THE 19 th CHESAPEAKE SAILING YACHT SYMPOSIUM. Tacking in the Wind Tunnel

THE 19 th CHESAPEAKE SAILING YACHT SYMPOSIUM ANNAPOLIS, MARYLAND, MARCH -1,9 Tacking in the Wind Tunnel Frederik C. Gerhardt, Yacht Research Unit, The University of Auckland, Auckland, New Zealand David Le Pelley, Yacht Research Unit, The University of Auckland, Auckland, New Zealand Richard G. J. Flay, Yacht Research Unit, The University of Auckland, Auckland, New Zealand Peter Richards, Yacht Research Unit, The University of Auckland, Auckland, New Zealand ABSTRACT In recent years a number of Dynamic Velocity Prediction Programs (DVPPs), which allow studying the behaviour of a yacht while tacking, have been developed. The aerodynamic models used in DVPPs usually suffer from a lack of available data on the behaviour of the sail forces at very low apparent wind angles where the sails are flogging. In this paper measured aerodynamic force and moment coefficients for apparent wind angles between and 3 are presented. Tests were carried out in the University of Auckland s Twisted Flow Wind Tunnel in a quasi-steady manner for stepwise changes of the apparent wind angle. Test results for different tacking scenarios (genoa flogging or backed) are presented and discussed and it is found that a backed headsail does not necessarily produce more drag than a flogging headsail but increases the beneficial yawing moment significantly. The quasisteady approach used in the wind tunnel tests does not account for unsteady effects like the aerodynamic inertia in roll due to the added mass of the sails. In the second part of paper the added mass moment of inertia of a mainsail is estimated by strip theory and found to be significant. Using expressions from the literature the order of magnitude of three-dimensional effects neglected in strip theory is then assessed. To further quantify the added inertia experiments with a mainsail model were carried out. Results from those tests are presented at the end of the paper and indicate that the added inertia is about 76 % of what strip theory predicts. NOTATION C F Force coefficient (-) C M Moment coefficient (-) C ω Drag constant for rotational motion (kg m²) c Chord length of mainsail (m) D Drag (N) E Foot of mainsail (m) F Force (N) f Boom height above centre of rotation (m) g Acceleration of gravity (m/s²) I Height of foretriangle (m) I Moment of inertia (kg m²) J Base of foretriangle (m) J added Added moment of inertia (kg m²) L Lift (N) LPG Perpendicular of longest jib (m) M Moment (N m) m added Added mass (kg) m Mass (kg) P Hoist of mainsail (m) q Dynamic pressure (N/m²) R Radius of pulley (m) S Reduction factor for added inertia V AW Apparent wind speed (m/s) V eff Effective wind speed (m/s) Centre of effort height (m) Z CE β AW Apparent wind angle ( ) β eff Effective wind angle ( ) Λ Aspect ratio (-) λ Taper ratio of mainsail (-) ρ Density of air (kg/m³) Ф Heel angle ( ) A F A M A N a b BAD Area of foretriangle (m²) Area of mainsail (m²) Nominal sail area (m²) Semimajor axis of ellipse (m) Semiminor axis of ellipse (m) Height of boom above deck (m) AC9 DVPP IACC IMS MHS VPP America s Cup 9 foot yacht Dynamic Velocity Prediction Program International America s Cup Class International Measurement System Measurement Handicapping System Velocity Prediction Program

1 INTRODUCTION Match races are usually won or lost on the upwind legs. A tacking duel in an America s Cup race can include more than 3 tacks per leg. Consequently a yacht that has been optimised to lose as little speed as possible during a tack has a huge advantage over its competitor. Over the last years a number of Dynamic Velocity Prediction Programs (DVPPs) have been developed to study the behaviour of yachts while tacking. These computer programs are usually based on numerically solving the equations of motion of the yacht according to Newton s second law. These equations consist of three force and three moment equations of the following types: (m+ m ) a = F + F (I + J ) ψ = M + M added,i i hydro,i aero,i i added,i i hydro,i aero,i Here the left-hand sides of the equations describe the inertia due the mass of the yacht and the added mass of the fluid surrounding it. The right hand sides represent the external hydrodynamic and aerodynamic forces (and moments) acting on the boat. To determine the external aerodynamic forces most DVPPs use modified versions of the aerodynamic model used in the International Measurement System Velocity Prediction Program (IMS- VPP) as described by Claughton, 1999. The IMS-VPP and its predecessor the MHS-VPP were developed for handicapping purposes based on steady state performance predictions and hence do not include sail force coefficients for apparent wind angles below 7 (Kerwin, 1978, Hazen, 198). However, during a tack the apparent wind angle becomes zero and therefore additional assumptions about the behaviour of the sail forces below β AW = 7 have to be made in most DVPPs. Basic quasi-steady time-domain studies of match races were first conducted by Oliver et al., 1987 and played an important role in designing the successful 1987 America s Cup challenger Stars and Stripes. The aerodynamic model used in these simulations was based on simple lifting-line theory. Larsson, 199 and Ottosson et al., report on a sailing simulator for match racing. The equations of motion of the yacht in four degrees of freedom (surge, sway, roll, yaw) are solved numerically. The quasi-steady aerodynamic model of the program is based on the IMS model. Masuyama et al., 1993, 1995, compared the calculated tacking behaviour of a yacht to full-scale results. Like Larson, Masuyama solves the equations of motion of the yacht in four degrees of freedom. For apparent wind angles larger than force coefficients from wind tunnel tests are used to describe the sail forces. Below β AW = Masuyama assumes the sail forces to change linearly i.e. the sail forces decrease from the close-hauled values at the start of the tack to zero and, then increase linearly again while bearing away. This behaviour had been observed in full-scale experiments. The model has subsequently been (1) refined (Masayuma and Fukasawa, 8) to account for the drag of flogging sails and for the time it takes to re-trim the sails on the new tack De Ridder et al., 4 presented an extension of Masuyama s model by formulating a set of equations containing only coefficients, which can be determined from theory or publicly available experimental data. The aerodynamic model is based on the IMS model. For low apparent wind angles the IMS coefficients are supplemented by self-defined values. To account for the drag of flapping sails the drive force below apparent wind angles of approximately 5 is negative. A user defined time-lag parameter accounts for trimming the sails after the tack. This model was subsequently modified and used to simulate tacking of an IACC yacht (Keuning et al. 7). The DVPP FRIENDSHIP-Equilibrium is also based on Masuyama s ideas but solves the equations of motion in six degrees of freedom (Richardt, 5). Harris, 5 developed a VPP capable of simultaneously simulating manoeuvring and seakeeping of a yacht while sailing upwind (e.g. tacking in waves). The quasi-steady aerodynamic model is based on panel code results and flogging of sails during a tack is accounted for by substituting the sail with a drag source if the angle of attack is below 5. Roux et al.,, 8 and Jacquin et al., 5, report on the development of a completely numerical VPP. As opposed to the conventional semi-empirical VPPs, their program calculates the hydrodynamic and aerodynamic forces acting on the sailing yacht solely from theory. The aerodynamic model used in this program was developed by Charvet et al., 1996 and is based on an unsteady lifting surface code. Charvet et al. numerically simulated the change in shape of a headsail during tacking, but no force or moment coefficients are presented. The choice of an inviscid panel model to simulate the separated flow around a flogging sail seems to be questionable. Recently Battistin and Ledri, 7 developed a tool for predicting the dynamic performance of sailing yachts. The aerodynamic model is again a modified version of the IMS model. An approach similar to Masuyama s is used to describe sail forces at low apparent wind angles. In summary it can be said that, when simulating tacking manoeuvres, most DVPPs use modified versions of the IMS-VPP aerodynamic model. A few DVPPs also use coefficients from wind tunnel tests or aerodynamic theory. These three approaches all suffer from the lack of available data on the behaviour of the sail forces at low apparent wind angles. So far no data seem to exist on the influence of different ways of handling the headsail during a tack (e.g. backing the headsail or releasing it early). The first part of this paper aims to fill parts of this gap by presenting sail force and moment coefficients from wind tunnel experiments. The second part of the paper then focuses on different ways to determine the most significant term on the left-hand side of equation (1), namely the added inertia in roll, J added,x.

