OPTIMISATION OF SPAN-WISE LIFT DISTRIBUTIONS FOR UPWIND SAILS

Transcription

1 3 rd High Performance Yacht Design Conference Auckland, -4 December, 8 OPTIMISATION OF SPAN-WISE LIFT DISTRIBUTIONS FOR UPWIND SAILS Peter Richards, pj.richards@auckland.ac.nz T. Junge, timm.junge@euroavia.de H. Hansen 3, hansen@friendship-counsulting.com D.J. Le Pelley 4, d.lepelley@auckland.ac.nz F. C. Gerhardt 5, fger3@aucklanduni.ac.nz Abstract. Wind tunnel experiments using a Real-Time Velocity Prediction Programme to investigate the optimal trim of a VO7 model under various simulated true wind speeds are reported. The results illustrate that the decision made depend upon the particular apparent wind direction and true wind speed. It is suggested that these can be sub-divided into three broad bands: low wind speeds where the total drag is minimised and the trim that provides the maximum thrust coefficient is chosen, moderate wind speeds where the heel angle has a strong effect and the optimum choice includes a reduction in lift coefficient and centre of effort height and strong winds where the heel angle and hence heeling moment is limited to the maximum acceptable value and the optimum loading distribution is strongly constrained by this limit. Extended Lifting Line Theory is used to further investigate the detailed loading distribution on an AC9 mainsail. The result illustrate the way in which the optimal distribution changes with varying conditions.. INTRODUCTION With aircraft wings and similar lifting surfaces performance optimisation is often linked with minimisation of the induced drag, which in turn leads to the well known elliptical lift distribution and a uniform downwash across the span. Jones [] has shown that if additional constraints are added, such that for example the bending moment at the wing-fuselage junction is limited to a value below that obtained with elliptical loading then the constrained optimal lift distribution for the same total lift is one that gives rise to a downwash distribution which varies linearly with span. While these basic ideas can be partially transferred to the study of upwind sails there are many additional constraints and limitations which need to be considered. These effects and constraints include: While the jib or genoa may be sheeted close to the deck there is always a foot gap below the boom and so there is an inevitable partial reduction of the lift distribution at the foot of the sails. There are physical limitations to the practical planform of the sails which in turn limit the possible lift distributions. This may be further constrained by the sail design rules for particular competitions. In addition there are real limits on the sectional lift coefficients that can be achieved on real sails. The atmospheric boundary layer results in a wind speed variation with height which tends to reduce the load distribution at low heights. The yacht is free to heel in response to heeling moments applied by the sails and keel. Some of these effects simply place limitations on the span-wise lift distributions which can be practically achieved, while the last has a strong effect on what distribution is optimal under particular sailing conditions. For close-hauled sailing the effect of heel can be considered in three general wind speed bands:. Low wind speeds: In such conditions the heel angle is small and obtaining the maximum thrust is simply a matter of minimising the induced drag for a given lift coefficient. If the water surface is considered as a mirror plane then this would be achieved by creating semi-elliptic loading, which decreases from a maximum at the water surface to zero at the masthead in an elliptic manner, essentially behaving in a manner similar to one wing on an aircraft. However in reality the foot gap, boundary layer profile and hull all mean that this is not practically achievable but does indicate the ideal to be aimed at.. Moderate wind speeds: Under such conditions the heel angle varies in response to the moments applied. If the yacht heels significantly the effective wind speed (perpendicular to the mast) is reduced and hence all the aerodynamic forces are smaller. Under such conditions it is the resulting thrust which needs to be maximised and not just a matter of minimising the induced drag. This means that it may be better to twist the sails and thereby lower the centre of effort, hence reducing the heel angle and increasing the effective wind speed and lift, even if as a result the induced drag is also increased. 