5th High Performance Yacht Design Conference Auckland, 10-12 March, 2015

PERFORMANCE ASSESSMENT AND OPTIMIZATION OF A C-CLASS CATAMARAN HYDROFOIL CONFIGURATION

Agathe Paulin1, agathe.paulin@gmail.com Heikki Hansen2, heikki.hansen@dnvgl.com

Karsten Hochkirch3, karsten.hochkirch@dnvgl.com Martin Fischer4, martin.ncl@gmail.com

Abstract. Recent breakthroughs in the America's Cup have put hydrofoil technology in the focus of high-performance sailing. This paper describes the performance assessment of two different centreboard hydrofoils designed for a C-class catamaran. Regarding the numerous design criteria resulting from sailing on hydrofoils, a reliable performance assessment tool helps to find the best compromise [1]. A Velocity Prediction Program (VPP) has been developed in order to facilitate the analysis of numerical simulation results obtained for appendages and the wing-sail in terms of speed potential of the C-Cat. The approach used to model the hydro- and aerodynamic forces on the catamaran is described, along with the challenges peculiar to a VPP model for sailing on hydrofoils. Different optimization schemes for trimming the wing-sail and hydrofoil configurations are tested to obtain the theoretically most efficient trim settings and to evaluate the different hydrofoil designs. Using the developed VPP model, realistic velocity prediction can be carried out. The catamaran flying on hydrofoils sails at up to 2.8 times the true wind speed in many conditions. Records and observations made during the International C Class Cup (ICCC) of 2013 confirm the predicted performance of the modelled C-Cat. Some remaining challenges in velocity prediction of vessels equipped with hydrofoils are highlighted.

1 Naval Architect, Department of Department of Naval Architecture and Ocean Engineering, TU Berlin, Germany 2 Team Leader, Fluid Engineering, DNV GL, Germany 3 Head of Department, Fluid Engineering, DNV GL, Germany 4 Designer / Naval Architect, NC Raceboats, New Caledonia

NOMENCLATURE

AoA Angle of Attack AWA Apparent Wind Angle AWS Apparent Wind Speed C-Cat C-Class Catamaran CFD Computational Fluid Dynamics ICCC International C Class Cup Shift Length of the hydrofoil emerged between the

hull and the water surface S-Pos Setting of the centreboard hydrofoil vertical position Rake Angle of hydrofoil from vertical in y-plane TWA True Wind Angle TWS True Wind Speed VMG Velocity Made Good VPP Velocity Prediction Program Vs Boat speed

1. INTRODUCTION

The "C" class is a high-performance developmental catamaran sailing class governed by a simple box rule. Overall length, beam and sail area must stay within the limits of the box rule and the boat has to be a catamaran, symmetric with respect to the centre plane of the boat [2]. C-class catamarans - or C-Cats - are small and very light boats used for match races known as the International C-Class Cup (ICCC). The class rules leave ample room for innovative concepts regarding the C-Cat sail and appendage configurations; one of its special features being the use of a rigid wing-sail instead of mast, boom and mainsail.

The goal of this work is to compare designs of centreboard hydrofoils for the C-Cat Groupama C by developing a tool capable of assessing the influence on the global performance of the boat. For this purpose a Velocity Prediction Program (VPP) is set up using the software FS-Equilibrium. The VPP calculates the variables describing the C-Cat's velocity, position and trim parameters by solving the equations of motion for given wind conditions in six degrees of freedom. The VPP model of the C-Cat sailing on hydrofoils is based on numerical simulation results for the appendages and the wing-sail. The work focuses on assessing the performance potential of two hydrofoil designs and on establishing trimming guidelines in order to obtain the best performance out of the hydrofoil design selected for the ICCC in September 2013.

2. C-Cat Force Balance on Hydrofoils

Hydrofoils are hull appendages designed to generate hydrodynamic forces capable of raising the yacht's body up and out of the water. Wetted surface is reduced, thereby decreasing frictional and wave-making resistance so that higher speeds can be reached. Yet the hydrofoils’ geometry and size necessary to produce significant hydrodynamic lift also cause additional drag, which harms the performance, especially at low speeds. This drawback has to be traded off against the system's ability to generate enough lift, which reduces hull drag for various wind conditions. When sailing on hydrofoils specifically, the vertical equilibrium of the boat depends on three forces (see Figure 1 and Figure 2). The gravitational force tends to

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make the boat sink, and is counterbalanced by the displacement of the hulls and/or the vertical lift produced by the foils. The different forces create bow-up or bow-down moments which result in an equilibrium pitch angle.

