A foil as dynamic ballast numerical simulation

1/3 Jean-françois Masset January 218 A foil as dynamic ballast numerical simulation contact : jfcmasset@outlook.fr Summary : 1. Presentation of the study 2. DB32 : the numerical boat of reference 3. The foil assembly 4. The 2 cases VPP used (upwind and downwind). The numerical simulation with foil, 6. Conclusions Acknowledgment Annexe 1 : DB32 data for 3 bulb ballast Annexe 2 : Foil formulations Annexe 3 : VPP formulations List of symbol

2/3 1. Presentation of the study The idea is to substitute all or fraction of a bulb ballast by a foil, hinged at the tip end of the keel, and able to produce a downforce lift. It is called dynamic ballast as this downforce is dependant of the boat speed. The present document proposes a numerical exploration of the idea, how this foil can reduce the heel angle, can delayed the reefing and affect the speed (taken into account the extra induced drag linked to this negative lift). At first a numerical hull inspired by a Melges 32 is built with the speadsheet application Gene-Hull, from which data and parameters can be used to feed a VPP. A simplified VPP is also built for 2 typical sailing cases, upwind with true wind angle (twa) between 39 to 43 and downwind with true wind angle 14, and validated for 3 boat displacements (according to three bulb ballasts) by comparison with the results of a Maxsurf VPP. This VPP aims to be a tool for comparison between various configurations with and without foils. A foil assembly is proposed which can orient the foil in the direction of the leeway when heeling occurs, and so avoid as much as possible that side forces act significantly on the assembly. Consequently, foil is assumed horizontal and its downforce lift is assumed vertical. This assembly as well as the variable trim of the boat tend to slightly increase the foil incidence when the boat heels, but otherwise no active command of the foil incidence is taken into account in this siimulation. The numerical simulation concerns 3 initial displacements cases, for which 2 foils surfaces are tested. Full results regarding speed, heel, displacement anf foil drag are discussed. 2. DB32 : the numerical boat of reference DB32 (stands for Dynamic Ballast 32) is a numerical hull+keel+rudder generated by Gene-Hull UE 2.1 (Hull generation spreadsheet application) and inspired from the Melges 32, sharing the following features in light weight conditions : Loa : 9,7 m (31,82') Lwl : 9,9 m (29,82') Beam B : 3, m (9,83') Draft T : 2,13 m (7') Displacement D : 1712 kg (377 lbs) Ballast : 77 kg (font keel + lead bulb) (178,6 lbs) >>> 2D linesplan (with waterline H in light weight conditions)

3/3 1 1-1 - 1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9 1 - -1-1 -2-2 1 8 6 4 2 1 8 6 4 2-16 -14-12 -1-8 -6-4 -2-2 2 4 6 8 1 12 14 16-16 -14-12 -1-8 -6-4 -2-2 2 4 6 8 1 12 14 16-4 -4-6 -6-8 -8-1 -1-12 -12-14 -14-16 -16-18 -18-2 -2-22 -22 2, 1, 1,,, -1-1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9 1 -, -1, -1, -2,

4/3 With this numerical hull, one can recover all the data needed to formulate the righting moment, the wetted surface, the drag components, etc in function of various bulb ballast as the goal is to replace a fraction of this bulb ballast by the foil downforce and induced drag. This extraction of data is done for 3 mass of the bulb ballast (+ 3 crew on board) so that to cover the whole range of the study : bulb ballast : 32 kg + 3 crew 24 kg >>> total displacement 12 kg bulb ballast : 77 kg + 3 crew 24 kg >>> total displacement 192 kg bulb ballast : 122 kg + 3 crew 24 kg >>> total displacement 242 kg >>> These data are detailed in Annexe 1 3. The foil assembly The foil is mounted with a short vertical wing hinged to the keel at z -17. This joint is inclined which has the effect, when heeling occurs, to orient the assembly in the direction of the leeway. For example, with an inclination of 11, a 2 heel angle causes a mounting orientation of 3.74 1 1-1 - 1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9 1 - -1-1 -2 Hinged inclined joint (here 11 ) -2

/3 The foil is mounted with an initial incidence to provide the downforce, this incidence is also lightly increasing with heel due to the assembly inclination : 11 inclination and 2 heel gives +,67 of incidence in addition to the initial one at heel. The foil profile used is a Naca 241, with a trapazoid plan form, tip chord = 2/3 central wing chord, foil span < boat beam, more data and formulations in Annex 2. Configurations showed with a 2 heel : 1 1 1 1-2 -1-1 - 1 1 2 2 3 - -2-1 -1-1 1 2 2 3 - -1-1 -1-1 -2-2 -2-2 2, 1, 1,,, -1-1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9 1 -, -1, -1, -2, The foil is showed oriented to 3,74 in the leeway direction due to the heel 2.

As a digression, one can notice that such assembly can also be envisaged with a bulb ballast, allowing to orient the streamlined bulb in the leeway direction and so reduce its drag : 6/3 4. The two cases VPP used (upwind and downwind) A specific VPP was developed with the objective to include as variables with speed and heel angle the total displacement and its related parameters (wetted surface, drag components, ), the righting moment (following the foil downforce), the foil drag itself, To simplify the VPP formulations, only two true wind angle (twa) cases are considered : upwind (with twa between 39 to 43 ) and downwind (with twa 14 ). The sails forces and corresponding heeling moment are adjusted on the basis of the VPP Maxsurf results for 3 displacements directly derived to three bulb ballasts (+ 3 crew centered) in order to cover the range of the foil downforce action : Bulb ballast : 32 kg (+ 3 Crew 24 kg) >>> 12 kg Bulb ballast : 77 kg (+ 3 Crew 24 kg) >>> 192 kg (the nominal configuration) Bulb ballast : 122 kg (+ Crew 24 kg) >>> 242 kg Other formulations involved for drag components are the usual ones proposed by «Principles of yacht design» (Larsson-Eliasson) and with data extracted from the numerical hull generated by Gene-Hull. For the righting moment and the wetted surface, the «hull with heel» module in Gene- Hull is used for 3 displacements and for heel angles between to 3. The VPP results for configuration with bulb ballats is checked with the Maxsurf one, the main objective being the right representation of the tendency when displacement varies more than the speed or heel exact values :

