beek et al. (1985) andRaven (1987, 1988) developed the Dutch NLR method (Labrujere etal. 1970), Xia & Larsson (1986) extended the Douglas method (Hess1972), arid Boppe et al. (1987) modified VSAERO (Maskew1982). In all cases a Dawson type of approach was chosen. Considerabledifferences do, however, exist between the methods. Van Beek et al.and Boppe et al. use a combination of sourcesand doublets on the free surface, whereas Xia & Larsson and Raven have stuck to Dawson's originalidea of using only sources. The free-surfaceboundary condition in the VSAERO generaliza- SCIENTIFIC METHODS IN YACHT DESIGN 379 tion is different from that of the other two; it is modified to fit VSAERO's way of satisfying the solid boundary condition by an internal zero-perturbation approach, rather than the regular Neumann condition. Other differences are related to the free-surface grid, finite differencing on the free surface, and wave damping at the lateral edges of the paneled part of the surface. In the following, some more details are given of the Xia & Larsson method. As in the original Hess (1972) method, the body is divided into "lifting" and "nonlifting" sections. The keel, which is obviously lifting, is divided into longitudinal strips (see Figure 17), each containing a number of panels. On the hull the panels may be arranged more arbitrarily. In Dawson's method the free-surface boundary condition is expressed in terms of derivatives of potentials along the double-model streamlines on the undisturbed surface. Therefore, a grid based on these streamlines may be convenient. During the development of the lifting method, it was found that such a grid could not be used for hulls at a yaw angle, since the bow and stern regions became inaccurately resolved. An algebraic body-fitted grid was therefore developed, and the boundary condition was rederived to include derivatives in the longitudinal and transverse directions of the grid. Each panel on all sections, including the free surface, has a constant- density source distribution, which is initially unknown. In addition, the panels on the lifting sections have an unknown doublet distribution. The latter is constant spanwise on each strip but is proportional to the arc length along the strip chordwise. The arc length is measured from the trailing edge on one side around the leading edge and back to the trailing edge on the other side. From this position, the doublet strength is continued unchanged on a few dummy panels placed in the wake behind the strip. In this way a constant bound vorticity on each strip is simulated, with free vortices shed at the common boundary between two strips. In the wake, the bound vorticity is zero. The unknowns of the problem are the source strengths and the doublet derivatives, one on each strip. A closed system of equations is obtained from the boundary conditions on the hull and the free surface and from the Kutta condition applied at each strip. The latter condition, which is nonlinear, is satisfied by specifying that the pressure is to be the same at the control points of the panels closest to the trailing edge on the two sides. As in the original Hess method, the problem is solved by a technique that avoids the solution of a large nonlinear system. A set of linear solutions is obtained under the assumption of zero doublet strength, except on one strip, where the derivative is unity. In each of these calculations, the onset flow is set to zero, but there is also one solution for the right onset flow with all doublets set to zero. All solutions are linearly combined with the 380 LARSSON

o CL Calculations a Catcutations A 0 CL o x102

x 102

}Measurements Fn = 0.35 40 Fn.°35G 4.0 0.3 0.3 0.0 0.0

d.° Measurements 2.0 2.0

0.0 0.0 5.0 a° 2.0 4.0 C0

Figure 19Comparison between measured and calculated lift and dragfor Antiope. (Left) lift vs leeway angle; (right) lift-drag polar. doublet derivatives as unknowns. One such linear combination for each Kutta condition yields a small nonlinear system that can be solved fairly easily. The free-surface condition is expressed in such a way in each linear calculation that it is satisfied in the combined solution. For more details, see Xia & Larsson (1986). Results from calculations for Antiope are given in Figures 19 and20. Four Froude numbers and five leeway angles have been calculated. In Figure 19 a small Froude number dependence on the lift can be seenin the computed results, but such a dependence is difficult to judgefrom the

0.05

Leeway angle

0.00 5.00 10.00 co .10

Figure 20Computed lift-drag polars for Antiope for varying Froude numbers. SCIENTIFIC METHODS IN YACHT DESIGN 381 measurements due to scatter. The predictions are nevertheless quite good considering the neglect of viscous effects. Avery regular pattern is observed in Figure 20, where lift-drag polarsare plotted for different Froude numbers. The slopes of the lines of constant leeway indicatea Froude- number dependence. Viscous Flow Calculations Very few calculations of the viscous flow around sailing yachts havebeen reported in the literature. The early attempt by Liden & Sallin (1975) has already been referred to, as have the Antiope calculations by the author in 1979. Apart from these, two recent calculations have also been reported. Both were aimed at predicting the location of separationnear the stern. Boppe et al. (1987) attempted touse two-dimensional boundary-layer theory at first but found it to yield results quite different from experimental data. They therefore turned to Navier-Stokes solutions and designeda grid that extended approximately halfa waterline length upstream and downstream of the hull and one length sideways and downward. The calculations were carried out by the method dueto Pulliam & Steger (1980), and the results are in qualitative agreement with data for theextent of the separated zone for a 12-m hull. The other set of separation predictionswas presented by the author (Larsson 1987) using three-dimensional boundary-layer theory(Larsson 1976). One purpose of the investigationwas to check the Reynolds-number sensitivity of the separation. The question had arisen whetheror not separation would occur on a small model, although it hadnot been observed either at full scaleor at a large (1: 3) model scale. The calculated skin-friction coefficient alongone critical streamline is shown in Figure 21.

CF x103

1 0.5 1.0 Midship L / 2 Stern Figure 21Skin-friction distribution along a representative streamlinefor a 12-m yacht. Results are given for four different scales. 382 LARSSON Results are given for four scales, and it it seen thatalthough there is a large droop close to the stern for the smallest scale,the skin friction is always positive. This indicates that separation will not occur. Arguments may be raised against each of the two setsof calculations reported. Obviously, the accuracy of boundary-layersolutions deteriorates close to separation, but it has still been frequentlyused in the past for rough estimates of the separation point, at least ontwo-dimensional and axisymmetric bodies (Chang 1970). There is every reason tobelieve that three-dimensional boundary-layer theory is adequatein other regions of the hull, as large curvatures are generallyavoided on sailing yachts. This theory has been proven to be very useful forship boundary layers, except close to the stern (Larsson 198 la). On theother hand, it is essential that the major effect of the three-dimensionality, namelystreamline convergence, is accounted for. Two-dimensional methods arebound to fail. A Navier-Stokes calculation might seemthe ideal way to find separation, but great caution must be exercised. Inorder to resolve the boundary layer properly, the grid must have several pointswithin the viscous sublayer, if wall functions are not used as the internalboundary condition. If wall functions are used, several points mustbe located inside the logarithmic region of the boundary layer (Broberg 1988).With a global grid like that used by Boppe et al. (1987), theserequirements cannot be met without excessive computer requirements. In fact,Boppe et al. state that the closest grid point was 0.04 ft away from thesurface. This distance is one to two orders of magnitude too great toresolve the boundary layer properly. With today's computers, it seemsthat the only way to overcomethis problem is to resort to a zonalapproach where potential flow/boundary layer/Navier-Stokes methods are matched.Such a system has recently been developed in the author's CFD group(see Larsson et al. 1989a,b).

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