Figure 1. Force balance on the C-Cat - side view

The proportion of the total weight carried by the hydrofoil lift is hereafter referred to as lift fraction. Different operating regimes of the sailing C-Cat can be distinguished:

Lift fraction is close to zero: buoyant forces dominate in carrying the boat's weight. Pitch stability is ensured by the hydrostatic properties of the hull(s), but significant wetted surface creates frictional and wave-making resistance.

Lift fraction is close to one: hydrofoil lift carries most of the boat's weight: the C-Cat is "skimming". Reduced wetted surface also entails less hydrostatic restoring moment and increases the risk of loss in pitch equilibrium.

Lift fraction equals one: the hydrofoils carry the total weight and the boat is fully foiling.

When fully foiling, the hull no longer creates buoyant forces and moments to balance heave and pitch so that these degrees of freedom need to be stabilised within very small margins (heave +/- 0.5 m, pitch +/- 1.5 degrees) by the appendages to achieve equilibrium. As the boat starts foiling, speed increases due to reduced drag. Consequently more lift is produced by the hydrofoils, which makes the boat ride higher. The foil is less submerged and produces less side force for a given AoA, so to maintain lateral equilibrium the leeway angle increases [3]. Figure 2 and Figure 3 show the transverse force components of the C-Cat. While larger leeway increases the AoA for generating side force, the effective AoA at the tip (horizontal part of the foil) decreases and the vertical lift is reduced. This way the ride height can be self-regulating.

Figure 2. Force balance on the C-Cat - front view

Pitch stability is achieved by the rudder elevators, which function similar to an elevator on an aircraft. An increase in pitch angle causes more AoA on the centreboard foil and the elevators. Due to the centre of gravity location, the additional pitching moment caused by the rudder elevators decreases the AoA so that equilibrium is achieved.

Figure 3. Force balance on the C-Cat - top view

A boat equipped with a hydrofoil configuration shall reach the fully foiling regime in the widest range of wind conditions possible. The driving factor for designing an efficient hydrofoil configuration is therefore obtaining the best lift to drag ratio while using the leeway to achieve heave stability. The examination of the force balance leads to the definition of design criteria for an efficient hydrofoil configuration. A reliable performance assessment is required to find the best compromise. It also appears that the stability of the boat's equilibrium condition is a significant aspect which should be investigated as well.

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2.1 Hydrofoil System of Groupama C

The hydrofoil system designed for Groupama C consists of the following elements:

Production of vertical lifting force from two retractable centreboard hydrofoils: one on each hull, the leeward hydrofoil is used alone while the windward foil is completely pulled up.

L-shaped rudders with symmetrical horizontal hydrofoils (rudder elevators) to ensure pitch stability while foiling.

A two person crew sails a C-Cat. This means the hydrofoils have to be managed by one crew member while the helmsman keeps steering the craft. The hydrofoil system used on Groupama C has a centreboard hydrofoil with an S-shaped head, which allows adjusting the cant angle (dihedral) by changing the foil extension. The hydrofoil strut can be pulled up or down. A bearing system is integrated into the hull, which pivots to follow the spine shape. By changing the foil extension, the average dihedral angle and the span of the hydrofoil change, influencing both magnitude and direction of the hydrodynamic lift force. The vertical position of the hydrofoils is measured by a value called “S-pos”: it corresponds to the length of the foil (in centimetres) which has been pulled up in the hull, see Figure 4. The horizontal angle of incidence of the foils, called “rake” can also be adjusted, see Figure 4. Its value corresponds to an angle varying a few degrees around zero, and is defined positive when more AoA is given to the foil for an identical bow-up pitch of the boat. Rake can be adjusted for the centreboard hydrofoils and rudder hydrofoils. The centreboard hydrofoil rake can be trimmed while sailing, but the rudder rake has to be adjusted on the ground and cannot be modified while sailing.