QUASI-STEADY TACKING IN THE WIND TUNNEL z To derive sail force coefficients for the low apparent wind angles occurring during a tack quasi-steady wind tunnel tests were carried out. In these tests the apparent wind angle β AW was changed from 3 down to in discrete steps of 5. Different sheeting scenarios were investigated (e.g. flogging headsail, backing the headsail)..1 Wind tunnel and measurement technique All test were conducted in the University of Auckland s Twisted Flow Wind Tunnel. This open circuit boundary layer wind tunnel was developed specifically for testing yacht sails (Flay, 1996). The current paper focuses on the behaviour of upwind sails at low apparent wind angles. In this case the twist in the apparent wind that the yacht sees is negligible. Therefore no twist was introduced into the onset flow in the experiments. Similarly, when testing upwind sails, the boundary layer profile that develops over the sea is of secondary importance only and was thus not replicated in the tests. The aerodynamic forces and moments were measured by means of a six-component balance under the wind tunnel floor. A detailed description of the Twisted Flow Wind Tunnel and the measurement techniques used in the wind tunnel can be found in Hansen, 6. x Figure 1 Yacht model used in tests y z. Boat model and sail plan A 1:17 scale model of a generic 7ft IRC yacht was used for the wind tunnel tests (Figure 1). Also shown in Figure 1 is the boat fixed coordinate system used in this paper to describe the sail forces. The origin of this righthanded system is located at the aft face of the mast at deck height. The tests were carried out under genoa and mainsail. The geometry of these sails is summarised in Figure and in Table 1. Using the values from Table 1 the nominal sail area can be determined using equation (): 1 AN = AF + AM = ( I J+ P E) = 1.15 m () x I LPG E J Figure Geometry of sail plan P BAD In the following sections all measured forces and moments are non-dimensionalised using this area and the wind tunnel dynamic pressure q = ρ/ V² AW. C C F = F q AN (3) M M = 3/ q AN Table 1:Values of the geometrical parameters; in [mm] J I LPG P E BAD 56 18 75 19 67 1