7/3 Comparison / upwind case (VPP : continue lines ; Maxsurf results : points) Boat speed (Knts) vs true wind (Knts) DB32 with crew 3 centered & twa 42 green : 12 kg ; red 192 kg ; blue 242 kg 8 7 6 4 3 2 1 4 3 3 2 2 1 1 Heel ( ) vs true wind (Knts) DB32 with crew 3 centered & twa 42 green : 12 kg ; red 192 kg ; blue 242 kg 2 4 6 8 1 12 14 16 18 2 2 4 6 8 1 12 14 16 18 2 Comparison / downwind condition (VPP : continue lines ; Maxsurf results : points) : Boat speed (Knts) vs true wind (Knts) DB32 with crew 3 centered & twa 14 green : 12 kg ; red 192 kg ; blue 242 kg 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 18 2 22 4 4 3 3 2 2 1 1 Heel ( ) vs true wind (Knts) DB32 with crew 3 centered & twa 14 green : 12 kg ; red 192 kg ; blue 242 kg 2 4 6 8 1 12 14 16 18 2 22

8/3. The numerical simulation with foil For the numerical simulation with foil, the principle is to add the foil downforce lift and drag, both function of the boat speed and heel (according to formulations detailed in Annexe 2), to the VPP formulations, wich has an impact on the displacement, on the righting moment and on the total drag. The righting moment takes into account that the vertical position of the hinged joint (z -1,7 m) is different from the bulb center of gravity (-2,). It takes also into account the position of the 3 crew : they are sit centered for light winds < 6 Knots, and then sit windward for winds > 6 knots. 3 initial displacement cases are sudied (12 kg, 172 kg, 192 kg), for each 2 foils surfaces (S = 1,2 m2 and S smaller in a search of a better efficiency), and two wind directions (upwind with twa 39 to 43, downwind with twa 14 ). The comparison is done with regard to two configurations with bulb, displacement 192 kg with the initial bulb 77 kg (>> ballast ratio 39,7 %) and displacement 242 kg with a heavier bulb 122 kg (>> ballast ratio 1%). In upwind conditions, the twa is choosen between 43 and 39 depending of the more or less drag, to optimise the VMG as a greater drag better combined with a lower twa, which partly compensate the related loss of speed. So here the comparison of the VMG is more relevant than the speed itself. A common incremental reefing is simulated, 1% sail area, 7%, %, which occurs when either the heel reachs ~ 2 or when a better speed is obtained with a lower sail area. The foil initial incidence at no heel is 3, leading to a Cz initial of about, to, (depending of the foil) and the incidence evolved towards about when 2 heel angle, leading to Cz about,7 to,7. Initial displacement Foil Comparison with D = 12 kg + foil ballast reduced to 37 kg ballast ratio : 24,1 % Case 2 : 172 kg ballast reduced to 7 kg ballast ratio 32,8 % Case 3 : 192 kg S : 1,2 m2 : L : 2, m Central chord :,6 m; Tip chord :,4 m Aspect ratio :, S :,8 m2 : L : 2,8 m Central chord :,36 m; Tip chord :,243 m Aspect ratio : 9,22 S : 1,2 m2 : L : 2, m Central chord :,6 m; Tip chord :,4 m Aspect ratio :, S :,72 m2 : L : 2,7 m Central chord :,32 m; Tip chord :,214 m Aspect ratio : 1,13 S : 1,2 m2 : L : 2, m Central chord :,6 m; Tip chord :,4 m Aspect ratio :, D = 192 kg nominal ballast 77 kg ballast ratio 39,7% D = 192 kg (nominal ballast 77 kg) (ballast ratio 39,7%) D = 192 kg (nominal ballast 77 kg) (ballast ratio 39,7%)

ballast nominal 77 kg ballast ratio 39,7 % S :,22 m2 : L : 2, m Central chord :,132 m; Tip chord :,88 m Aspect ratio : 18,18 9/3 & D = 242 kg (nominal ballast 122 kg) (ballast ratio 1 %) Case 1 : D 12 kg Upwind sailing Speed (knots) versus real wind (knots) (continue line : speed V ; dashed line : VMG) : red 192 kg with bulb, ballast 77 kg & twa 43 blue 12 kg with foil 1,2 m2 & twa 39 blue 12 kg with foil,8 m2 & twa 39 8 7 6 4 3 2 1 8 7 6 4 3 2 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Heel ( ) 12 kg with foil 1,2 m2 & twa 39 12 kg with foil,8 m2 & twa 39 4 4 3 3 3 3 2 2 2 2 1 1 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26

1/3 Displacement (kg) 12 kg with foil 1,2 m2 & twa 39 12 kg with foil,8 m2 & twa 39 3 28 26 3 28 26 24 22 2 18 16 24 22 2 18 16 14 2 4 6 8 1 12 14 16 18 2 22 24 26 14 2 4 6 8 1 12 14 16 18 2 22 24 26 Foil drag / total drag ratio (%) 12 kg with foil 1,2 m2 & twa 39 12 kg with foil,8 m2 & twa 39 4 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 4 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Case 1 : D 12 kg Downwind sailing Speed (knots) versus real wind (knots) (continue line : speed V ; dashed line : VMG) : red 192 kg with bulb, ballast 77 kg & twa 14 blue 12 kg with foil 1,2 m2 & twa 14 blue 12 kg with foil,8 m2 & twa 14 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 18 2 22 24 26 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 18 2 22 24 26

Heel ( ) 12 kg with foil 1,2 m2 & twa 14 12 kg with foil,8 m2 & twa 14 4 4 11/3 3 3 2 2 1 1 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Displacement (kg) 12 kg with foil 1,2 m2 & twa 14 12 kg with foil,8 m2 & twa 14 34 32 3 28 26 24 22 2 18 16 14 2 4 6 8 1 12 14 16 18 2 22 24 26 34 32 3 28 26 24 22 2 18 16 14 2 4 6 8 1 12 14 16 18 2 22 24 26 Foil drag / total drag ratio (%) 12 kg with foil 1,2 m2 & twa 14 12 kg with foil,8 m2 & twa 14 4 4 3 2 1 2 4 6 8 1 12 14 16 18 2 22 24 26 3 2 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Comments on case 1 :