Figure 4. Trimming possibilities of the hydrofoil system

2.2 Performance Parameters

The force balance is influenced by the position of the boat (speed, heel, pitch, leeway, sink) but also by parameters which can be adjusted or trimmed by the crew:

Crew position: the C-Cat's weight (approx. 155 kg) is smaller than the crew's weight (approx. 170 kg). Hence the lateral and longitudinal position of the crew plays a significant role in the equilibrium.

Trim of the wing-sail: the C-Cat is equipped with a rigid wing-sail which consists of two rigid panels. Trimming possibilities are described in Figure 5. Both wing panels can be twisted, which results in a difference in angle of incidence between the head of the wing-sail and its foot (W1 Twist, W2 Twist). In addition the incidence (W1 Angle) and camber (W2 Angle) of the wing-sail can also be adjusted.

Setting of the appendages: as described in previous section.

Figure 5. Trimming possibilities of the rigid wing-sail

3. VELOCITY PREDICTION

3.1 Special Features of the C-Cat VPP Model

The goal of the performance assessment is to obtain the optimal velocity for a constant wind condition with the constraint that the forces and moments are in equilibrium. Given optimal velocity for each wind condition the hydrofoil designs can be compared based on their best achievable speed. Efficient trim will lead to the fastest sailing condition under the equilibrium constraint. In practice, this is regulated by adjusting all the parameters together. Hence the parameter interdependency shall be modelled accurately to describe the actual behaviour of the boat and allow reliable performance assessment. The software FS-Equilibrium accomplishes this task. The algorithm that solves the force balance is nested in an optimization routine which maximizes the boat's speed.

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In addition to the 5 state variables (section 2.2) describing the position and motion of the craft, there are 10 independent variables defining the boat’s trim: 4 wing-sail settings, 2 parameters describing the longitudinal and transverse position of the crew, and 4 describing the hydrofoil trim: centreboard hydrofoil S-pos and rake, and rudders angle and rake. In order to solve the equilibrium in 6 degrees of freedom all variables need to be defined in one of the following ways:

as free variables to balance the boat for a chosen degree of freedom,

as optimization variables which are adjusted within feasible range in order to optimize speed for a calculated equilibrium condition,

to remain constant or evolve as functions of other variables.

Given: The force model for the C-Cat sailing on hydrofoils TWS, TWA conditions Start values for the position of the boat and performance parameters:

x (xDoF1, xDoF 2,..., xDoF 6,

FREE VARIABLES

x1, x2,..., xn )OPTIMIZATION

Maximize goal function:

G(x)VS

By choosing variables:

x1, x2,..., xn

Opt

imiz

atio

n R

outi

ne

Subject to:

0)(

0)(

xM

xF

Constraints of optimization problem (force balance) solved by choosing variables:

xDoF1, xDoF 2,..., xDoF 6

Figure 6. Structure of the VPP model set-up using FS-Equilibrium

Figure 6 shows the structure of the VPP model. The velocity prediction requires both a realistic definition of the role played by the different parameters and an accurate model of the forces acting on the boat. Using this optimization process, the most efficient trim set-up according to the wind conditions is obtained, and therefore the best performance. Two main challenges for modelling the C-Cat sailing on hydrofoils are:

Implementing a force model based on inter-polated numerical simulation results,

Simulating the C-Cat in different operating regimes varying from displacement to foiling mode (lift fractions from 0 to 1).

3.2 Integration of Numerical Simulation Results

In order to analyse the speed potential of the boat at the design stage, CFD results for the appendages and the wing-sail are used as input for the performance assessment. For the appendages and the wing-sail, CFD simulations provide the forces and moments on the elements placed in constant inlet flow for a series of calculation cases: several inlet flow conditions and several settings of the element position in the flow depending on the parameters as depicted in Table 1. In the VPP, the CFD calculation cases are interpolated to obtain a continuous data set over the entire analysis range. Thus the aero- and hydrodynamic forces and moments can be calculated in FS-Equilibrium for any set of values given to the input parameters, and not only for the combinations used for the CFD simulations.