.3 Wind tunnel test results for low apparent wind angles and comparison with IMS coefficients The majority of today s DVPPs utilise the IMS aerodynamic model to describe the sail forces. This force model is based on tabulated values of lift and drag produced by each sail as a function of the apparent wind angle. The lowest wind angle covered by the tables is β AW = 7. When beating to windward fast modern yachts usually sail at much tighter angles, typically around 19. The first part of the tests thus aimed at obtaining values of the sail forces for apparent wind angles lower than 7. In order to allow for some overlap with the IMScoefficients the apparent wind angle in the tests was decreased from 3 in steps of 5. In this section the range of 15 β AW 3 is discussed. In all of these tests the sails were trimmed for maximum drive force. Below 15 the genoa started to stall and flog, while the fully-battened main still kept its shape. Below 5 the mainsail then also started to lose its shape and flog. The tests for β AW < 15 are presented in sections.5 and.6. In the IMS model the lift and drag of the sails and the corresponding centre of effort heights are given for zero heel only. The effect of heel on the sail aerodynamics is accounted for by the heeled-plane model or effective angle theory. In effective angle theory, as discussed by Jackson, 1996, Campbell, 1997 and in more detail by Hansen, 6, it is assumed that the sails are insensitive to flow along their span (i.e. parallel to the mast) and that only the flow component in a plane perpendicular to the mast produces lift and drag. Geometric projection of the apparent wind vector into a plane normal to the mast yields an effective wind angle and an effective wind speed: tanβ = tanβ cosφ eff AW eff AW AW V = V 1 sin β sin φ The influence of heel on lift and drag is then accounted for by calculating effective lift and drag coefficients according to: with q eff = ρ/ V² eff. C F,eff ( β eff ) = q C ( β ) = eff F A M M,eff eff 3/ qeff AN Effective lift and drag coefficients from the wind tunnel tests are shown in Figure 3. The windage of the models hull and rigging was determined in separate bare pole tests and then substracted from the test results, i.e. the curves shown in the figure represent the lift and drag of the sails alone. N (4) (5) CDeff, CLeff 3 1 Lift Drag IMS F = = 15 = 3 genoa flogging 3 6 9 beff [ ] Figure 3 Effective lift and drag coefficients from wind tunnel tests. Sails trimmed for maximum drive force The symbols in Figure 3 represent measured force coefficients and the dashed lines are the coefficients predicted by the IMS aerodynamic model. It can be seen that the transition from the IMS curves to the measured upright coefficients (Ф =, square symbols) is relatively smooth and the measured coefficients simply extend the IMS predictions down to β AW = 15. The agreement between IMS model and wind tunnel measurements is somewhat better for the lift than for the drag. The large crosses at β eff = 1 represent a test run where the sails were also trimmed for maximum drive force but, due to the low apparent wind angle, the genoa was on the brink of flogging and large areas of separated flow could be observed. In Figure 3 it can also be seen that effective angle theory does not really collapse coefficients that were measured at different heel angles onto the same line. This is particularly obvious in the case of the lift coefficients recorded at Ф = 15 and Ф = 3. As discussed by Hansen, 6 this apparent failure of effective angle theory is attributed to sail-hull interaction. This interaction leads to forces parallel to the mast, which are ignored in effective angle theory. In a recent Seahorse article, Teeters, 7 found only an insignificant change in drive force over the heel range of Ф = to Ф = 3 at apparent wind angles ranging from β AW = to 3. The model used for the wind tunnel tests discussed in the article is of the Whitbread 6-type and has a length/beam ratio of approximately 3 whereas the length/beam ratio of the IRC yacht in Figure 1 is 4.5. Length/beam ratios of models used for commercial testing by the University of Auckland s Yacht Research Unit (YRU) are usually in the range of 3.8 (Volvo 7) to 5 (IACC). As discussed in the Seahorse article itself this very wide beam might be the reason for the peculiar

behaviour of the drive force measured on the Whitbread type model. In our present experiments and many other commercial YRU wind tunnel tests a definite change in drive and side force with heel angle has always been observed. Figure 4 shows the height of the sail force centre of effort above the foot of the mast. It can be seen that for the smaller heel angles the IMS model predicts the height correctly but for Ф = 3 the agreement is not so good. In the IMS model the centre of effort height is taken as the application point of the aerodynamic forces and used to work out the heeling moment produced by the side force. As discussed by Hansen, 6 the concept of expressing the aerodynamic moments by specifying only a vertical and a longitudinal position of the centre of effort is problematic because the sail forces do not act on the symmetry plane of the yacht but usually well outboard on the leeward side. For an accurate representation of measured sail forces one would therefore have to specify the x, y, z coordinates of the point of application of the aerodynamic forces. An alternative approach is to represent the overall aerodynamic forces and moments with reference to one fixed point. This is the approach taken in this paper. All force and moment coefficients presented here are with reference to a boat fixed (i.e. heeled), right handed coordinate system, the origin of which is located at the aft face of the mast at deck height (Figure 1). The positive x- axis points forward and y is positive to port. Tabulated force and moment coefficients for the set of tests discussed in this section can be found in the appendix (Table A1)..4 Windage of hull and rigging An important part of the overall drag of a yacht is the windage of the hull-topsides and the rigging. In the original MHS and first IMS models, topside windage was assumed to be independent of apparent wind angle β AW and was calculated based on maximum beam and average freeboard (i.e. the frontal area) of the yacht. Similarly the drag of mast and rigging was assumed to be independent of β AW as well, and was calculated from the mast diameter and the mast height. This crude model has subsequently been refined to account for, amongst other things, the change in windage with apparent wind angle (Claughton, 1999). Richards et al., 6 also discuss ways of predicting the windage of a yacht. They propose the following expression to account for the change in windage with apparent wind angle. C = C cos β + C sin β (6) D,windage D,min AW D,max AW where C D,min and C D,max are the drag coefficients at β AW = and 9 respectively. Figure 5 summarises the results from bare pole tests with the IRC 7 model yacht. The solid symbols in the figure represent the measured windage drag. The coefficients shown are based on the nominal sail area of the model. The dashed line is based on Equation (6) with the experimentally determined values of C D,min =.3 and C D,max =.. The dotted line represents the constant hull and mast drag according to the simple IMS model. It can be seen that Equation (6) describes the behaviour of the windage reasonably well while the simple IMS model overestimates the drag for low apparent wind angles..1.4.8 Z CE /(P+BAD). IMS F = = 15 = 3 CD,windage.4 Measured Eqn. (5) Simple IMS 1 3 4 5 beff [ ] 5 1 15 5 3 baw [ ] Figure 4 Measured centre of effort heights. Figure 5 Windage of hull and rigging at zero heel

