System-based modelling of a foiling catamaran

Ocean Engineering 171 (2019) 108–119

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System-based modelling of a foiling catamaran

T

Boris Horel Ecole Centrale Nantes, LHEEA Res. Dept. (ECN and CNRS), Nantes, France

A R T I C LE I N FO

A B S T R A C T

Keywords: Foiling catamaran System-based modelling Maneuverability Stability Dynamical velocity prediction program (DVPP)

In the last decades, the use of foils on sailing yacht has highly increased. Whether they are mono or multihull, yachts are using foils to reduce their drag forces and then, to increase their speeds in a large range of wind and sea conditions. Several CFD-based studies have already been carried out in order to optimize the foil's shape and location on the hull, but feedbacks on the yacht's behavior is mainly given by the crew when sailing at sea. The aim of the present paper is to propose a complementary and faster approach that could help to predict and quantify the yacht behavior in calm water and in waves while sailing under foil's action. This approach is well known as a system-based modelling and is a mathematical method that leads to understand the complexity of a system from the study of its interactions in their entirety. The paper will present the ability of the system-based approach to predict the attitude of a catamaran while performing maneuvers such as turning circles with 35 degrees of rudder deflection and zigzag tests 10-10 and 20-20 for different initial Froude numbers and foil's shapes.

1. Introduction Whether they are mono or multihull, yachts such as IMOCA, Q23, Gunboat G4 or AC45 are now using foils in order to improve their performances. The use of foils in naval hydrodynamics is not a recent innovation since the first known foiling catamaran is the Catafoil of Robert Gilruth in 1938, the Amateur Yacht Research Society (1970). The major innovation lies in the capacity of the crew to adjust the foils' position and orientation in order to increase the yacht speed. In the last decades, this was made possible thanks to the development of hightechnologies. Until recently, hulls have been mainly classified in two types: displacement and planing. With the use of foils, it appears that a third mode of sailing needs to be identified: the foiling mode where the hull is no longer in contact with the water and where the yacht behavior is no longer influenced by the hydrodynamic loads on the hull but by the loads on the appendices. This new mode of sailing makes yacht faster but also more unstable. For the standard maneuvers of this kind of yacht, the International Maritime Organization (IMO) or the American Bureau of Shipping (ABS) have not yet established criteria. Then, the foiling mode or hydrodynamic flight has to be studied in order to evaluate the capacity of the current numerical tools to predict the yacht behavior under foil's action. Depending on the precision of the predictions, standards and recommendations could be established based on the definition of stability criteria for foiling yacht.

In this paper, the presented method is the so-called system-based method that leads to understand the complexity of a system from the study of its interactions in their entirety. The use of this low timeconsuming method to evaluate the stability of the foiling yachts is quite new and is a complementary solution to computational fluid dynamics (CFD) and sea trial testing. Numerical results will be presented for the following maneuvers: turning circles with 35 degrees of rudder deflection for several initial Froude numbers and two foil's shape and zigzag tests 20-20 for L-foil shape. These results are compared with the maneuvering features of the hull without foils to evaluate their effects on the dynamical stability of the yacht. In order to evaluate the ability of the method to take into account aerodynamic loads on a wing sail, the results of a zigzag test 10-10 on a foiling catamaran with L-foil is also presented. Finally, results of straight line course-keeping test in following waves while foiling in strong wind and under active control of the foil's rake angle is presented. 2. Mathematical model 2.1. Dynamical model The 6 degrees of freedom (DOF) dynamical model derives from the Newton's second law with the assumptions that the yacht is symmetrical in (xbObyb) and (ybObzb) planes and that the origin Ob of the yacht

E-mail address: [email protected]. https://doi.org/10.1016/j.oceaneng.2018.10.046 Received 6 September 2017; Received in revised form 21 August 2018; Accepted 24 October 2018 0029-8018/ © 2018 Elsevier Ltd. All rights reserved.

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Nomenclature CB G g h Ix Ixz Iy Iz KR m p q r

Sr Sw Sw0 T TD TWS U u v w δ ρ ρa η

Center of buoyancy (m) Center of gravity of the yacht hull (m) Gravity acceleration (m.s−2) Foil submergence (m) Roll moment of inertia (kg.m2) Roll and yaw product of inertia (kg.m2) Pitch moment of inertia (kg.m2) Yaw moment of inertia (kg.m2) Rudder gain constant Yacht mass (kg) Roll angular rate (rad.s−1) Pitch angular rate (rad.s−1) Yaw angular rate (rad.s−1)

complexity of the force modelling. As previously mentioned, the dynamical formulation of the 6DOF mathematical model comes from the general non-linear maneuverability equations and the total forces applied on the ship hull are written as the superposition of seven external forces: gravity (Grav), hydrostatic (HS), hydrodynamic interactions (HD), damping (Damp), control (Ctrl), propulsion (P) and waves (W) expressed at the center of gravity of the ship. Thus, a strong maneuverability model that takes into account the foils and wave effects is obtained.