Inlet Flow Force coefficients

Wing-sail

Apparent wind > AWS, AWA

Wing-sail trim > Angle & twist of both rigid elements

Appendages

Velocity of the boat > Vs, leeway angle

Height of the appendage root above-water > Shift Centreboard > S-pos, rake Rudder > Rudder angle, rake

Effective AoAs seen by the appendages and AWA take pitch and heel angles of the C-Cat into account

Table 1. Input parameters influencing the flow around the wing-sail and the appendages (see parameter definition in Figure 4 and Figure 5)

For the appendages and the wing-sail, response volumes of lift and drag coefficients are created as functions of the effective AoAs seen by the modelled element, which are defined as difference between inlet flow direction and the incidence given to the element, with a contribution of pitch and heel angles of the boat. The following method is used to process the simulation results:

First, the forces and moments are expressed in the coordinates related to the undisturbed flow, so that they are decomposed into components perpendicular and parallel to the flow, ie. into a "lift and drag" description [4].

The force origin used for CFD calculations on the appendages is at hydrofoil/water interface, ie. moving along the hydrofoil from one simulation case to another. The moments of each calculation case are expressed at the same body-fixed origin, as depicted in Figure 7.

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The forces and moments are then normalized by the dynamic pressure of their corresponding inlet flow.

Finally, the normalized coefficients and their corresponding combinations of input parameters obtained for CFD calculations are gathered in a table. This data is fitted in FS-Equilibrium using a multidimensional B-Spline to describe the response volume.

This way, the VPP is not confined to the discrete cases used in the CFD calculations and can update the aero- and hydrodynamic force and moments while solving the equations of motion. Modifying the aero- and hydrodynamic flow around the C-Cat and the values of the input parameters, the forces and moments can be obtained by multiplication of the interpolated force and moment coefficients with the dynamic pressure of the updated flow condition.

Figure 7. Change in origin to import numerical simulation results into the VPP model

3.3 Modelling the Foiling C-Cat

Whether the appendages are completely submerged or not, the hydrodynamic forces on the appendages are influenced by the same parameters: horizontal and vertical AoAs and inlet flow speed. However, with growing lift fraction, the appendage starts to emerge partially. The height of the appendage above the water, called shift and defined in Figure 7, also influences the forces. Hence two sets of CFD results using a potential code were computed. Free-surface boundary conditions are taken into account when computing flows around partially emerged appendages, whereas in the case of completely submerged appendages, rigid surface boundary conditions are considered: the water-plane defines a symmetry plane of the flow. Since this method cannot capture spray drag, an empirical spray drag was added to the computed drag. In the VPP, as long as the appendage under consideration is fully submerged and covered by the buoyant hull, the simulation results without free-surface effects are used. When the body-fixed force origin of the appendage (as defined in Figure 7) nears the water-surface 10

centimetres before the appendage root gets lifted out of the water, free-surface effects are progressively taken into account. This transition force model is defined through a linear interpolation over shift between identical simulation cases obtained with both methods. This way the force and moment output is continuous when the VPP switches between the different force models, based on the shift parameter, which facilities robust convergence towards an equilibrium condition. For race yachts, knowing the most efficient trim set-up according to the wind conditions is crucial for achieving the best performance. The position of the crew is one of the keys in the force balance, since the crew's weight is bigger than the boat's weight. The VPP is therefore set up to imitate the behaviour of the crew, which is constantly steering with the rudder, changing its position in the trapeze, and trimming the wing-sail. The free variables used to solve the force balance, as defined in section 3.1 and Figure 6, are set-up using the following scheme:

longitudinal forces determine the C-Cat's speed, side forces are balanced by the leeway angle, vertical forces by the sink (directly linked to the

shift and hence the operating regime), moments around the longitudinal axis by a

variable modelling the change of the crew's lateral position and the depowering of the sail

moments around the lateral axis by the pitch angle,

and moments around the vertical axis by the rudder angle.

When the lateral forces create a heeling moment, the crew compensates them by changing its transversal position on the trapeze and depowering the wing-sail if necessary. This is why the free variable linked to the moment balance around the longitudinal axis is a combination of the crew lateral position and the depowering of the wing-sail. The heeling angle is being considered as an optimization parameter.

3.4 Verification of Optimization Techniques

As shown in Table 2, numerous parameters influencing the force model are not solved while balancing the boat and can be used to optimize the boat speed. Due to the large number of optimisation variables and the interdependency of the forces influenced by these trim parameters, a sensible or even optimum equilibrium condition may not be found by optimisation algorithms when considering the complete set of trim parameters as optimization variables at the same time.