12/3 The idea for this case is to test the minimum ballast option which can ease the trailing on roads, i.e. for the DB32 a displacement in light weight condition (sailing condition without crew) that is reduced of 4 kg, to 1312 kg instead of 1712 kg, obtained with a keel-foil ballast of 37 kg instead of 77 kg (the foil being built within this 37 kg). The initial displacement with 3 crew is 12 kg, this give an initial ballast ratio of 37/12 = 24 %. Two foil dimensions are tested : 1,2 m2 and,8 m2. The first check concerns the heel : for upwind sailing, despite the low ballast, the foils action can maintain the heel angles in quasi the same values as the reference boat (more or less 1 ), and reefing is triggered for the same wind values. for downwind sailing, we have less heel in the whole range of winds, and delayed reefing. With the 1,2 m2 foil, the heel reduction is noticeable between 1 to 16 knots of wind, up to less (1 instead of 2 ). As regard the speed : for upwind sailing, when one considers the VMG, the loss with the foil 1,2 m2 is one knot or more for winds between 8 to 14 knots, i.e. 2% to 2% loss. The loss is reduced to 1% to 1% for stronger winds > 2 knots. With foil,8 m2, the VMG maximum loss is 17% (4 knots instead of 4,8 knots for 1 knots wind) and reduced to less than 1% for winds > 16 knots. For downwind sailing : the loss of speed increases progressively with wind speed, up to 17% with foil S 1,2 m2 and 9% with foil S,8 m2, e.g. 12 knots instead of 13,2 knots for 24 knots of wind, associated with heel of 17 instead of 21. Check of the displacement : for upwind sailing : the maximum displacement resulting from the foil downforce increases progressively to reach about the one (192 kg) of the reference boat for both foils. For downwind sailing : the maximum displacement is a lot higher, especially with the foil S 1,2 m2, the displacement reachs 324 kg for wind 2 Knots, so an equivalent of 19 crew on board, associated with a heel angle of 17. It is an impressive figure but still feasable as regard the sheer line position above the waterline (sketch here below). With foil,8 m2, the maximum displacement is about 28 kg. 14 12 1 8 6 4 2-18 -14-1 -6-2 2 6 1 14 18-2 -16-12 -8-4 -4 4 8 12 16-6 Configuration for displacement 324 kg and 17 heel

13/3 As regard the foil drag / total drag ratio : for both sailing conditions, drag ratio of the foil S 1,2 m2 is clearly more important (up to 3%) than the one of foil S,8 m2 (up to 2%). Case 2 : D 172 kg Upwind sailing Speed (knots) versus real wind (knots) (continue line : speed V ; dashed line : VMG) : red 192 kg with bulb, ballast 77 kg & twa 43 blue 172 kg with foil 1,2 m2 & twa 4 blue 172 kg with foil,72 m2 & twa 4 8 7 6 4 3 2 1 8 7 6 4 3 2 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Heel ( ) 172 kg with foil 1,2 m2 & twa 4 172 kg with foil,72 m2 & twa 4 4 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Displacement (kg) 172 kg with foil 1,2 m2 & twa 4 172 kg with foil,72 m2 & twa 4 3 3 4 3 3 2 2 1 1 28 26 24 22 2 18 16 28 26 24 22 2 18 16 14 2 4 6 8 1 12 14 16 18 2 22 24 26 14 2 4 6 8 1 12 14 16 18 2 22 24 26

14/3 Foil drag / total drag ratio (%) 172 kg with foil 1,2 m2 & twa 4 172 kg with foil,8 m2 & twa 4 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 3, 3, 2, 2, 1, 1,,, 2 4 6 8 1 12 14 16 18 2 22 24 26 Case 2 : D 172 kg Downwind sailing Speed (knots) versus real wind (knots) (continue line : speed V ; dashed line : VMG) : red 192 kg with bulb, ballast 77 kg & twa 14 blue 172 kg with foil 1,2 m2 & twa 14 blue 172 kg with foil,72 m2 & twa 14 14 12 1 8 6 4 2 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Heel ( ) 172 kg with foil 1,2 m2 & twa 14 172 kg with foil,72 m2 & twa 14 4 4 3 3 2 2 1 1 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26

Displacement (kg) 172 kg with foil 1,2 m2 & twa 14 172 kg with foil,72 m2 & twa 14 34 32 3 28 26 24 22 2 18 16 14 2 4 6 8 1 12 14 16 18 2 22 24 26 Foil drag / total drag ratio (%) 172 kg with foil 1,2 m2 & twa 14 172 kg with foil,72 m2 & twa 14 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 34 32 3 28 26 24 22 2 18 16 1/3 14 2 4 6 8 1 12 14 16 18 2 22 24 26 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Comments on case 2 : For this case, the keel-foil ballast is 7 kg instead of 77 kg, so a reduction of 2 kg which leads to an initial displacement with 3 crew of 172 kg. It is an intermediate solution between the case 1 (12 kg) and boat of reference (192 kg). This gives an initial ballast ratio of 7/172 = 33 %. Two foil dimensions are tested : 1,2 m2 and,72 m2. As regard the heel : for upwind sailing, the foils action can give equivalent or a slight reduction of the heel angles in comparison with the reference boat, and reefing can be slightly delayed. for downwind sailing, we have less heel in the whole range of winds, and delayed reefing, but in comparable proportion with case 1 : for example, with the 1,2 m2 foil, the heel reduction is 6 (14 instead of 2 ) with wind 14 knots, to be compared with the reduction with case 1. As regard the speed : for upwind sailing, when one considers the VMG, the loss with the foil 1,2 m2 is one knot max between 8 to 14 knots, i.e. 21% loss. The loss is reduced to 12% or less for stronger