3. High wind speeds: Under strong winds the sails will be trimmed so that the heel angle is fixed at its maximum acceptable value. This corresponds to a certain heeling moment and so the optimisation problem becomes one of maximising the thrust Associate Professor, Yacht Research Unit, Department of Mechanical Engineering, University of Auckland Visiting Student, Yacht Research Unit, Department of Mechanical Engineering, University of Auckland 3 Former PhD student, Yacht Research Unit, Department of Mechanical Engineering, University of Auckland 4 Wind Tunnel Manager, Yacht Research Unit, Department of Mechanical Engineering, University of Auckland 5 Current PhD student, Yacht Research Unit, Department of Mechanical Engineering, University of Auckland

2 subject to the constraint of a particular maximum heeling moment. This problem is similar to that considered by Jones [] but the optimisation seeks to maximise the thrust subject only to a constraint on the moment, rather than on both the total lift and moment. In this paper both experimental and theoretical studies will be used to illuminate these often complex interactions.. WIND TUNNEL STUDIES Although standard wind tunnel force measurements are unable to provide details of the span-wise load distribution, the resulting forces and moments can provide some information which indicates the general form of those distributions and the ways in which they change.. Experimental techniques The yacht used in this study was a :4 scale model of a generic Volvo Ocean 7 (VO7) with mainsail and jib, see Figure. The model was tested in the University of Auckland s Twisted Flow Wind Tunnel (TFWT) [], where the forces and moments are measured by a sixcomponent balance located below the turntable. The tunnel is 7 m wide and 3.5 m high and is equipped with turning vanes which create a twisted flow that represents the variation of apparent wind direction with height. Both the variation of wind speed and direction with height were adjusted to match typical VO7 upwind sailing conditions. In order to allow more natural trimming during wind tunnel testing the University of Auckland has developed a Real-Time Velocity Prediction Program (RT-VPP) [3]. As illustrated in Figure, the Real-Time VPP LabVIEW application takes the balance measurements of forces and moments and passes them to FRIENDSHIP-Equilibrium in coefficient form, which solves for the equilibrium condition in up to six-degrees of freedom employing semi-empirical descriptions to model the hydrodynamic forces. The sails can now be adjusted based on the predicted performance of the yacht with the current sail trim. In addition the RT-VPP calculates the expected heel angle for the current trim and adjusts the model s attitude to reflect this. As part of this process the user defines the full-scale true wind speed which is to be considered by the VPP and so if a high true wind speed is defined the model will be heeled to larger angles, even though the actual wind speed in the tunnel remains the same. In such situations it is the velocity scale factor (ratio of tunnel speed to full-scale speed) which has altered. By using this facility it has been possible to investigate the changes in trim required to maximise boat speed as the full-scale true wind speed is increased. Force balance Aerodynamic forces LP-Filter Pressure transducer LP-Filter Computer A/D card RT-VPP LabVIEW application Pitot tube Reference pressure FS-Equilibrium Parallel port Result file Accelerometer Heel angle (φ) Electric motor Heel angle (φ) Controller Figure. Schematic description of the implementation of the Real-Time VPP (RT-VPP) in the TFWT Figure. Volvo 7 model used in the wind tunnel study. Preliminary study Liddle and O Brien [4] have conducted a preliminary study of the depowering of the VO7 model yacht for a few apparent wind angles. During this study the model was allowed to heel under the control of the RT-VPP as the sails were trimmed. Once the optimal trim was found the yacht was rotated back to one particular heel angle so that photographs could be taken. Figure 3 shows two of these images superimposed, both are for an apparent wind angle (AWA) of 3 but they represent low and moderately high wind speeds. It is clear from this image that at low wind speeds the chosen trim has the boom close to the centreline and the sail relatively untwisted. However in stronger winds the sail has been eased and allowed to twist. Analysis of the stripes on the sails allowed them to quantify these changes. In particular the second stripe up the sail, at about mid-mast, has been analysed and the difference between the angle of the