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Force balance variables Optimization variables

Fx Vs Aerodynamic trim > W1 Angle, W1 Twist, W2 Twist

Fy Leeway angle

Fz Shift (Sink) Hydrodynamic trim > S-pos, Rake of main hydrofoil and rudders

Mx Crew transverse position, depowering(W2 Angle)

My Pitch angle Crew longitudinal position, Heel angle Mz Rudder angle

Table 2. Utilization of the variables in the C-Cat VPP model

A study on optimization schemes is carried out to find a suitable methodology for the C-Cat’s performance assessment. As a result, the optimization of the trim parameters influencing the hydrodynamic forces and those influencing the aerodynamic forces are separated. Focus is first laid on the optimization of the centreboard hydrofoil trim. A semi-empirical force model of the wing-sail is used instead of the CFD based model, and several combinations of optimization variables are tested:

Optimizing heel angle, for an arbitrary constant hydrofoil trim

Optimizing heel angle and S-pos, the vertical position of the hydrofoil

Optimizing heel angle and both trim parameters of the hydrofoil, S-pos and rake

In a second step, the wing-sail model based on CFD results and its trim parameters are investigated. So, the heel angle and trim parameters W1 Angle, W1 Twist, and W2 Twist are optimized for a constant hydrofoil set-up1. This set-up (S-pos = 10 cm, rake = 2°) corresponds to the optimal setting obtained using the semi-empirical wing-sail force model for the majority of TWAs at a TWS of 6 m/s. Looking at the values of the different parameters describing the equilibrium conditions, the conclusion can be drawn that if hydrodynamic and aerodynamic variables are optimized separately, the simulations are robust and the results are sensible. The findings in Figure 8 show that optimizing only the S-pos parameter in addition to only the heel angle does not make a significant difference in the performance, whereas optimizing the foil rake enables to reach full foiling regime downwind.

1 The wing-sail trim parameters are introduced in Figure 5, and the parameter W2 Angle is obtained from the variable used to balance the heeling and righting moments by lateral movement of the crew and depowering the wing-sail.

Figure 8. Comparison of optimization techniques

Figure 8 also shows that the CFD based aerodynamic model leads to a better performance, particularly upwind, than the semi-empirical aerodynamic model even without modifying the hydrofoil trim. The resulting boat state parameters and crew position for the equilibrium conditions remain close to those found with the semi-empirical aerodynamic model. The difference in resulting performance suggests that the parameters chosen for the semi-empirical model were too conservative and underestimate the performance characteristics of a rigid wing-sail. One drawback is that the wing-sail model is based on computations carried out for a narrow range of AWAs. Even if with fast-sailing boats, apparent wind is generally "close-hauled", it is not always the case running downwind. Hence simulations are not computed beyond TWA = 140°. The methodology chosen for the performance assessment lets the VPP handle the wing-sail trim. FS-Equilibrium solves the force balance using the camber of the sail (W2 Angle) together with the lateral crew position for heeling moment equilibrium while the optimization algorithm adjusts the other wing-sail trim parameters to enhance speed. On the actual boat, the crew also adjusts the sail trim more actively than the foil trim to control stability and maximise speed. Wing-sail trim has also a larger number of parameters to be optimized compared to the hydrofoils. Systematic trim variations are carried out for the hydrodynamic trim parameters.

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4. PERFORMANCE ASSESSMENT

4.1 Method

Both centreboard hydrofoil designs tested are depicted in Figure 9. The innovative "double-S" shaped design sls03 is compared with the more conservative "V" shaped design g2b.

Figure 9. Spines of the centreboard hydrofoils tested for the performance assessment (front view, foils presented with maximal span)

As described in the previous section, the methodology of letting the VPP algorithm optimize wing-sail trim and systematically compute a matrix of centreboard hydrofoil set-ups is employed for the performance assessment. The force balance leading to the optimal speed for each wind condition (TWS, TWA) is then selected from the data set. Both hydrofoil configurations can therefore be compared at their best trim. The hydrofoil trims tested vary within the following ranges, which are feasible on the C-Cat:

Hydrofoil rake in deg: [-2; 0; 2; 4] S-pos in cm: [0; 10; 20; 25; 30; 40; 50]

These ranges are made up of relatively few values due to the required computation time to cover the whole matrix for both hydrofoil configurations. Since the trim settings are adjusted during a race typically only with an accuracy of a few centimetres or degrees, the results given by the VPP are considered adequate to describe the general behaviour of the C-Cat and to provide guidelines on the best trimming configuration. The C-Cat is designed to sail in TWS up to 10 m/s (approx. 20 kts). The simulations were therefore carried out for three true wind velocities of 3 m/s, 6 m/s and 9 m/s.