16/3 winds > 16 knots. With foil,8 m2, the VMG maximum loss can be roughly half reduced : 12% at the max and reduced to less than 4% for winds > 16 knots. For downwind sailing : as for the case 1, the speed loss increases progressively with wind speed, but the loss is a bit less : up to 1% with foil S 1,2 m2 and 7, % with foil S,72 m2, e.g. 12,2 knots instead of 13,2 knots for 24 knots of wind, associated with heel of 17 instead of 21. Check of the displacement : for upwind sailing : the maximum displacement resulting from the foil downforce increases progressively to reach respectively about 22 kg (with S 1,2 m2) and 28 kg (with S,72 m2), so above the 192 kg of the reference boat. for downwind sailing : the maximum displacements are quite the same as for case 1, i.e. over 32 kg with the foil S 1,2 m2 and 28 kg with the foil,72 m2. As regard the foil drag / total drag ratio : for both sailing conditions, drag ratio of the foil S 1,2 m2 is clearly more important (up to 33%) than the one of foil S,72 m2 (up to 1-16 %). Case 3 : D 192 kg Upwind sailing Speed (knots) versus real wind (knots) (continue line : speed V ; dashed line : VMG) : red 192 kg with bulb, ballast 77 kg & twa 43 green 242 kg with bulb, ballast 122 kg & twa 42, blue 192 kg with foil 1,2 m2 & twa 39 blue 192 kg with foil,22 m2 & twa 42, 8 8 7 6 4 3 2 1 7 6 4 3 2 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Heel ( ) 192 kg with foil 1,2 m2 & twa 39 blue 192 kg with foil,22 m2 & twa 42, 4 3 3 2 2 1 1 4 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26

Displacement (kg) 192 kg with foil 1,2 m2 & twa 39 blue 192 kg with foil,22 m2 & twa 42, 3 3 17/3 28 26 24 22 2 18 16 28 26 24 22 2 18 16 14 14 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Foil drag / total drag ratio (%) 192 kg with foil 1,2 m2 & twa 39 blue 192 kg with foil,22 m2 & twa 42, 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Case 3 : D 192 kg Downwind sailing Speed (knots) versus real wind (knots) (continue line : speed V ; dashed line : VMG) : red 192 kg with bulb, ballast 77 kg & twa 14 green 242 kg with bulb, ballast 122 kg & twa 14 blue 192 kg with foil 1,2 m2 & twa 14 blue 192 kg with foil,22 m2 & twa 14 14 12 1 8 6 4 2 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26

18/3 Heel ( ) blue 192 kg with foil 1,2 m2 & twa 14 blue 192 kg with foil,22 m2 & twa 14 4 4 3 3 2 2 1 1 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 2 4 6 8 1 12 14 16 18 2 22 24 26 Displacement (kg) blue 192 kg with foil 1,2 m2 & twa 14 blue 192 kg with foil,22 m2 & twa 14 3 33 31 29 27 2 23 21 19 17 1 2 4 6 8 1 12 14 16 18 2 22 24 26 3 33 31 29 27 2 23 21 19 17 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Foil drag / total drag ratio (%) blue 192 kg with foil 1,2 m2 & twa 14 blue 192 kg with foil,22 m2 & twa 14 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 Comments on case 3 : 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2 22 24 26 The idea for this case is to add a foil to the nominal configuration without mass reduction of the keel, so the keel-foil ballast is still 77 kg and the initial displacement with 3 crew 192 kg. The initial ballast ratio is 77/192 = 39,7 %. For this case, we add to the comparison the reference

19/3 boat with an heavier bulb, for a ballast of 122 kg instead of 77 kg, so a displacement of 242 kg and a ballast ratio of 122/242 = 1 % (this option is showed with green curves in the results here above). Two foil dimensions are tested : 1,2 m2 and,22 m2. As regard the heel : for upwind sailing, the 1,2 m2 foil can give a reduction of the heel angles (especially between to 1 knots of wind) and reefing can be delayed of 2-3 knots of wind. With the small,22 m2, equivalent heel angles and reefing can be slightly delayed of 1 knot of wind. for downwind sailing, the 1,2 m2 can give a lot less heel in the whole range of winds, and delayed reefing : for example, heel reduction is 7 (13 instead of 2 ) with wind 14 knots. This heel reduction is greater that the one with an heavier bulb ballast of 122 kg. With the,22 m2 foil, slight reduction of the heel but reefing can also be delayed. As regard the speed : for upwind sailing, when one considers the VMG, the loss with the foil 1,2 m2 is about one knot max as for case 2 between 8 to 14 knots, i.e. 21% loss. The loss is reduced to <, knot for stronger winds > 18 knots. With foil,22m2, the VMG maximum loss is minimal :,2 knots to,1 knots for wind > 16 knots. Moreover, for wind < 8 knots, the VMG is greater than with the heavier bulb ballast option. For downwind sailing : with foil 1,2 m2 the speed loss remains important, increases progressively with wind speed, up to 1% for 24 knots of wind. With foil,22 m2, we have a gain of speed between 16 and 24 knots of wind, up to +, knots i.e. + 4%. Check of the displacement : for upwind sailing : the maximum displacement resulting from the foil downforce increases progressively to reach respectively about 24 kg (with S 1,2 m2) and 28 kg (with S,22 m2). for downwind sailing : the maximum displacement is impressive with foil 1,2 m2, over 33 kg when wind is 24 knots. With foil,22 m2, the maximum displacement is 23 kg. As regard the foil drag / total drag ratio : for both sailing conditions, drag ratio of the foil S 1,2 m2 is up to 3 %, and with foil,22 m2 remains at %. 6. Conclusions A foil, hinged at the tip of the keel wing and mounted to provide a downforce lift, can act as a dynamic ballast, increasing the righting moment with speed. The proposed foil assembly, with a hinged joint slightly inclined, when the boat heels, can orient automatically the foil in the leeway direction and with same order of magnitude for the angle, and consequently it was assumed that the foil plan remains horizontal and that the downforce lift is vertical. But the induced drag of the foil is not negligeable and usually cannot avoid a reduction of the speed. So the use of such «passive» foil (i.e. without active command of its incidence) is relevant only for others objectives than speed :