frame is combined with the center of gravity G, i.e. xG=yG=zG=0. According to these assumptions, the behavior of the yacht can be predicted solving the following simplified equations of motion:

X = m [u˙ + qw − rv]

(1)

Y = m [v˙ + ru − pw]

(2)

Z = m [w˙ + pv − qu]

(3)

K = Ix p˙ − Izx (r˙ + pq) + (Iz − Iy ) qr

(4)

M = Iy q˙ + (Ix − Iz ) rp + Izx (p2 − r 2)

(5)

N = Iz r˙ − Izx (p˙ − qr ) + (Iy − Ix ) pq

(6)

Rudder area (m2) Wetted surface (m2) Static wetted surface (m2) Draft (m) Time constant for differential controller (s) True wind speed (m.s−1) Yacht speed (m.s−1) Surge velocity (m.s−1) Sway velocity (m.s−1) Heave velocity (m.s−1) Rudder deflection (deg) Density of water (kg.m−3) Density of air (kg.m−3) Wave elevation (m)

⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯→ ⎯⎯⎯⎯→ ⎧ FDamp ⎫ ⎧ FGrav ⎫ ⎧ FHS ⎫ ⎧ F ⎫ ⎧X K ⎫ + = ⎯⎯⎯⎯⎯⎯⎯→ + ⎯⎯⎯⎯⎯→ + ⎯⎯⎯⎯⎯⎯HD Y M ⎨Z N ⎬ ⎨ ⎬ ⎨ ⎬ ⎨ →⎬ ⎨ ⎯⎯⎯⎯⎯⎯⎯⎯→ ⎬ ⎩ ⎭G ⎩ MGrav ⎭G ⎩ MHS ⎭G ⎩ MHD ⎭G ⎩ MDamp ⎭G ⎯⎯⎯→ ⎯⎯⎯⎯⎯→ ⎯→ ⎯ ⎧ FW ⎫ ⎧ F ⎫ ⎧ FP ⎫ + + ⎯⎯⎯⎯⎯⎯Ctrl + ⎯⎯→⎬ ⎨ ⎯⎯⎯⎯→ ⎬ ⎨ M→ ⎬ ⎨ ⎯M Ctrl ⎭G ⎩ P ⎭G ⎩ MW ⎭G ⎩

In equations (1)–(6), X, Y, Z, K, M and N are respectively the total surge force, sway force, heave force, roll moment, pitch moment and yaw moment. The moments are expressed at the center of gravity G. The components of forces and moments are expressed in the yacht's reference frame as defined in Fig. 1. The roll, pitch and yaw angles are defined in Fig. 2. According to the definition of the orientation parameters, a vector expressed in the earth's reference frame can be expressed in the yacht's reference frame following equation (7).

(11)

2.2.2. Gravity forces The gravity force is modelled as a vertical force in the earth's reference frame and is expressed at the center of gravity of the yacht. The crew members are modelled as an external force due to a mass mcrew located at the center Gcrew. The influence of the crew on the total inertia of the yacht is taken into account using the Huygens theorem.

0 0 ⎞ xb x cos ψ − sin ψ 0 ⎞ cos θ 0 sin θ ⎛ 1 ⎛ ⎞ ⎛ ⎞ ⎛ 0⎞ ⎛ y0 = ⎜ sin ψ cos ψ 0 ⎟ 0 cos ϕ − sin ϕ ⎟ yb 0 1 0 ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ z0⎠ ⎝ 0 0 1 ⎠ ⎝− sin θ 0 cos θ ⎠ ⎝ 0 sin ϕ cos ϕ ⎠ ⎝ z b ⎠ (7)

⎯⎯⎯⎯⎯⎯→ → → g ⎧ FGrav ⎫ ⎧ ⎫ ⎧m. g ⎫ = + mcrew . ⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯ ⎯⎯⎯⎯⎯⎯ → → ⎨ M ⎬ ⎨ 0 ⎬ ⎨ GGcrew ∧ → g⎬ Grav ⎭G ⎩ ⎭G ⎩ ⎭G ⎩

Formula 8 to 10 give the expressions of the roll angular rate p, the pitch angular rate q and the yaw angular rate r depending on the attitude of the yacht.

2.2.3. Hydrostatic forces In the literature, two main methods are used to compute the hydrostatic forces in time: the linear method using a hydrostatic stiffness matrix and a nonlinear method that uses the integration of the hydrostatic pressure pHS on the wetted surface. In order to take into account the nonlinearities of the wave profile when the yacht is sailing in a seaway, a method of immersed surface capture developed by Horel et al. (2013) is used. Then, the hydrostatic component of equation (11) can be written as follow:

p = −sin θψ˙ + ϕ˙

(8)

q = sin ϕ cos θψ˙ + cos ϕθ˙

(9)

r = cos ϕ cos θψ˙ − sin ϕθ˙

(10)

Based on previous work from White (2007), the differential equations (1)–(6) are solved in time by integrating the acceleration vector using a fourth order Runge-kutta method. In some case, depending on the required rapidity/precision ratio and when the rapidity is preferred, an explicit Euler method can be used.