4.2 Results

Figure 10 presents the results obtained after carrying out the simulations using the methodology described in the previous section. The sls03 foil performs better than g2b for all conditions apart from upwind courses at TWS of 9 m/s. The C-Cat equipped with sls03 is sailing very efficiently even in small wind speeds of TWS = 3 m/s, reaching velocities of three times the speed of the wind.

Figure 10. Speed of C-Cat with optimal trim settings for both hydrofoil configurations tested

At TWS ≥ 6 m/s, where the C-Cat is fully foiling, this configuration reaches velocities of about 2.8 times the true wind speed. Both configurations show a significant increase in performance between TWS of 3 m/s and 6 m/s as they reach the fully-foiling regime at all TWAs for TWS from 6 m/s. The large increase in speed can be attributed to the hydrofoils completely carrying the hull out of the water (lift fraction = 1). With the sls03 hydrofoil, the findings obtained at TWS = 3 m/s show that the trade-off between the increase in drag at low speed and the production of lift is balanced. This design is able to produce more vertical lift in small wind speeds than the design g2b, but also has the ability to sail at smaller leeway angles (see Figure 11).

Figure 11. Polar plots of lift fraction and leeway angle for both hydrofoil configurations, TWS = 3 m/s

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Figure 11 shows that the vessel sails at leeway angles between -0.5 degrees (downwind) and 3.2 degrees (upwind). This indicates that the hydrofoils generate sufficient side force at small leeway angles even in conditions where the foil almost reaches the fully-foiling regime. The leeway to generate the required side force is smaller for sls03. Some combinations of hydrofoil trim and wind conditions lead to sailing states where the program reaches the edge of the feasibility range defined for one or several variables. In those cases the VPP might have problems to converge. On the actual boat, very large leeway angles would make the hydrofoil stall, which is why a limitation on the valid range for leeway angle is implemented in the VPP (as for the hydrofoil shift and all other variables). The maximal feasible value of 6° is set for the leeway angle to ensure that the trim setting for the equilibrium conditions found will not lead to stall of the hydrofoil. A faster sailing condition might exist for greater values of leeway angles but since the hydrofoil behaviour with regards to stall is not assessed, these sailing conditions are not considered for the performance assessment. The same conservative range limits are applied to significant hydrofoil shift values, which could trigger ventilation when boat “flies” too high. When a foil ventilates air gets drawn down into the water because of the low pressure on the suction surface, leading to a drastic loss of lift [5], which can result in the boat “falling” off the foils into the water causing a sudden deceleration. Therefore conservative and constant feasible ranges are chosen for the VPP model variables since ventilation and stall are not captured by the CFD models used. To model the performance and stability of the C-Cat more realistically in strong wind conditions, a wider operating range of the hydrofoils should be considered. This can be achieved by using CFD models capturing stall and ventilation. Another option is to generate additional simulation data with the current CFD model to extend the valid ranges and limit the feasible values by penalty functions. For instance, a penalty function on the maximal value of the hydrofoil lift coefficient as the maximal AoA tolerated without stalling is approached or a penalty function on the maximum lift coefficient as the hydrofoil tip gets close to the water surface. The pitch angle of the boat remains almost constant for all TWSs and both hydrofoil configurations. It varies in the case of sls03 in a range of less than 2°. This would mean that imposing a zero-pitch condition to the VPP could possibly enable the software to solve the moment balance around the transversal axis with a different parameter, such as the longitudinal position of the crew. This approach represents in a more realistic way the behaviour of the crew sailing on a small light catamaran.