.5 Flogging sails If the sails and in particular the headsail are not sheeted in far enough, they start to flap or flog (Figure 6). This situation inevitably occurs during a tack below a certain apparent wind angle but can also be brought about by cutting loose the headsail earlier than necessary. In this section measured aerodynamic forces and moments for apparent wind angles below 15 are presented. In order to keep the results independent of the yacht and as generic as possible, all measurements presented in this and the next section were carried out with the boat upright. Using the Real-Time VPP developed by Hansen, 6, tests have also been carried out with the boat heeled to its correct equilibrium angle. For the case of flogging sails the resulting heel angles were generally small (Ф < 9 ). Therefore the difference between upright and heeled coefficients will only be small as well. The measured force and moment coefficients are plotted in Figure 7 and Figure 8 and are also reproduced in the appendix (Table A). Three sets of data are presented in the figures. Firstly, there is a data set in which the sails were trimmed for maximum drive force (solid square symbols). This case was discussed in the previous section. The corresponding curve terminates at β AW = 15 where the genoa starts to flog. Secondly, a set in which the genoa was backed (triangular symbols). This situation is discussed in the next section. Thirdly, a set where the genoa and, below β AW = 5, also the mainsail flogs (open square symbols). These data can be seen as the extension of the maximum drive force set to lower wind angles. At β AW = 1 there is no way of avoiding a flapping genoa and, given that the main is re-trimmed, this then also represents a trim for maximum drive force. The dashed arrows in the diagrams indicate this transition from one set of data to the other. In order to analyse the affect of cutting loose the genoa at angles where it still has some shape the diagrams also contain flogging-values for β AW = 15 and. At β AW = the drive force coefficient is C Fx = -C D =.. This is also the value given by Keuning at al., 7. In full-scale tests Masayuma and Fukasawa, 8 recorded a value of only C D =.1. It is currently believed that this discrepancy could be caused by differences in sail plan or cloth weight and further experiments to clarify this are planned. It can be seen from the top diagram in Figure 7 that cutting the genoa loose early decreases the drive force significantly, at β AW = 15 it even becomes negative. If one does not release the headsail but only luffs, the drive force follows the line indicated by the arrow and becomes negative at about β AW = 1. Furthermore it can be observed that, because the genoa flogs below 15, the slope of the drive force curve decreases. From the middle diagram in Figure 7 it can be seen that cutting the headsail loose decreases the side force and that even with the headsail flogging the side force, now mostly produced by the mainsail, is still significant. The vertical force component is presented in the bottomdiagram. As discussed by Hansen, 6 this vertical force is partly caused by sail-hull interaction and partly by the headsail itself. Due to the forestay being angled backwards and the sheeting point being to leeward of the centreline, a plane through the three corners of the headsail is not perpendicular to the deck plane. This can produce a small vertical force component. As can be seen from the figure the vertical force increases with increasing apparent wind angle because the headsail produces more lift. From β AW = 3 onwards the endplate effect of the deck (sealing the gap underneath the headsail) is lost and the vertical force decreases again. Similarly cutting the headsail at apparent wind angles of 15 or decreases the vertical force. At zero heel the rolling moment (Figure 8, top diagram) is mainly caused by the aerodynamic side force and therefore follows a similar trend. The pitching moment (middle diagram) is caused by the drive force and the vertical force. It becomes negative (bow up) at low apparent wind angles when the sails produce a negative drive force. If the sails are trimmed for maximum drive force the yawing moment with reference to the foot of the mast is very small and positive (Figure 8, bottom). For a standard yacht this corresponds to the desired small amount of weather-helm. If the headsail is cut the mainsail produces all the side force and the yawing moment increases (i.e. more weather helm). In summary it can be said that cutting the headsail loose earlier than necessary leads to a significant loss in drive force and that probably the only reason why one might do this is to increase the yawing moment to pushes the yacht through a tack. Whether this increase in yawing moment is worth the drive force penalty can only be determined from VPP simulations. Figure 6 Flogging genoa

.6 1.5 F x M x.4 1...9 CFx CMx.6 -. -.4 -.4 genoa flogging max Fx genoa backed poorly backed baw [ ] 4 F y.3.9.6 baw [ ] 4 M y genoa flogging max Fx genoa backed -.8.3 CFy -1. CMy -1.6 - genoa flogging max Fx genoa backed baw [ ] 4 -.3 -.6 genoa flogging max Fx genoa backed baw [ ] 4.6.9 F z M z.4.6..3 CFz CMz -. -.4 genoa flogging max Fx genoa backed baw [ ] 4 -.3 -.6 genoa flogging max Fx genoa backed poorly backed baw [ ] 4 Figure 7 Measured force coefficients Ф = Figure 8 Measured moment coefficients Ф =

.6 Backing the headsail Backing of the headsail is a technique used to deliberately increase the yawing moment during tacking. When backing the headsail, releasing the old sheet is delayed. This is illustrated in Figure 9 for a yacht that was originally sailing on a starboard tack, changed heading and is now on port tack with β AW = 5. Backing the genoa in this situation will push the boat around more quickly. Somehow contrary to intuition backing does not rely on the drag of the headsail area that is exposed to the wind, but on the lift produced by the sail. Preliminary tests in the wind tunnel have shown that there is an optimum way of backing the genoa for each apparent wind angle. A good and a poor backing-trim are illustrated in Figure 9 and in Figure 1 respectively. For the good trim the genoa is relatively flat and aerofoilshaped. As can be seen from the tell-tales the flow around the genoa is attached and the lift produced by this sail will be relatively high while the pressure drag will be low. On the other hand the shape of the poorly backed genoa in Figure 1 is full and almost baggy. Due to separation a sail trimed in such a way will produce little lift and much drag. At low apparent wind angles the lift is the only force component that can produce a yawing moment, therefore the yawing moment of the good trim can be expected to be higher than the yawing moment of the poor trim. Similarly the increased pressure drag of the poor trim will lead to a larger drive force penalty. In the case shown in the photographs, the good trim produced an 8 % higher yawing moment at a drive force penalty that was only 6 % of the one of the poor trim. In the wind tunnel test results presented here, the backed headsail has consequently been trimmed to produce as much lift and therefore as big a yawing moment, as possible. In all cases this also led to the smallest drive force penalty. Whether this ideal backing-trim can also be achieved on the water and during a fast tacking manoeuvre will, however, very much depend on the skills of the crew. The measured force and moment coefficients for the mainsail and the backed genoa are plotted in Figure 7 and in Figure 8 and are also tabulated in the appendix (Table A). The most striking feature of the top diagram in Figure 7 is that the (negative) drive force under a correctly backed genoa is about the same as the drive force at the same apparent wind angle with the sail flogging. Most likely this is a result of the lower pressure drag of a backed, aerofoilshaped sail compared to the drag of a flogging sail. The diamond in the figure illustrates the case of a poor trim. The effect of backing can clearly be seen in the side force and yawing moment diagrams (Figure 7 middle and Figure 8 bottom).vertical force and pitching moment, however, seem to be largely unaffected by backing. At low apparent wind angles the large (negative) side force due to backing leads to an increased rolling moment (Figure 8, top). Judging from the yawing moment and drive force diagrams it seems to be advantageous to always back the headsail when coming out of a tack rather than to leave it flogging. y V AW x Figure 9 Backing the genoa, good trim β AW = 5 y V AW x Figure 1 - Backing the genoa, poor trim β AW = 5 A more detailed analysis producing quantitative results and instructions of when to cut the headsail loose or whether to back it or not will be boat specific and will have to include the use of a DVPP. The main goal of the tests presented in the last sections was to derive sail force coefficients that can be used for such DVPP simulations of different tacking strategies.