3 chord line at this height and the boom angle used as a measure of the twist. Figure 4 shows the way in which both the boom and twist angles have been allowed to increase as the true wind speed strengthens. This graph also illustrates the difficulty in finding the exact optimal trim. In many situations several similar trims will give almost identical predicted boat speeds. This is particularly evident in the changes between 6 and 7 m/s, where the boom angle has been reduced but the twist in the sail increased. As a result the angle of the flow onto the centre of the sail will have decreased slightly. Figure 4 shows the corresponding changes in some of the aerodynamic characteristics. As the true wind speed increases the general angle of attack of the sails has been reduced and the lift coefficient follows. In addition the increased twist of the sails means that the unloading is particularly occurring near the head of the sail and so the centre of effort height also reduces, although not as dramatically. Subsequently, a very similar set-up has been used in a more extensive study of VO7 trim. In this study more apparent wind angles and a broader range of true wind speeds have been investigated. This data will be used to illustrate various points in sections. and.3. Sail Trim for V T = 4 m/s Sail Trim for V T = m/s Figure 3. Volvo 7 model trimmed for maximum boat speed at an AWA of 3 with true wind speeds of 4 and m/s. Angle (degrees) Twist Angle Boom Angle True Wind Speed (m/s) CoE Height / Mast Height True Wind Speed (m/s) Figure 4. Changes in the boom angle and twist angle and lift coefficient and centre of effort height with true wind speed..3 Optimal trim in light winds With rigid aerofoils, such as glider wings, although there may be a wide range of aerodynamic characteristics possible at the design stage, once the aircraft is constructed, the lift and drag force coefficients become simply functions of the angle of attack. In contrast with sails there is always a wide range of shapes available to the trimmers and hence a range of aerodynamic characteristics. Figure 5 illustrates some of lift-drag combinations possible, where all the results from the more extensive study are plotted. In obtaining this data, the forces have been reduced to coefficient form through calculations such as F CF = () q eff A Ref where F represents the relevant force (Lift L or Drag D), A Ref is the reference sail area, which was taken as the total sail area of.53 m at model scale, and q eff is the effective dynamic pressure calculated in the heeled plane perpendicular to the mast. In order to handle the force and moment changes due to heeling the effective angle concept is employed, which assumes that the sails are insensitive to the span-wise flow that results from heeling as described by, for example, Jackson [5]. The lift and drag forces plotted in Figure 5 are those in the heeled plane CoE Height / M ast Height 3

4 The effective dynamic pressure at a height of m may be related to the air density (ρ), the reference true wind speed (V T ) at m full-scale, the boat speed (V S ), the true wind angle (β T ) and the heel angle (φ) through: q eff ( V sin( β )cos( φ) ) + ( V + V cos( β )) ) ρ = () T T S T T or in the wind tunnel where the apparent wind speed (V A ), at a height equivalent to m in full-scale, and the apparent wind angle (AWA, β T ) are explicitly modelled: q eff ( V sin( β )cos( φ) ) + ( V cos( β )) ) ρ = (3) A A A A The cloud of data shown in Figure 5 can be bounded by an envelope, represented by the solid line, which is the locus of the most extreme lift-drag pairs possible. (c) V S V A Total C D Trim point for maximum /C D.5.5 Total C D Trim point for maximum thrust coefficient with AWA = Total C D Figure 5. Lift-drag polar for the VO7 model together with an approximate boundary envelope, the tangent showing the point of maximum /C D ratio and (c) the tangent indicating the point of maximum thrust coefficient for an AWA of 7. The next problem is to determine which of these points offers the best performance. For a glider the minimum glide angle is obtained by choosing the angle