4.3 Trimming the C-Cat

One important assessment made studying the findings for both configurations is that different hydrofoil settings can lead to a very similar performance for a given wind condition. The centreboard hydrofoil rake plays a more important part in the performance than the S-pos. The results show that the configurations with negative or zero degrees of foil rake do not enable the C-Cat to always reach the fully foiling regime nor the highest velocities, whereas the maximal lift fraction is obtained with almost all S-pos values in TWS of 6 and 9 m/s. The fact that S-pos is not monotonically increasing or decreasing with TWA or TWS shows that an optimal value has to be determined. This behaviour can be explained by the fact that the hydrofoil position does not vary much for a range of S-pos values between 20 and 30 cm due to its "double S" shape (as shown in Figure 4). Further investigations could be carried out to find a way to have this variable calculated as optimization variable while solving the equilibrium. TWS 3 m/s 6 m/s 9 m/s TWA S-pos Rake S-pos Rake S-pos Rake 40° 40 cm - 2° 25 cm 4°

20 cm 2°

50° 30 cm

0° 20 cm

2°

4° 60°

4°

70° 20 cm 25 cm 10 cm 2°

80° 30 cm

20 cm 20 cm 90°

10 cm

50 cm

0° 100° 110°

20 cm 20 cm

120°

130° 10 cm

Table 3. Optimized sls03 centreboard hydrofoil trim

Table 3 presents the hydrofoil settings corresponding to the maximal speed for the hydrofoil sls03. For TWS = 3 m/s, a rake of 4° leads to the best performance for a wide range of TWAs. In general the optimum rake reduces with increasing TWA since the faster water flow over the foil generates sufficient lift at smaller AoA, which is the combination of rake and pitch. Table 4 shows the optimal trim values for hydrofoil g2b, which shows similar trends.

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TWS 3 m/s 6 m/s 9 m/s TWA S-pos Rake S-pos Rake S-pos Rake 40°

20 cm 0°

30 cm

4°

0 cm 2° 50°

4°

50 cm 4°

60°

10 cm

40 cm

2° 70° 30 cm

40 cm 80°

40 cm 90° 100° 110° 20 cm 4° 120° 30 cm

0 cm 0° 130° 20 cm -2° 40 cm

Table 4. Optimized g2b centreboard hydrofoil trim

In light wind conditions, the g2b design cannot generate enough lift efficiently, as seen in Figure 11 by the small resulting lift fraction. Hence the optimal S-pos position between 10 and 20 cm is driven by minimising foil drag.

4.4 Optimizing the Rudder Hydrofoils Settings

The influence of the rudder rake as a function of the wind velocity is investigated separately to the hydrofoil trim, because it can only be adjusted on land when setting up the rudders. As mentioned in section 2, the crew's weight strongly influences the equilibrium of the C-Cat. Hence in this study the longitudinal crew position was taken as constant in order to isolate the influence of the rudder rake on the pitch equilibrium. This investigation is carried out with the faster hydrofoil configuration (i.e. sls03), with optimized trim. It appears that the best hydrofoil settings do not vary much with rudder rake, considering a constant wind speed. Positive rake (previously defined in section 2.2) aims at giving more vertical AoA to the rudder hydrofoil, and consequently at generating more upwards lift for a given pitch angle and boat speed. With growing wind force and hence boat velocity, the centreboard hydrofoil generates more lift, which creates more bow-up moment. If the rudders generate more lift as well and thereby a bow-down moment, pitch is reduced, meaning less vertical AoA and hence less lift for the main hydrofoil, which stabilises the C-Cat. Tests are carried out in a first step using the semi-empirical sail module. The following configurations are tested:

rudder rake in deg: [-2; 0; 1] TWS in m/s: [3; 6; 9]

From the results the optimal rudder rake is chosen for each true wind speed, and used as a starting point for further simulations using the CFD-based wing-sail model. Several rudder rake trims are calculated with the

wing-sail as well so that the results obtained can be cross-checked. Figure 12 presents the polar plots obtained for this investigation. For TWS = 3 m/s, the performance of the different foil rakes is very similar for many TWAs, but rake = 0° show the best performance for reaching conditions with TWAs between 100° and 130° and is hence selected. For TWA = 6 m/s rake = 0° shows the best performance for most TWA, especially downwind where the speed is significantly larger compared to the other settings. At small TWAs rake = -2° is fastest, but shows detrimental performance for larger TWAs. Hence rake = 0° is recommended. For TWS = 9 m/s, rake = 0° shows the best performance upwind and the maximum downwind velocity made good. This can be explained by the increase in bow down moment from the wing-sail at higher speeds, which would require less rudder rake. For reaching conditions with TWAs between 70° and 125° rake = 1° is superior. For up- and downwind races rake = 0° should be used, while for other race courses rake = 1° is recommended since the set-up provides a more stable and smooth performance over the entire TWA range requiring less centreboard hydrofoil trimming manoeuvres.