3 UNSTEADY AERODYNAMIC MOMENTS In quasi-steady wind tunnel tests one obviously cannot measure time dependent sail forces and moments. Probably the largest terms thus neglected in the above tests are the added mass and the added moment of inertia due to rolling acceleration of the yacht during the tack. Added mass or virtual mass is the inertia added to a system because an accelerating body must push aside and hence accelerate some volume of surrounding fluid as it moves through it (see Saunders, 1957 for more details on the concept of added mass). The second part of this paper focuses on the added moment of inertia of a mainsail of a yacht that rolls. In section 3.1 the added moment of inertia of a square-headed mainsail is estimated by means of strip theory. In section 3. expressions from the literature are then used to assess the significance of the three-dimensional effects neglected in strip theory. Section 3.3 finally summarises experiments that were carried out to determine the added mass moment of inertia of a model mainsail. 3.1 Approximating the added moment of inertia of a mainsail by strip theory Assuming that the added moment of inertia of the mainsail is primarily a function of the sail planform, an idea of its magnitude can be obtained from strip theory. In strip theory, three-dimensional effects are neglected and the sail is replaced by a flat plate of the same planform. This equivalent flat plate is then divided into an infinite number of strips and the added mass of each strip is calculated from potential flow theory. Integrating the individual added masses over the plate yields the overall added mass and added moment of inertia. Here ρ denotes the density of the fluid surrounding the plate. This basic expression can be used to work out the added moment of inertia acting on a mainsail. Figure 1 depicts an idealised mainsail of a yacht that rolls at an angular rate of φ around an axis that is located at f below the boom. λ E z P φ, φ df c(z) E dz Figure 1 Geometry of rolling mainsail The elemental added mass force acting on the strip located at z from the axis is: df = dmadded z φ (8) The moment produced by this force around the axis is: f a* c π = =ρ φ= φ (9) 4 dm df z c (z) z dz dj added And the added moment of inertia of the whole sail finally becomes: dz Figure 11 - Infinitely long flat plate that is accelerated Figure 11 shows an infinitely long, flat plate subject to an acceleration a* that acts perpendicular to the plate. The added mass of a strip of width dz can be determined from potential flow theory (e.g. Saunders, 1957). It is: dmadded π 4 =ρ c dz (7) J added,x f+ P π =ρ c (z) z dz 4 (1) This expression can now be used to obtain an idea of the magnitude of the added moment of inertia of a mainsail. Typical values of the geometry parameters for an AC9 mainsail are given in Table. Table : Approximate geometry of AC9 mainsail Foot length Hoist Boom height Taper ratio E P f λ 11 m 35.4 m 3. m.46 f

With these values and as shown in the appendix the added moment of inertia of the AC9 sail becomes: Jadded,x = 833 1 kg m 3 This factor is a function of the aspect ratio Λ of the ellipse. Λ = 4a 4 a A = π b ellipse (13) This is quite large. For comparison, the mass moment of inertia of the keel bulb of an AC9 yacht is approximately. I D M = (6 m) 19 kg = 684 1 kg m bulb,x 3 Tuckerman gives the following expression for S(Λ) 16 S( Λ ) = πλ + πλ E ( 3) 16 K (14) Where D is the draft of the yacht and M the mass of the bulb. It can be seen that added moment of inertia of the mainsail is of the same order of magnitude (and even larger) than the mass moment of inertia of the bulb. It hence becomes obvious that the mainsail will have a significant influence on the rolling response and consequently on the tacking behaviour of a yacht and it seems appropriate to investigate the added moment of inertia of the mainsail of a rolling yacht in more detail. 3. Significance of three-dimensional effects The strip theory based on Figure 11 and Equation (7) assumes the sail to be infinitely long and thus neglects the three-dimensional flow around the head and the foot of the sail. An initial idea of the significance of these threedimensional effects can be obtained by analysing the added moments of inertia of an elliptical plate rotating around its smaller axis b, this situation is illustrated in Figure 13. For the case of such an elliptical plate, analytic expressions that also include three-dimensional effects, are given by Tuckerman, 196. This exact solution can be compared to the strip-theory approximation for the same elliptical plate. The difference between exact and strip theory solution then indicates how much the flow around the head and foot of the ellipse reduces the added inertia. The dashed lines in Figure 13 show an equivalent mainsail of the same area A and the same hoist P = a. According to strip theory (Equation 1) the added moment of inertia of the elliptical plate sketched in Figure 13 becomes: J 4 15 3 added,x =ρ π a b (11) In this expression K and E are complete elliptical integrals of the first and second kind respectively. π/ 1 K = dϕ 1 k sin ϕ E π/ 1 k sin d with k as the elliptic modulus: = ϕ ϕ a b 16 (15) k = = 1 (16) a πλ These elliptical integrals can be solved numerically and the function S(Λ) can be plotted (Figure 14). A ellipse = A sail z P = a The corresponding exact solution after Tuckerman can be written as: 4 =ρ π Λ (1) 15 3 Jadded,x,exact a b S( ) Here S( Λ ) can be interpreted a factor describing the influence of the 3D effects. b Figure 13 Rotating elliptical plate and equivalent sail