of attack that gives the highest lift-drag ratio. For this data maximum L/D is determined by drawing a tangent from the origin to the envelope as illustrated in Figure 5. However with a yacht the normal objective is to sail faster and so from the aerodynamic perspective this primarily means maximising the thrust, which, as illustrated in Figure 5(c), can be constructed by finding the tangent perpendicular to the boat speed vector, which is at the apparent wind angle relative to the horizontal in these figures. It is recognised that this is a simplified illustration since it ignores second order effects such as the changes in hydrodynamic resistance due to side force. However these more complex interactions are taken into account through the hydrodynamic models in the RT- VPP. Shown in Figure 6 are the trim points selected during wind tunnel testing for low true wind speeds and various apparent wind angles. All of these are very close to the envelope drawn around the cloud of data points shown in Figure 5. In particular the trim point for an apparent wind angle of 7, as illustrated in Figure 6, is very close to the trim indicated by the tangent. It may also be noted that the trim points chosen for the various apparent wind angles form a systematic series where as the angle increases the chosen trim moves towards higher lift and drag values. It may also be noted that the various points are more spread out than would be anticipated from tangents to the envelope shown. This occurs because the point cloud shown in Figure 5 is a reasonable approximation for apparent wind angles in the middle of the angle range but is less accurate for those at either end. In order to accurately analyse these points in this graphical fashion, it would be necessary to measure in detail the lift-drag combinations that can be accomplished at each apparent wind angle. Certainly the envelope for would be different from that at 36 due to the change in the relative location of the jib and mainsail with respect to the apparent wind. During trimming with the RT-VPP a variety of trims are investigated for each AWA and estimates of the corresponding boat speed are displayed for the trimmer, however no data is logged until the trimmer is satisfied that the desired optimum has been found. All of the chosen trim points in Figure 6 represent the minimum drag for a particular lift. Any additional drag above this minimum always results in a reduction in the resulting thrust. In this regard the total drag is made up from both the induced and parasitic drag on the sails and the windage on the hull and rig. In principle minimising the induced drag is, as discussed for example by Marchaj [6], a matter of achieving a near semi-elliptical load distribution similar to that illustrated as the ideal curve in Figure 7. If this can be achieved then a tip vortex is only shed from the head of the sails and the induced drag coefficient will be given by: 4

5 C Di CL = eπar where the aspect ratio AR is defined in terms of the actual sail area A Ref and mast height (H), from the sea surface, through H AR = A Ref and the efficiency factor e, in the denominator of equation (4), has an ideal value of since the sea surface acts as a mirror plane. (4) (5) In reality the load distribution will decrease near the sea surface due to a number of effects including: V S V A Envelope AWA deg AWA 4 deg AWA 7 deg AWA 3 deg AWA 33 deg AWA 36 deg.5.5 Total C D The mainsail does not extend below the boom and so the loading below this level is inevitably reduced. The apparent wind speed reduces with height and changes direction, tending towards the reversed boat speed at very low levels. The hull is not an efficient lifting device. However the windage measurements of Richards et al. [7] show that when the sails are present there is an enhanced pressure difference across the hull, relative to that measured by a bare poles test, which can increase the lift coefficient by up to.7. With a freeboard of approximately.47h, the lift coefficient per unit height for the hull is.48, which is only slightly less than the lift coefficient for the total model and indicates that a substantial proportion of the pressure difference is maintained down to the sea surface. Further, Krebber and Hochkirch [8] have numerically simulated the load distribution on the yacht DYNA. These results suggest that a real load distribution might be more like that shown in Figure 