Figure 12. Influence of the rudder rake on speed depending on wind conditions

As far as negative rudder rakes are concerned for TWS = 9 m/s, calculated equilibrium conditions vary too much to be retained for analysis. In light wind conditions however, the assumption is made that giving the rudders negative rake enhances the performance. By generating downwards lift thereby creating more bow-up moment, the vertical AoA of the centreboard hydrofoil increases

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along with its lift. However, the analysis does not confirm this assumption. It appears in the polar plots of the C-Cat’s speed for TWS = 3 m/s, that setting a negative rudder rake up harms the performance given a constant crew longitudinal position. This can be explained by studying the lift fraction and the pitch of the C-Cat. Regardless of how much rudder rake is set up, in small wind conditions the moments around the transverse axis always balance in a zero pitch condition. This means a variation in rudder rake does not influence the AoA of the appendages enough to generate more lift. Zero rudder rake is therefore the best choice in the case of small wind conditions. Table 5 summarizes the results of this study.

TWS 3 m/s 6 m/s 9 m/s Rudder rake 0° 0° 1°

Table 5. Optimized rudder rake as a function of wind speed

5. CONCLUSIONS

Using the VPP model developed for the purpose of this work, realistic velocity predictions and performance assessments can be carried out. Records and observations made during the ICCC of 2013 confirm the predicted performance of the modelled C-Cat. With the scripted pre-processing of the CFD results, the performance potential of a hydrofoil design developed for the C-Cat can be quickly assessed, and further steps in the design process can be taken. Two hydrofoil designs were compared and the “double S” shape was shown to have better efficiency regarding lift production than the more conventional design with a "V" shape, making a significant difference especially in light wind conditions. Optimal trim settings for the different sailing conditions were obtained as sailing guidelines for the crew. The trim of the rudder rake has to be set up on the ground and remain constant while sailing. An investigation on the influence of the wind force on this setting was carried out. It appears that modifying this angle in a range of ±1° can make a significant difference in the performance. But since the rudder foils generate significant lift only with adequate velocity, the rudder rake has to be seen more as a means to stabilize the fast sailing C-Cat than a solution to enhance its performance at low speed. Conservative and constant feasible ranges have been chosen for the VPP model variables to ensure that stall and ventilation of the hydrofoils do not occur. In future work these parameter operating ranges can be widened by using CFD models capturing stall and ventilation. Alternatively, additional data can be generated with the current CFD model and feasible values can be limited by means of penalty functions. This way, more robust simulation results can be obtained and compared to a wider range of real conditions and the experience of the crew.

Further developments in the VPP model could enable analysis of the dynamic stability of the C-Cat [1]. The velocities and corresponding trim settings found by the VPP model correspond to stable sailing states, but the response of the C-Cat to small perturbations could be investigated. For example, the likeliness and consequences of stall and ventilation of the centreboard hydrofoil can then be assessed.

Acknowledgements

The authors like to thank Groupama Sailing Team for their cooperation, the project input and their interest in this work.

References

1. F. Furrer (2010), “Developing a Simulation Model of a Catamaran using the Concept of Hydrofoils”, ETH, Zurich, Switzerland

2. The International C-Class Championship 2013 Website (2013), http://www.theflyingboats.com [Online; last accessed September the 28th, 2014].

3. T.A. Bardenand & J.R. Binns (2012), “On the Road to Establishing Ventilation Probability for Moth Sailing Dinghies”, 18th Australasian Fluid Mechanics Conference, 3-7 December 2012, National Centre for Marine Engineering and Hydrodynamics Australian Maritime College, Hobart, Australia.

4. I.H. Abbott & A.E von Doenhoff (1959), Theory of Wing Sections: Including a Summary of Airfoil Data, Dover Publications, New York, United States of America.

5. F. Lefaudeux (1998), “New Advances in Sailing Hydrofoils”, RTO AVT Symposium on Fluid Dynamics Problems of Vehicles Operating near on the Air-Sea Interface, 5-8 October 1998, Amsterdam, The Netherlands.

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