2/3 When the objective is to have a minimum displacement, with a low keel-foil ballast ratio in the range of 24% (i.e. the 12 kg case), the positive result is that we can compensate the lower ballast by the dynamic downforce of a foil providing equivalent heel angles, but it is at the cost of 1% to 17% speed loss in upwind conditions and less than 9% in downwind conditions (results with foil S,8 m2 and Lspan= 2,8 m). When the objective is to increase dynamically a ballast ratio from an initial value of e.g. 4% (i.e. the 192 kg case), a foil can give a reduction of the heel angle (but especially in downwind condition, up to 7 ) and a delayed reefing. The heel reduction effect as the speed loss depend of the foil surface. When using a small enough foil (here S,22 m2 and Lspan 2, m), we can have a full positive result, i.e. both heel and speed gains, in downwind conditions when wind speed is over 16 knots, and quasi no speed loss in upwind conditions (,1 to,2 knots VMG). An heavier bulb, for a higher ballast ratio (e.g. 1% instead of 4%), when comparing to the above small foil option, leads to : less speed but lower heel angle when wind < 8 knots, less heel angle and equivalent speed for wind 8 to 16 knots, better speed (upwind) / lower speed (downwind) when wind > 16 knots. Results are also provided with an intermediate displacement (172 kg case) and an initial ballast ratio of 33%. These results show the kind and the order of magnitude of the trade-off when using such foil, and the tools developed can help defined the optimal foil dimension for a given specification. Axis of investigation to improve the efficiency can be : the use of a foil with an active device for the lift adjustement (flap) can improve these results in particular for the light winds < 1 knots where the speed loss is usually maximum while the heel reduction is not crucial. To add sails area when adding a foil, to compensate the speed loss instead of searching for less heel angle Acknowledgment My warmest thanks to Alain Lebeau who supported me in this approach, realizing the Maxsurf VPP computations and all the 3D views which illustrate this document.

21/3 Annexe 1 : DB32 data for 3 bulb ballasts Hydrostatics data in light weight condition without crew (Gene-Hull outputs) : 2.1 Hull Loa (m) 9,7 Lwl (m) 9,9 > Lwl/D^(1/3) 7,82 >> ft 31,82 29,82 B (m) 3, at X (% Lwl) 34, >> ft 9,83 Bwl (m) 2,1 at X (% Lwl) 39, > Bwl/B,716 >> ft 7,4 Freeboards (m) > Aft Midship Fore Tc (m),181 at X (%Lwl),,68,78,92 >> ft,9 >> ft 2,23 2,6 3,2 Displacement at H (m3) 1,7 at Xc (m) 4,232 Xc (%Lwl) 46,6 Zc (m) -,7 >> lbs 349 w. seawater 12 kg/m3 >> ft -,23 Displacement at H-3cm (m3) 1,1772 at Xc (m) 4,299 Xc (%Lwl) 47,3 Zc (m) -,6 Displacement at H+3cm (m3) 1,99671 at Xc (m) 4,169 Xc (%Lwl) 4,86 Zc (m) -,8 Cp (%) 7,2 Sf (m2) 13,74 at Xf (m) 3,967 Xf (%Lwl) 43,64 >>> Xc Xf (%Lwl) 2,92 >> ft2 147,87 >> ft 13,1 Angle immersed sheer li ( ) 27, at section C4 (4% Lwl) Sw (m2) 14,3 >Sm/D^(2/3) 1,39 >> ft2 11,4 Shull (m2) 32,8 at X (m) 414,78 Z (m),14 >> ft2 34,34 >> ft 136,83 >> ft,46 Sdeck (m2) 21,24 at X (m) 37,1 >> ft2 228,9 >> ft 1231,99 2.2 Keel Vol. keel (m3),628 at X (m) 4,436 X (%Lwl) 48,8 Z (m) -,99 Mass keel(kg) 44,6 >> ft 14, >> ft -3,24 >> lbs 97 Vol. Bulb(m3),2948 at X (m) 4,246 X (%Lwl) 46,71 Z (m) -2, Mass bulb(kg) 334,6 >> ft 13,93 >> ft -6,6 >> lbs 738 Draft oa (m) 2,13 Sw (m2) 2,9 Sxz (m2) 1,17 >> ft 6,99 >> ft2 31,27 >> ft2 12, LCR (m) 4,4 >> ft2 48,81 2.3 Rudder(s) Number 1 method : keel profile extended to the waterline, 2% c at 4% draft oa Volume (m3),98 at X (m) -,1 X (%Lwl) -1,1 Z (m) -,6 Sw (m2),86 >> ft -,33 Sxz (m2),41 per rudder >> ft2 9,2 >> ft2 4,4 2.4 Hull + Keel + Rudder(s) Displacement at H (m3) 1,6717 at Xc (m) 4,214 Xc (%Lwl) 46,36 Zc (m) -,14 (kg) 1712 >> ft 13,83 >> ft -,46 >> lbs 3774 Ballast (kg) 77 at Xg (m) 4,34 Xg (%Lwl) 47,9 Zg (m) -1,42 >> lbs 178 >> ft 14,29 >> ft -4,67 >> % Ballast 4,3 Sw (m2) 17,8 >Sw/D^(2/3) 12,64 >> ft2 191,6 Then, using Gene-Hull module for hull with heel, data for 3 displacements (12 kg, 192 kg, 242 kg) and heel angles from to 3 are extracted, these 3 displacements being derived from the variation of the ballast bulb, respectively 32 kg, 77 kg and 122 kg.

22/3 Heel ( ) D3 4kg D crew 3 D3 + 4kg, 1,3668 1,847 2,24372 H (cm) 1,378-1,6696-4,6413 Lwl (m) 9,48 9,6 9,9 Bwl (m) 2,11 2,19 2,26 Sw (m2) 17,6 18,74 19,89 Xc (m) 4,26 4,19 4,14 Yc (m),,, Z c (m) -,6 -,8 -,9 >> Yb (m),,, Ym crew3 c,,, GZ (m),,, RM (kn.m),,, Ym crew3 w,22,17,13 GZ (m),22,17,13 RM (kn.m) 3,18 3,18 3,18 Mass Tot (kg) 12 192 242 Heel ( ) D3 4kg D crew 3 D3 + 4kg, 1,3668 1,847 2,24372 H (cm) 2,1638-1,16-4,1449 Lwl (m) 9,4 9,1 9,6 Bwl (m) 2,6 2,17 2,22 Sw (m2) 16,9 18,36 19,47 Xc (m) 4,22 4,16 4,11 Yc (m) -,2 -,18 -,17 Z c (m) -,7 -,8 -,9 >> Yb (m) -,18 -,16 -,16 Ym crew3 c -,2,2, GZ (m),16,19,2 RM (kn.m) 2,37 3,6 4,8 Ym crew3 w,2,19,18 GZ (m),38,3,34 RM (kn.m),4 6,72 7,96 Mass tot (kg) 12 192 242 Heel 1 ( ) D3 4kg D crew 3 D3 + 4kg 1, 1,3668 1,847 2,24372 H (cm) 4,16,6164-2,941 Lwl (m) 9,21 9,47 9,4 Bwl (m) 1,91 2,6 2,19 Sw (m2) 16,34 17,86 18,94 Xc (m) 4,9 4,6 4,3 Yc (m) -,38 -,3 -,33 Z c (m) -,7 -,9 -,1 >> Yb (m) -,34 -,32 -,31 Ym crew3 c -,4,4,9 GZ (m),3,36,4 RM (kn.m) 4,48 6,93 9,36 Ym crew3 w,18,2,22 GZ (m),2,3,3 RM (kn.m) 7,6 1,6 12,49 Mass tot (kg) 12 192 242