7 labelled Possible?. In which case the foot of the sail will also shed a trailing vortex as illustrated in the flow visualisation of Krebber and Hochkirch [Figure 9 in reference 8]. This partial unloading does mean that the efficiency factor e will be less than and is often around. [9] to.3 [], although in these references the efficiency factor is incorporated into an effective rig height. V S V A Envelope VT = 3 m/s VT = 6 m/s VT = 8 m/s VT = m/s VT = m/s.5.5 Total C D Relative mast height (z/h) Boom Height Sheer Height Ideal Possible? Normalised load distribution (c) Figure 7. The ideal load distribution and a sketch of what might be possible in reality.5 Cl V S V A 7.5 Envelope t =.8 t =.75 VT = 3 m/s VT = 8 m/s.5.5 Total CD.5.5 Flat Plate Flat Plate Polynomial Camber = 5.8% Camber = 5.8% Polynomial Camber = 8% Camber = 8% Polynomial Camber =.% Camber =.% Polynomial Figure 6. The trim points chosen with the RT-VPP for various apparent wind angles in light winds, the trim points chosen for various true wind speed with AWA = 7, and (c) the modifications to the enveloped as a result of increased twist at 8 m/s...4 CDp Figure 8. Lift-parasitic drag polars for D sails of different cambers, data from Dayman []. 5

6 While a semi-elliptical load distribution would minimise the induced drag this must be achieved without creating extra parasitic drag. Figure 8 shows the lift-parasitic drag data obtained by Dayman [] using D sail models. This data shows that if the sectional lift coefficient is too high for a particular camber then the parasitic drag increases rapidly. This data can be modelled by: C C Dp Dp min C ( C L =.46*( C Dp min L C ( C max ) =.6*( Y max Dp min +.9*( CL CL( C =.+.6* Y / c L )) Dp min 4 )) + C D min / c).43*( Y max / c) where Y max /c is the camber of the section, C Dpmin is the minimum drag coefficient and (C Dpmin ) is the corresponding lift coefficient at which this occurs, both of which tend to increase with camber. These results show that the parasitic drag increases rapidly as - (C Dpmin ) tends towards.. Hence the desire to achieve a near elliptical lift distribution has to be tempered with the reality that, with a particular planform, the desired distribution may be either unachievable or may incur a parasitic drag penalty that outweighs any induced drag gain..3 Optimal trim with increasing wind speed At low wind speeds maximising thrust is primarily a matter of minimising drag for a particular lift. However as the wind speed increases then the heel begins to have a significant influence. Figure 6 shows the trim conditions chosen with AWA = 7 but with increasing true wind speeds. This data shows that as the wind speed increases the lift coefficient chosen decreases, but the associated drag coefficient is higher than the minimum achievable. Figure 9 shows some of the associated data obtained from the RT-VPP (data points) along with results of the analytic modelling (lines) to be discussed in section 3.. The RT-VPP results show that even though the effective wind speed increases steadily the sails are being eased to limit the forces generated and also twisted in order to lower the centre of effort, and thereby keeping the yacht more upright. These changes in sail trim can also be seen in Figure. Exactly which trim condition is chosen is the result of complex interactions which take account of many factors, including the increase in drag which results from lowering the centre of effort. Jackson [5] points out that if the sails are twisted in order to lower the centre of effort then the induced drag will be increased such that: C (6) CL = ( ct ) (7) e π AR Di + where the twist parameter t is calculated from t = z z CoE CoE Ideal Force (N) (c) Speed (m /s) E+4 5.E+4 4.E+4 3.E+4.E+4.E+4.E+ Centre of Effort Height (m) 5 5 Effective Apparent Wind Speed RT-VPP Effective Apparent Wind Speed Analysis Boat Speed RT-VPP Boat Speed Analysis True Wind Speed (m/s) Thrust RT-VPP Thrust Analysis Side Force RT-VPP Side Force Analysis Heeling Moment RT-VPP Heeling Moment Analysis True Wind Speed (m/s) Vertical CoE Height RT-VPP Vertical CoE Height Analysis Heel Angle RT-VPP Heel Angle Analysis True Wind Speed (m/s) 5.E+5 4.E+5 3.E+5.E+5.E+5.E+ Figure 9. The effect of true wind speed on various factors at maximum boat speed with an AWA = 7. Effective apparent wind speed and boat speed, Thrust and side forces and heeling moment and (c) Centre of effort height and heel