The aspect ratio of the equivalent ellipse needs to equal the one of the AC9 sail: 4a P P Λ= = = = 4.41 A A E (1 +λ) ellipse sail approach based on a rotational acceleration of the sail has been used herein. The experimental setup is sketched in Figure 15. The dashed line in Figure 14 indicates this aspect ratio of 4.41. The corresponding reduction factor is S =.79. This means that the three-dimensional flow around the tips of the ellipse reduces the added moment of inertia to 79 % of the value that is predicted by strip theory. It can also be seen from the figure that only for aspect ratios larger than, say Λ = 18 the difference between strip theory and exact solution becomes insignificant. In the limit of Λ (infinitely long plate) Tuckerman s solution is identical to strip theory, S = 1. I pulley m g R φ, φ, φ I shaft I sail, J sail 1. Figure 15 Schematic drawing of experimental setup 1.8 S.6.4 model support. AC9 aspect ratio 5 1 15 L Figure 14 Reduction factor S as function of aspect ratio 3.3 Experimental determination of added moment of inertia of a mainsail In order to further quantify the added moment of inertia of a mainsail of a rolling yacht the authors have also attempted to measure the added inertia of a model sail. These experiments are summarized in the next paragraphs. 3.3.1 Experimental Setup Nazarov et al., 6 have determined the added mass of hull-keel combinations by means of straight-line accelerated towing tank tests. In order to determine the added moment of inertia of a model mainsail a similar sail model Figure 16 Experimental setup A model of a mainsail was mounted on a vertical shaft. One end of a string was attached to a pulley on the shaft while the other end was attached to a weight of mass m. At t = the weight was released and the angle Ф and its rate were recorded. From these time histories the added moment of inertia of the mainsail can be determined. Figure 16 shows the actual experimental setup. The shaft and pulley can be seen in the foreground. The weight is in the centre of the photograph and the sail model and supporting structure are in the background.

Assuming that the added moment of inertia is primarily a function of the sail planform, the sail model was manufactured from a flat sheet of 3 mm MDF and did not have any camber. The 1:17 model geometry was based on the AC9 mainsail from Table. A shaft-mounted potentiometer was used to record the angle Ф and an accelerometer on the sail recorded the angular acceleration. Both signals were digitally sampled and analysed on a PC. 3.3.3 Experimental results Figure 17 shows the results of tests with two different weights (5 kg and 1 kg). The black dots represent measured values of the angle Ф and the grey continuous lines are curves according to Equation (19). In the 1 kg case the mainsail reached its end-stop at 7 after 6 seconds, from thereon the recorded angle stayed constant. 3.3. Data reduction 3 If the mass moments of inertia of sail and shaft are known, the added mass moment of inertia of the sail can be worked out from the time history of the angle Ф. The differential equation describing the motion of the system sketched in Figure 15 is: 5 fitted curve measured φ Isail,x + Ishaft + Jsail,x + mr + Cωφ = mgr M (17) f F [ ] 15 Here J sail,x is the added moment of inertia that we want to measure, I sail,x and I shaft are the mass moments of inertia of sail model and shaft C ω is a constant describing the drag of the sail in rotational motion and M f is a term describing the friction in the bearings of the system. Initial experiments without the sail model have shown that the friction term is very small. Hence it will be neglected in the following analysis so M f =. Equation (17) can then be rewritten as: 1 5 1 kg 5 kg 4 6 8 1 t [s] φ + φ = (18) I* Cω m g R Figure 17 Time histories of angle Ф and fitted curves As shown in the appendix the solution of this equation in terms of the angle Ф is: I* m g R φ (t) = ln cosh t mgr I* ω (19) In this expression ω denotes the terminal angular velocity for t. mgr ω =φ (t ) = () C As expected the terminal velocity is a function of the driving moment m g R and the drag of the sail only and independent of the inertia terms. Equation (19) contains two unknown quantities, the inertia of the system I* and the drag constant C ω. By fitting a curve of the type given by Equation (19) to the measured time histories of the angle Ф these two parameters can be determined. The mass moments of inertia of sail and shaft can be calculated from standard theory and the added moment of inertia becomes: J I* I I mr sail,x = sail,x shaft (1) ω For both the 5 kg and the 1 kg cases, the parameters used to generate these curves are: I* = 5.5 kg m C = 4 kg m ω The mass moment of inertia of the shaft and the sail model was calculated to be: I + I = 4.786 kg m sail,x shaft With a pulley radius of R = 37.5 mm the added moment of inertia of the model sail can be calculated from Equation (1). Averaging over several such measurements results in: Jsail,x =.45 kg m The corresponding strip theory value from Equation (1) for the model sail is: Jadded,x =.59 kg m