23/3 Heel 1 ( ) D3 4kg D crew 3 D3 + 4kg 1, 1,3668 1,847 2,24372 H (cm) 7,4931 3,6176,1144 Lwl (m) 8,8 9,26 9,49 Bwl (m) 1,77 1,94 2,8 Sw (m2) 1,29 16,88 18,2 Xc (m) 3,92 3,92 3,92 Yc (m) -,3 -, -,47 Z c (m) -,8 -,1 -,11 >> Yb (m) -,47 -,46 -,44 Ym crew3 c -,,6,14 GZ (m),42,2,7 RM (kn.m) 6,17 9,94 13,46 Ym crew3 w,16,22,27 GZ (m),63,68,7 RM (kn.m) 9,24 13, 16,3 Mass tot (kg) 12 192 242 Heel 2 ( ) D3 4kg D crew 3 D3 + 4kg 2, 1,3668 1,847 2,24372 H (cm) 12,397 8,226 4,198 Lwl (m) 8, 8,83 9,18 Bwl (m) 1,66 1,77 1,91 Sw (m2) 14,32 1,98 17,41 Xc (m) 3,77 3,78 3,79 Yc (m) -,63 -,61 -,9 Z c (m) -,9 -,1 -,12 >> Yb (m) -,6 -,6 -, Ym crew3 c -,7,8,18 GZ (m),49,64,73 RM (kn.m) 7,19 12,21 17,9 Ym crew3 w,13,24,31 GZ (m),69,79,8 RM (kn.m) 1,17 1,19 2,7 Mass tot (kg) 12 192 242 Heel 2 ( ) D3 4kg D crew 3 D3 + 4kg 2, 1,3668 1,847 2,24372 H (cm) 18,836 13,9173 9,392 Lwl (m) 8,31 8,6 8,6 Bwl (m) 1, 1,7 1,81 Sw (m2) 13,7 1,39 16,3 Xc (m) 3,6 3,6 3,67 Yc (m) -,71 -,69 -,68 Z c (m) -,1 -,11 -,13 >> Yb (m) -,63 -,63 -,63 Ym crew3 c -,9,1,22 GZ (m),4,73,8 RM (kn.m) 7,94 13,93 2,1 Ym crew3 w,11,2,34 GZ (m),73,88,97 RM (kn.m) 1,82 16,81 22,89 Mass tot (kg) 12 192 242

24/3 Heel 3 ( ) D3 4kg D crew 3 D3 + 4kg 3, 1,3668 1,847 2,24372 H (cm) 26,73 21,3448 16,4663 Lwl (m) 8,1 8,3 8,6 Bwl (m) 1,1 1,62 1,74 Sw (m2) 13,36 14,77 1,82 Xc (m) 3,4 3, 3,6 Yc (m) -,78 -,76 -,7 Z c (m) -,1 -,12 -,13 >> Yb (m) -,68 -,69 -,69 Ym crew3 c -,1,12,26 GZ (m),8,81,9 RM (kn.m) 8,8 1,46 22,44 Ym crew3 w,8,26,38 GZ (m),77,9 1,7 RM (kn.m) 11,33 18,21 2,19 Mass tot (kg) 12 192 242 RM curves are then derived from these data : Mass tot(kg) 12 Mass tot(kg) 192 Mass tot(kg) 242 Heel angle RM crew3 c RM crew3 w RM crew3 c RM crew3 w RM crew3 c RM crew3 w ( ) kn.m kn.m kn.m kn.m kn.m kn.m, 3,18, 3,18, 3,18 2,37,4 3,6 6,72 4,8 7,96 1 4,48 7,6 6,93 1,6 9,36 12,49 1 6,17 9,24 9,94 13, 13,46 16,3 2 7,19 1,17 12,21 1,19 17,9 2,7 2 7,94 1,82 13,93 16,81 2,1 22,89 3 8,8 11,33 1,46 18,21 22,44 2,19 D 12 kg D 192 kg D 242 kg 3 2 1 RM (kn.m) vs Heel ( ) Crew 3 center & windward ; Keel 32 kg 3 2 1 RM (kn.m) vs Heel ( ) Crew 3 center & windward ; Keel 77 kg 3, 2, 1, RM (kn.m) vs Heel ( ) Crew 3 center & windward ; Keel 122 kg 1 1 2 2 3 1 1 2 2 3, 1 1 2 2 3 From all these data, hull wetted surface and righting moment RM formulations are established, function of both the heel and the displacement, and introduced in the VPP. Other variable parameters like Lwl, Bwl, etc... are also used to formulate the residuary drag component.

2/3 Annexe 2 : Foil data and formulations The foil profile used for the simulation is a Naca 241 : http://airfoiltools.com/airfoil/details?airfoil=naca241-il Curves for Re =, E6 and 1, E6 Formulations used: i = > Cl =,2 ; i = 7,1 > CL = 1, >>> CL 2D =,2 +,16 i >>> with application of the lifting line theory : CL = (,2 +,16 i) / (1 + 2 / π ARe) Cd = Cd + CL2 / π ARe, with ARe = Effective aspect ratio For a symetric trapazoid plan form, of span L and chords C1 (central) and C2 (tip) : ARe = L / ((C1+C2)/2) Cd ~ 2,4 Cf