angle, as determined in the wind tunnel (data points) and from the analysis (lines) described in Section 3. and, following Jones [], suggests c = 8 and an ideal centre of effort height at.4h. Figure 6(c) shows the changes to the lift-drag envelope that might be expected with twist level commensurate with the centre of effort heights indicated in Figure 9(c) at V T = 3 and 8 m/s. The various interaction which influence the optimal trim will be discussed further in section 3..4 Optimal trim with high wind speed In high winds the optimal trim may be primarily constrained by the need to limit the heeling moment to an acceptable level. However even then, there will be various load distributions which will give the maximum allowable heeling moment, but only one will provide the maximum thrust Heel Angle (degrees) Moment (Nm) 6

7 (c) (d) (e) (f) Figure. Stern photographs of the model yacht at an AWA =7 as it is trimmed for true wind speeds of 3 m/s, 6 m/s, (c) 8 m/s, (d) m/s and (e) m/s. (f) Shows the sail stripes for the same set of speeds superimposed on an upright model trimmed for the highest wind speed. 3. ANALYTIC STUDIES 3. Trim Optimisation In order to further investigate the choice of sail trim under various conditions a spreadsheet was developed which incorporated the following features:. The two independent variables studied were the choice of lift coefficient and the twist parameter t, which was used to determine the centre of effort height.. Polynomials were fitted to: the variation of heel angle with heeling moment, the variation of boat speed with thrust and the parasitic drag coefficient variation with lift coefficient, using data obtained from the RT-VPP. 3. Equation (7) was used to account for the increased induced drag with twist. 4. Effective angle theory, through equation (), was used to take account of the changes in dynamic pressure caused by heeling. 5. An iterative solution was used to determine the equilibrium conditions corresponding to the chosen variables. This analysis has been used to investigate the maximum thrust obtainable with an apparent wind angle of 7 and various true wind speeds. The associated results at maximum thrust are shown in Figure 9 (lines) and demonstrate that this analysis is replicating the major features of the RT-VPP wind tunnel study. However additional insight can be gained from the variations of thrust with lift coefficient and twist as illustrated in Figure. Here it may be observed that at a low wind speed, Figure, increasing the twist has a general detrimental effect and that maximising thrust is simply a matter of choosing a lift coefficient that is as high as possible without incurring an excessive parasitic drag penalty. 7

8 Twist factor t Thrust (N) Twist.5 factor t Heel (deg) Figure. Heel angle as a function of lift coefficient and twist with AWA = 7 for a true wind speeds of 6 m/s (c) Twist factor t Twist factor t Thrust (N) Thrust (N) Extended Lifting Line Code In order to investigate in more detail the optimal spanwise lift distribution an Extended Lifting Line Code (ELLC) is being developed. This formulates the loading distribution along the span of the sail by developing a model of the sail geometry and the flow field around the sail. In the current study the sail geometry is taken from an AC9 mainsail design. The geometry is analyzed to yield the chord-length, camber, twist, and quarter-chordline as a function of span. Thin aerofoil theory is applied to calculate the local aerofoil s zero lift angle of attack. The sail is then shrunk to its quarter-chord-line, and the downwash induced by the bound and trailing vortex system of the sail and its image are analysed on the threequarter-chord line. Figure shows the setup of the vortex system currently used to model the aerodynamics of a single mainsail. Figure. Thrust coefficient as a function of lift coefficient and twist with AWA = 7 for true wind speeds of 4 m/s, 6 m/s and (c) 8 m/s. At moderate wind speeds, Figure, the heel angles have a significant influence and there is a clear shift of the region of maximum thrust to more moderate lift coefficients and twist. Figure shows the associated heel angles, where the general trend is one of increasing heel with high lift coefficient and high centre of effort, (i.e. low twist). The drop in heel angle at very high lift coefficients is the result of the reduced thrust and hence boat speed when the parasitic drag is high. y, η U x, ζ Γ b Γ t P( x, y) Increasing the true wind speed further, Figure (c), shifts the peak of the thrust surface to even lower lift coefficients and higher twist levels. If too high a lift coefficient is chosen or too little twist, the increase in heel, resulting from an incremental increase in lift coefficient, decreases both the effective wind speed and the effective wind angle. The former limits the resulting change in lift, while the latter rotates the lift away from the thrust direction and produces a net decrease in thrust. Γ bi Figure. The vortex system used in the Extended Lifting Line Code. Γ ti 8