Comparing these two results shows again that strip theory overestimates the added inertia of the mainsail because it neglects the three-dimensional flow around the head and foot of the sail. The experimentally derived reduction factor describing the error of strip theory thus becomes:.45 S= =.76.59 This agrees well with the estimation S =.79 from section 3.. It should however be noted that in our tests the inertia of shaft and sail was 1 times bigger than the added inertia that was to be measured and that the mass moments of inertia of the setup had to be calculated. Hence the accuracy of our tests was only in the order of.5 kg m². More accurate results could probably be obtained from similar tests in water where the added inertia exceeds the ordinary mass inertia. 4 CONCLUSIONS In the first part of this paper experimentally derived sail force and moment coefficients for apparent wind angles between 3 and were presented and discussed. Coefficients for such low apparent wind angles are required when analysing the tacking performance of yachts with the help of DVPPs. To derive these coefficients, tests were carried out in the University of Auckland s Twisted Flow Wind Tunnel in a quasi-steady manner for stepwise changes in the apparent wind angle. To facilitate simulations of different and realistic tacking scenarios, experiments with flogging and backed sails were also carried out. In the first set of tests, comprising apparent wind angles between 3 and 15, the sails where trimmed to deliver maximum drive force. It was found that near β AW = 3 the IMS coefficients describe the lift and drag of the sails reasonably well. The method of accounting for the effect of heel on sail aerodynamics by projecting the apparent wind vector into a plane perpendicular to the mast ( heeled-plane model or effective angle theory ) was found to underestimate the decrease in sail forces. The windage drag of the model under bare poles was found to change significantly with apparent wind angle but this change can be accounted for by simple theories from the literature (Richards et al., 6). In a second set of tests ( < β AW < ) the sail forces and moments under flogging sails were investigated. At β AW = the drag coefficient of the flogging sails was measured to be C D =.. This compares well to the drag predicted by Keuning at al., 7. Masayuma and Fukasawa, 8 however recorded a value of C D =.1 in full-scale tests. It is currently believed that this discrepancy could be caused by differences in sail plan or cloth weight and further experiments to clarify this are planned. Furthermore the wind tunnel test results show that cutting the headsail loose earlier than necessary obviously leads to a significant loss in drive force on the one hand but also to an increase in yawing moment pushing the yacht into the tack on the other hand. Whether this increase in yawing moment, at the cost of a decrease in drive force, leads to a gain in overall tacking performance can only be determined from DVPP simulations. In a third set of tests ( < β AW < ) the technique of backing the headsail was investigated. It was found that a correctly backed headsail does not produce more drag than a flogging headsail but increases the beneficial yawing moment significantly. Unsteady added mass forces and moments could not be measured in these quasi-steady wind tunnel tests, but were addressed in the second part of the paper. The added moment of inertia in roll of the mainsail of an AC9 yacht was approximated by two-dimensional strip theory. It was shown that this added inertia is of the same order of magnitude as the inertia of the keel bulb and will therefore considerably influence the tacking behaviour of the yacht. Using analytical expressions from the literature that also include three-dimensional flow effects, the added inertia in roll of the mainsail was approximated and was found to be about 79 % of the value predicted by the two-dimensional strip theory. Experiments to determine the added inertia of a mainsail model have been carried out and the results showed that the added inertia was about 76 % of what strip theory predicted. It follows that the difference between predictions based on strip theory and measured values of the added inertia in roll is probably too large to be neglected when analysing the tacking performance of yachts. ACKNOWLEDGEMENTS The corresponding author gratefully acknowledges the financial support of the Yacht Research Unit. REFERENCES Battistin, D., Ledri, M., A Tool for Time Dependent Performance Prediction and Optimization of Sailing Yachts, SNAME 18th CSYS, Annapolis, MD, 7. Campbell, I., Optimisation of a Sailing Rig Using Wind Tunnel Data, SNAME 13th CSYS, Annapolis, MD, 1997.

Charvet, T., Hauville, F., Huberson, S., Numerical Simulation of the Flow Over Sails in Real Sailing Conditions, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 63, p. 111-19, 1996. Claughton, A., Developments in the IMS VPP Formulations, SNAME 14th CSYS, Annapolis, MD, 1999. De Ridder, E.J., Vermeulen, K. J. Keuning, J. A, A Mathematical Model for the Tacking Manoeuvre of a Sailing Yacht, HISWA 18th HISWA Symposium on Yacht Design and Construction, Amsterdam, 4. Flay, R. G. J., A twisted flow wind tunnel for testing yacht sails, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 63, p. 171-18, 1996. Hansen, H., Enhanced wind tunnel techniques and aerodynamic force models for yacht sails, PhD thesis, University of Auckland, Auckland, 6. Harris, D. H., Time Domain Simulation of Yacht Sailing Upwind in Waves, SNAME 17th CSYS, Annapolis, MD, 5. Hazen, G., A Model of Sail Aerodynamics for Diverse Rig Types, New England Sailing Yacht Symposium. New London, CT, 198. Jackson, P., Modelling the Aerodynamics of Upwind Sail, Journal of Wind Engineering and Industrial Aerodynamics, Vol. 63, p. 17-34, 1996. Jacquin, E., Roux, Y., Guillerm, P. E., Alessandrini, B., Toward numerical VPP with the full coupling of hydrodynamic and aerodynamic solvers for ACC yacht, SNAME 17th CSYS, Annapolis, MD, 5. Kerwin, J.E., A Velocity Prediction Program for Ocean Racing Yachts revised to February 1978, Department of Ocean Engineering, Massachusetts Institute of Technology, Rep. 78-11, Cambridge, MA, 1978. Keuning, J. A., Katgert, M., Mohnhaupt, A., The Use of a Manoeuvring Model for the Optimization of the Tacking Procedure of an IACC Sailing Yacht, RINA The Modern Yacht, Southampton, 7. Larsson, L., Scientific Methods in Yacht Design, Ann. Rev. Fluid Mech., Vol., p. 349-385, 199. Masuyama, Y., Nakamura, I., Hisayoshi, T., Ken, T., Dynamic Performance of Sailing Cruiser by Full-Scale Sea Trials, SNAME 11th CSYS, Annapolis, MD, 1993. Masuyama, Y., Fukasawa, T., Sasagawa, H., Tacking Simulations of Sailing Yachts, Numerical Integration of Equation of Motion and Application of Neural Network Technique, SNAME 1th CSYS, Annapolis, MD, 1995. Masayuma, Y., Fukasawa, T., Tacking Simulations of Sailing Yachts With New Model of Aerodynamic Force Variation, RINA 3rd High Performance Yacht Design Conference, Auckland, 8. Nazarov, A., Dolinsky, D., Agishev, I., Experimental Research of Sailing Craft Added Masses for Hull-Keel Combinations by Acceleration Tank Tests, RINA nd High Performance Yacht Design Conference, Auckland, 6. Oliver, J.C., Letcher, J. S. Jr, Salvesen, N., Performance predictions for Stars & Stripes, SNAME Transactions Vol. 95, p. 39-61, 1987. Ottosson, P., Brown, M., Larsson, L., The Effect of Pitch Radius of Gyration on Sailing Yacht Performance, RINA High Performance Yacht Design Conference, Auckland,. Richards, P., Le Pelley, D., Cazala, A., McCarty, M., Hansen, H., Moore, W., The Use of Independent Supports and Semi-Rigid Sails in Wind Tunnel Studies, RINA nd High Performance Yacht Design Conference, Auckland, 6. Richardt, T., Harries, S., Hochkirch, K., Maneuvering Simulations for Ships and Sailing Yachts using FRIENDSHIP-Equilibrium as an Open Modular Workbench, International EuroConference on Computer Applications and Information Technology in the Maritime Industries, Hamburg, 5. Roux, Y., Huberson, S., Hauville, F., Boin, J. F., Guilbaud, M. Malick, B., Yacht Performance Prediction: Towards a numerical VPP, RINA High Performance Yacht Design Conference, Auckland,. Roux, Y., Durand, M., Leroyer, A., Queutey, P., Visonneau, M., Raymond, J., Finot, J. M., Hauville, F., Purwanto, A., Strongly Coupled VPP and CFD RANSE Code for Sailing Yacht Performance Prediction, RINA 3rd High Performance Yacht Design Conference, Auckland, 8. Saunders, H. E., Hydrodynamics in ship design, SNAME, New York, 1957. Teeters, J., The story so far, Seahorse, July 7. Tuckerman, L.B., Inertia Factors of Ellipsoids for Use in Airship Design, NACA TR 1, 196.