26/3 2,4 is the approximative ratio between the wetted surface and the wing surface Cf is the friction coefficient according to ITTC 7 formula with (C1+C2)/2 used for the Reynolds computation. Foil Downforce : Foil Drag : Ffoil= CL I/2 ρ S V2 Dfoil = Cd I/2 ρ S V2 Foil data used for the simulation : with initial displacement 192 kg : Foil 1 : Surface :,22 m2 Span : 2, m Effective aspect ratio : 18,18 (Central chord 132 mm ; Tip chord 88 mm) i (at heel ) : 3 >>> CL =,49 hinged inclination of the foil assembly : 11 hinged height: z -1,7 m Foil 2 : Surface : 1,2 m2 Span : 2, m Effective aspect ratio : 1 (Central chord 6 mm ; Tip chord 4 mm) i (at heel ) : 3 >>> CL =,4 hinged inclination of the foil assembly : 11 hinged height: z -1,7 m with initial displacement 172 kg : Foil 1 : Surface :,72 m2 Span : 2,7 m Effective aspect ratio : 1,13 (Central chord 32 mm ; Tip chord 214 mm) i (at heel ) : 3 >>> CL =,34 hinged inclination of the foil assembly : 11 hinged height: z -1,7 m Foil 2 : Surface : 1,2 m2 Span : 2, m Effective aspect ratio : 1 (Central chord 6 mm ; Tip chord 4 mm) i (at heel ) : 3 >>> CL =,14 hinged inclination of the foil assembly : 11 hinged height: z -1,7 m with initial displacement 12 kg : Foil 1 : Surface :,8 m2 Span : 2,8 m Effective aspect ratio : 9,22 (Central chord 36 mm ; Tip chord 243 mm) i (at heel ) : 3 >>> CL =,31 hinged inclination of the foil assembly : 11

27/3 hinged height: z -1,7 m Foil 2 : Surface : 1,2 m2 Span : 2, m Effective aspect ratio : 1 (Central chord 6 mm ; Tip chord 4 mm) i (at heel ) : 3 >>> CL =,14 hinged inclination of the foil assembly : 11 hinged height: z -1,7 m Foil assembly formulations : 1 1-1 - 1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9 1 - -1-1 Hinged angle α -2-2 Incidence angle i 1 1-2 -1-1 - 1 1 2 2 3 - -1 Heel angle φ -1-2 -2

28/3 2, 1, 1,,, -1-1 1 2 2 3 3 4 4 6 6 7 7 8 8 9 9 1 -, -1, -1, Orientation λ of the foil -2, Orientation λ of the foil (in the leeway direction) : sin λ = sin α sin φ Incidence i of the foil (in the vertical plan), taking into account two effects which both increase the incidence : due to the foil assembly, due to the trim, both increasing with heel according to : i = i + (α - α) +,9 (φ/2), with sin α = sin α cos φ, where : α : hinged inclination (11 in this simulation) i : initial incidence of the foil (3 in this simulation) φ : heel angle Heel φ ( ) 1 2 3 «Leeway orientation» λ ( ) Incidence variation δi ( ) >>incidence i ( ) 1,9 3,74,48 3,62 3,62 1,7 4,7 2,84,84 Annexe 3 : VPP specific modelling / main formulations This VPP is built with a set of formulations which should reflect the displacement variation for each line of computation, i.e. each real wind force, this displacement variation resulting from the foil downforce variation with boat speed. For that objective, 3 reference Maxsurf VPP for 3 bulb ballasts (32 kg, 77 kg and 122 kg) are performed and used (to determine sailing forces and moments,, eventually to validate the specific VPP). For simplification (of the sailing forces formulations in particular), two sailing cases are studied : upwind, with twa 43 to 39 downwind, with twa 14 Are determined from Maxsurf VPP through semi-empirical formulations specific for each sailing

29/3 condition (upwind twa 39 to 43, downwind twa 14 ) : Sailing forces forward and side, Heeling arm and moment, Are determined from Gene-Hull hydrostatics outputs and with formulations proposed by Larsson- Eliasson in «Principles of Yacht design» 2 nd Edition 2 or by Hoerner in «Fluid-dynamic drag» 196 : Righting moment, Drag components, *** A3,1 Sailing thrust and side forces The Maxsurf VPP is used with Melges 32 sails data and DB32 hull outputs: Mainsail surface : 41.3 m2 Jib surface : 23.6 m2 Asymétrique spi :121. m2 Jib triangle : I : 12.43 m ; J : 3.29 m Mainsail triangle : P : 13.4 m ; E : 4.72 m For simplification, 2 different semi-empirical formulations are used for upwind and downwind conditions and taken into account the flattness adjustement as done by the VPP for the 3 cases, according to the schemes here below : the points are the results of the VPP maxsurf, the continue lines are the formulations adopted color code : green D 12 kg ; red D 192 kg ; blue D 242 kg Twa 42 ; Flatt (42, φ, D) Twa 42 ; Flats (42, φ, D),9,8,7,6,,4,3,2,1,,, 1 1 2 2 3 3 4 1 1 2 2 3 3 4 Twa 14 ; Flatt (14, φ, D) Twa 14 ; Flats (14, φ, D),4,4,3,3,2,2,1,1,2,2,1,1,3,3,2,2,1,, 1 1 2 2 3 3 4,1,, 1 1 2 2 3 3 4

3/3 Upwind : Ft = R x (twa/42)^, x (awa/18xπ) x Va^2 x Flatt (42, φ, D) Fs = R x (twa/42)^, x (awa/18xπ)x Va^2 x Flats (42, φ, D) Downwind : Ft = R x (awa/18xπ) x Va^2 x Flatt (14, φ, D) Fs = R x (awa/18xπ) x Va^2 / Cos (awa-9) x Flats (14, φ, D) With : Ft = thrust force (kn) ; Fs = side force (kn) R : reefing coefficient (1 for 1% sail surface,,7 for 7%,, for %) awa : apparent wind angle ( ) Va : apparent wind (knots) *** A3,2 Rigthing moment RM The formulation for RM take into account the Gene-Hull results for the 3 displacement cases : the points are the results of the VPP maxsurf, the continue lines are the formulations adopted color code : green D 12 kg ; red D 192 kg ; blue D 242 kg 3, 2, 2, 1, 1,, Twa 42 Twa 14 3, 2, 2, 1, 1,,, 1 1 2 2 3 3 4, 1 1 2 2 3 3 4 Comment : there is a good agreement for heels between 1 and 22. For heel > 22, RM modelling becomes higher than the Maxsurf points : it is likely due to the computation at trim fixed to zero, actually the center of buyancy moves backward, the rear hull goes up and that decreases the hull contribution to the RM. *** A3,3 the friction component of the drag Usual formulation is used with friction coefficient Cf from the ITTC 7 formula (,7 / (Log Re -2)^2 ) and considering the wetted surfaces of the hull (variable with displacement and heel), of the keel and of the rudder : Df =, x 12 x Vb^2 x (Sw hull (φ,d) x Cf hull + Sw keel x Cf keel + Sw rudder x Cf rudder), Shull (φ,d) being built with the data given in Annexe 1