9 The theory utilised is based on Weissinger s [3] extension of the original Prandtl Lifting Line Theory, which extends the concept of the lifting line to sweptback wings of arbitrary planform. The concept of a continuous lifting line does not account for the nonplanarity of the wake shed by the bound vortex system. Here the wake is modelled as a planar vortex sheet, which has the direction of the apparent wind. An extension of the theory to a non-planar wake which has the direction of the local apparent wind, if wind twist and sheer are included, may not be implemented easily into the analytic solution and is thus postponed to future work. Nevertheless, the calculation of the local lift coefficient is based on the local geometric angle of attack, which takes into account the sail twist. The downwash integrals are derived by introducing a Cartesian coordinate system such that the positive x direction points downstream of the onset flow and the y direction points outwards toward the head of the sail. The y direction is centred at half span. Additionally, a dummy coordinate system ζ,η is introduced. The z-coordinate reaches out of the plane of the paper. Accordingly, a downward induced velocity is negative in a right hand system. This co-ordinate system is oriented in the manner commonly used in lifting line texts rather than that used in earlier sections. The velocity induced by the sail at a point P(x,y) in the plane of the sail can be calculated using the Biot-Savart Law: 4π Γ r ds r w i = 3 which gives the fluid velocity at any point in space a distance r from a vortex of strength Γ. The total downwash is then the sum of the downwash induced by all vortices through superposition. The ELLC sought to find the lift distribution that would provide the maximum thrust subject to the constraint of a specific heeling moment for each heel angle. The essential features of the model included: For a specified true wind speed the corresponding apparent wind speed was calculated based on assumed ratios of boat speed to true wind speed. The apparent wind angle was set to 7 and the foot gap to 5.7% of the luff length. For each heel angle the corresponding heeling moment was calculated based on a simple heeling balance using assumed bulb weight and location. Allowance was made for the mainsail only providing part of the side force under normal upwind sailing with a headsail. Parasitic drag was included by using the local chord length to determine the sectional lift coefficient. The data from Dayman [] (Figure 8) was then used to estimate the sectional parasitic drag coefficient for the appropriate local camber. (8) Increasing heel spanwise coordinate y b [ ] Increasing heel spanwise coordinate y b [ ] Figure 3. Optimised non-dimensional loading distributions for true wind speeds of 4 m/s (V A = 8.33 m/s) and 6 m/s (V A =.65 m/s) and heel angles from to 3 in intervals. There are limits placed on the maximum sectional lift coefficient that is permitted and the sail is prevented from carrying negative lift. Figure 3 shows the loading distributions determined for two true wind speeds. These show that if the yacht is to remain more upright, particularly with the stronger wind in Figure 3, then the head of the sail must be progressively unloaded. In addition in the stronger winds the magnitude of the non-dimensional loading distribution, which is normalised by the apparent wind speed, is reduced. These results once again show that in moderate to strong winds the optimal distribution is a balance between lowering the centre of effort and reducing the overall loading Increasing heel spanwise coordinate y b [ ] Figure 4. Optimised induced angle distributions for a true wind speeds 6 m/s and heel angles from to 3 in intervals. 9