APPENDIX Force and moment coefficients from wind tunnel tests All force and moment coefficients presented in this paper are with reference to a boat fixed (i.e. heeled), right handed coordinate system the origin of which is located at the aft face of the mast at deck height. The positive x-axis points forward and y is positive to port (Figure 1). In all tests the model was sailing on a port tack, when used for tacking simulations the signs of side force, yawing moment and rolling moment need to be inverted if the yacht sails on a starboard tack. The windage of hull and rigging has been substracted from the measured values and is thus not included in the coefficients presented here. The following equations relate the values of side force and drive force given in the tables below to the lift and drag values of section.3: C = C cosβ + C sinβ C = C sinβ C cosβ L Fy AW Fx AW D Fy AW Fx AW Table A1 Coefficients for maximum drive force trim (A1) baw [ ] F [ ] Side F. Drive F. Vertical F. Pitch M. Roll M. Yaw M. 5 15-1.49.4.35.3 1.13.4 3 15-1.6.53.37.31 1.19.5 3 3-1.33.45.49.34 1.15.4 5 3-1.19.33.46.6 1.8.5 5-1.68.44.1.7 1.19. -1.43.9.19.16 1..3 15-1.13.14.9.11.83.8 3-1.81.6.15.38 1.3. 1 8.3 -.46 -.5.3 -..36.1 Table A - Coefficients for different headsail trims, Ф = baw [ ] Genoa trim Side F. Drive F. Vertical F. Pitch M. Roll M. Yaw M. 3 Max Drive -1.81.6.15.38 1.3. 5 Max Drive -1.68.44.1.7 1.19. Max Drive -1.43.9.19.16 1..3 15 Max Drive -1.13.14.9.11.83.8 Flogging -.94.4..9.71.14 15 Flogging -.78 -.4 -.1.5.61.1 1 Flogging -.5 -.8.7 -.3.43.9 5 Flogging -.8 -.11.4 -.5.4.6 Flogging -.6 -..1 -.11.8.1 Backed -1.1..4.11.71 -.1 15 Backed -.99 -.8.7.5.69 -.13 5 Backed -.8 -.1.6 -.1.57 -.19 Backed -.37 -.16. -.6.3 -.1 boom height (z = f) to the mast top (z = P+f) gives: π 1 3 Jadded,x = ρ E P [6λ + 3λ+ 1 + f P λ + λ+ + f P λ +λ+ 5 (3 1) 1 ( 1)] (A3) Inserting the values from Table for E, P, f and λ yields: Jadded,x = 833 1 kg m 3 For a triangular sail that rotates around the boom: λ = and f =. Equation (A3) reduces to: π Jadded,x = ρ E P 1 3 This is the expression used by Masayuma et al., 1995 to determine the added moment of inertia of a mainsail. Solution of Equation (18) Rewriting Equation (18) in terms of the angular velocity ω = d φ /dt gives: ω I* + Cω ω = m g R dω mgr C 1 ω = ω dt I* mgr Substituting: (A4) mgr ω = (A5) C into Equation (A4) and separating the variables gives: ω mgr dω = dt ω ω I* ω (A6) Integrating both sides and rearranging yields: Added moment of inertia of AC 9 mainsail The chord-length of the sail shown in Figure 1 is given by: (z f ) c(z) = E 1 (1 λ) P (A) Inserting this into Equation (1) and integrating from mgr t I* ω mgr t I* ω e 1 mgr ω (t) =ω =ω tanh t I* ω e + 1 (A7) This can be integrated once more and then becomes equation (19): I* m g R φ (t) = ln cosh t (19) m g R I* ω