31/3 As a remind, foil drag formulation is given in Annex 2, inc. its friction component. *** A3,3 Specific drag of the bulb For a good comparison of the foil versus the bulb, it is necessary to assess at the best the form drag of this bulb in addition to the friction one, using a formulation proposed by Hoerner in «Fluidynamic Drag» for 3D streamline form : when considering the wetted surface of the bulb, the Cf coefficient is increased by a multiplying factor ΔCf = 1 + 1, (d/l)^3/2 + 7 (d/l)^3, d and L being respectively the equivalent diameter and lenght of the streamline bulb. For the 3 keel-bulb cases : 12 kg : no bulb in addition to the keel wing >>> Sw keel 2,71 m2, Δcf = 192 kg : Sw keel : 2,71 m2 + ΔSw bulb :,19 m2 with Δcf : 1,86 242 kg : Sw quille : 2,71 m2 + ΔSw bulb :,71 m2 with Δcf : 1,18 *** A3,4 residuary component of the keel Dr Formulation using the Delft series reported in «Principles of Yacht Design» with DB32 hull data provided by Gene-Hull outputs : 1 9 8 7 6 4 3 2 1 Residuary drag Dr/Mg (%) versus Froude Fn,,1,1,2,2,3,3,4,4,,,6,6,7,7 Blue points are from Delft series / parent model N 1 (V mid-heavy displacement hull), obtained with data Lwl, Bwl, Cp,etc, and given up to Fn,4 Red points are from Delft series / parent model N 2 (U light displacemet hull), idem hull data and given from Fn,47 to,7,47 In green, the adopted formulation taken into account the two series of points but without the peak at Fn,4-,4 as this peak is not apparent in the Maxsurf results. It is assumed rational as the DB32 is a U light weight design.

32/3 *** A3, Extra drag due to heel According to the formulation proposed by «Principles of yachy design» : Dh =, x 12 x Vb2 x Sw(φ= ) x Ch x Fn2 x (φ/18xπ)2, with Heel resistance coefficient Ch : Ch = ( 6,747 (Tc/T) + 2,17 (Bwl/Tc) + 3,71 (Tc/T) (Bwl/Tc) ) 1,91 1-3 Vb (m/s) speed boat Sw (m2) wetted surface φ ( ) heel angle >>> Dh (kn) Extra drag *** A3,6 Induced drag From Maxsurf VPP, an average proportion of Fs is adopted for both upwind and downwind (a bit conservative for downwind) : Di =,18 Fs Exluding the foil : as a remind, the induced drag of the foil is within Dfoil here above *** A3,7 VPP results for the 3 displacements and comparison with the Maxsurf VPP results Upwind with Twa 42 Boat speed (Knts) vs true wind (Knts) DB32 with crew 3 centered & twa 42 green : 12 kg ; red 192 kg ; blue 242 kg Heel ( ) vs true wind (Knts) DB32 with crew 3 centered & twa 42 green : 12 kg ; red 192 kg ; blue 242 kg 8 7 6 4 3 2 1 2 4 6 8 1 12 14 16 18 2 4 3 3 2 2 1 1 2 4 6 8 1 12 14 16 18 2

33/3 Downwind with Twa 14 Boat speed (Knts) vs true wind (Knts) DB32 with crew 3 centered & twa 14 green : 12 kg ; red 192 kg ; blue 242 kg Heel ( ) vs true wind (Knts) DB32 with crew 3 centered & twa 14 green : 12 kg ; red 192 kg ; blue 242 kg 14 4 12 4 1 8 3 3 2 6 2 4 1 2 2 4 6 8 1 12 14 16 18 2 22 1 2 4 6 8 1 12 14 16 18 2 22

34/3 List of symbol ARe : foil effective aspect ratio awa : apparent wind angle B : Hull beam Ballast : weight of keel + bulb + if any foil downforce lift Bwl : hull beam at the water line Cd : foil Drag coefficient Cf : friction coefficient Ch : Heel resistance coefficient CL : foil Lift coefficient CL 2D : the 2D lift coefficient of a foil profile Cp : Hull prismatic coefficient C1 : foil central chord C2 : foil tip chord D : Displacement Df : Friction drag of the boat (excluded the foil, in Dfoil) Dfoil : foil drag Dh : Extra drag due to heel Di : Induced drag of the boat (excluded the foil, in Dfoil) Dr : Residuary drag Fn : Froude number Ffoil : foil downforce lift Flatt, Flats : Flatness functions for thrust and side sailing forces Fs : foil side force Ft : foil thrust force GZ : lever arm between buoyancy and gravity vertical forces HM : heeling moment i : flow incidence angle on the foil (i : initial incidence when heel is zero) L : span lenght of the foil Loa : Length overall Lwl : lenght water line R : reefing cooefficient Re : Reynolds number RM : righting moment RM crew3 c, RM crew3 w : righting moment with 3 crew sit centered, sit windward S : wing surface of the foil

3/3 Sf : hull floatation surface Sw : wetted surface T : Draft overall Tc : Hull draft twa : true wind angle V, Vb : boat speed Va : apparent wind speed VMG : Velocity made good Xc, Yc, Zc : coordinates of the hull center of buoyancy Xf : X coordinate of floatation surface geometrical center Yb: Y coordinate of the total buoyancy center of the immerged bodies (hull + keel + rudder) Ym crew3 c : Y coordinate of the total mass center of gravity, with 3 crew sit in the center Ym crew3 w : Y coordinate of the total mass center of gravity, with 3 crew sit windward α : hinged joint angle in the vertical plane projection (α initial angle when heel is zero) λ : «leeway orientation» angle of the foil assembly φ : heel angle ρ : sea water volumic mass 12 kg/m3