10 Thrust (N) Extended Lifting Line Theory has been used to further investigate the detailed loading distribution on an AC9 mainsail. The result illustrate the way in which the optimal distribution changes with varying conditions. They also show that where the centre of effort is to be lowered, this is best achieved by a loading distribution which gives an induced angle that varies linearly with height. Acknowledgements apparent wind speed V Figure 5. Maximum thrust force determined for various apparent wind speeds and heel angles. Figure 4 shows the corresponding induced angles for a true wind speed of 6 m/s. These results are consistent with the work of Jones [] who showed that if the bending moment at the wing-fuselage junction is limited to a value below that obtained with elliptical loading then the constrained optimal lift distribution, for the same total lift, is one that gives rise to a downwash distribution which varies linearly with span. The situation is slightly different in this case but the downwash distributions are nearly linear. It may be noted that with the lower heel angles the head of the sail is very lightly loaded and so the core of the vortex shed from the head of the sail is also lowered and in some cases produces negative induced angles on the sail above this core. Figure 5 shows the overall set of thrust values determined from the ELLC modelling. This once again shows that at the lower apparent wind speeds, the maximum thrust is obtained at a moderate heel angle, but as the wind speed increases this optimal heel angle increases until the heel becomes limited by the maximum acceptable angle for the particular yacht. 4. CONCLUSIONS A,ref m s Wind tunnel experiments using a Real-Time Velocity Prediction programme have investigated the optimal trim of a Volvo Ocean 7 model under various simulated true wind speeds. The results illustrate that the decision made depend upon the particular apparent wind direction and true wind speed. It is suggested that these can be subdivided into three broad bands: Low wind speeds where the total drag is minimised and the trim that provides the maximum thrust coefficient is chosen. Moderate wind speeds where the heel angle has a strong effect and the optimum choice includes a reduction in lift coefficient and centre of effort height Strong winds where the heel angle and hence heeling moment is limited to the maximum acceptable value and the optimum loading distribution is strongly constrained by this limit. [ ] heel angle The authors would like to acknowledge the assistance of Emirates Team New Zealand for use of the AC9 model. References. Jones, R.T. (95) The Spanwise Distribution of Lift for Minimum Induced Drag of wings having a given Lift and a given Bending Moment. NACA TN 49.. Flay, R. G. J. (996) "A Twisted Flow Wind Tunnel for Testing Yacht Sails." Journal of Wind Engineering and Industrial Aerodynamics, Vol. 63, Hansen, H., Jackson, P. S., and Hochkirch, K. (3) "Real-Time Velocity Prediction Program for Wind Tunnel Testing of Sailing Yachts." The Modern Yacht Conference, RINA, Southampton. 4. Liddle, S.G. & O Brien, K.M. (6) Depowering of Yachts, Project in Mechanical Engineering Reports 6-ME39 and 53, University of Auckland. 5. Jackson, P. S. () "An Improved Upwind Sail Model for VPPs." The 5th Chesapeake Sailing Yacht Symposium, SNAME, Annapolis, Marchaj, C.A. (99) Sail Performance Techniques to Maximize Sail Power, International Marine, USA. 7. Richards, P.J., Le Pelley, D., Cazala, A., McCarty, M., Hansen, H. & Moore W.E. (6) The use of independent supports and semi-rigid sails in wind tunnel studies, nd High Performance Yacht Design Conference, Auckland, 4-6 February 6 8. Krebber, B. & Hochkirch, K. (6) Numerical Investigation of the Effects of Trim for a Yacht Rig, nd High Performance Yacht Design Conference, Auckland, 4-6 February 6 9. Claughton, A. (999) "Developments in the IMS VPP Formulations." The 4th Chesapeake Sailing Yacht Symposium, SNAME, Annapolis, -.. Campbell I.M.C. (997) "Optimisation of a Sailing Rig Using Wind Tunnel Data." The 3th Chesapeake Sailing Yacht Symposium, SNAME, Annapolis, Dayman, B. (953) Experimental Aerodynamics of a twodimensional sail, Thesis, California Inst. of Tech. Pasadena, CA.. Kerwin, J. E. (978) A Velocity Prediction Program for Ocean Racing Yachts revised to February 978. Report No. 78-, Massachusetts Institute of Technology, Cambridge. 3. Weissinger, J. (947) The Lift Distibution of Swept-Back Wings, NACA TM.