⎯⎯⎯⎯→ ⎯⎯⎯⎯→ ⎯⎯⎯⎯→ ⎧ ⎫ ⎧ FHS ⎫ FHS ⎧ FHS ⎫ = = ⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯→ ⎬ →⎬ → ⎬ ⎨ ⎯⎯⎯⎯⎯ ⎨ ⎨ ⎩ MHS ⎭G ⎩ 0 ⎭CB ⎩ GCB ∧ FHS ⎭G with:

2.2. Force model 2.2.1. General formulation Based on previous work from Horel (2016), the system-based method is used as part of a dynamical velocity prediction program (DVPP). The accuracy of the prediction is highly dependent on the

Fig. 1. Coordinate systems: earth (b0), wave (bw) and yacht (bb). 109

(12)

(13)

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Fig. 4. Type of foils, a) L-foil, b) V-foil.

Fig. 2. Yaw angle ψ, pitch angle θ and roll angle φ

⎯⎯⎯⎯→ FHS = −

Table 1 Gains of the controllers.

⎯⎯⎯→

∫ ∫ pHS (z ) d Sw Sw

(14)

pHS (z ) = −ρgz

(15)

Since a yacht is used to experience large amplitude motions, the choice of a numerical method is made in order to take into account nonlinear effects such as large roll angles in the calculation of hydrostatic forces. However, the linear and nonlinear hydrostatic models were implemented in order to reduce the CPU time when the yacht experiences motions with low amplitude variations. In the DVPP, the hull geometry is defined from a STL file. The main advantage of such a file format is to be a mesh format. It means that the hull is made of a finite number of facets Nf and the hydrostatic pressure is known at the center Gf of each facet. Then, the pressure can be integrated on the wetted surface in a discrete manner.

⎯⎯⎯⎯→ N ⎯⎯⎯⎯→ ⎧ ⎫ ∑1 f d FHS ⎧ FHS ⎫ = →⎬ ⎯⎯⎯⎯→ ⎬ Nf ⎯⎯⎯⎯⎯→ ⎨ ⎯⎯⎯⎯⎯ ⎨ ⎩ MHS ⎭G ⎩∑1 GGf ∧ d FHS ⎭G

Gain

Values

Gain

Values

Gain

Values

Kr TD

4 rad/rad 0.005 s

Kf TDf

2 rad/m

2.5 × 10−4 s

KTf TDTf

2.5 × 10−3 s

4 rad/rad

with:

⎯⎯⎯⎯→ ⎯n d FHS = −pHS (z G ). dSw. ⎯→ f

(17)

2.2.4. Damping forces The damping forces need to be modelled in order to take into account the energy dissipation between the yacht and the fluid. According to ITTC (2011), for the roll motion, these forces are mainly due to an eddy making component, a frictional component and a wave-making component. When results from experimental or CFD decay tests are available, an analysis of the maxima of the decay curve can be conducted to identify the linear B1 and nonlinear B2 coefficients. The resulting damping model in roll is a so-called linear + quadratic model.

(16)

a)

b)

c)

d)

Fig. 3. a) Surge force XHD against speed u, b) Surge force XHD against drift angle β for Fn = 0.4, c) Sway force YHD against drift angle β for Fn = 0.4, d) Yaw moment NHD against drift angle β for Fn = 0.4. 110

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Fig. 5. Dynamic effects on wind triangle (Fossati et al. (2010)).

Fig. 8. Mean roll angle.

Table 2 Yacht features. Features

Values

Length overall, LOA Width, B Draft of the bare hull, T Yacht mass, m Crew mass, mcrew Wing mass, mwing Longitudinal position of the center of gravity of the hull, xCoG Roll moment of inertia, IGx Pitch moment of inertia, IGy Yaw moment of inertia, IGz L-foil Tip's chord Tip's span V-foil Chord Span

13.45 m 8.47 m 0.22 m 1400 kg 525 kg 445 kg 6.47 m 39.7 × 103 kg m2 27.7 × 103 kg m2 66.9 × 103 kg m2 0.2 m 1.6 m 0.37 m 2.4 m

Fig. 9. Transversal stability modes.

When heave and pitch data are also available, the damping model can be expressed as follows:

⎯⎯⎯⎯⎯⎯⎯→ B1 ϕ˙ + B2 ϕ˙ |ϕ˙ |⎫ ⎧ 0 FDamp ⎫ 0 MDamp = → ⎨ ⎯⎯⎯⎯⎯⎯⎯⎯ ⎬ ⎨ MDamp ⎬ 0 Z ⎩ ⎭G ⎩ Damp ⎯⎯x ,⎯→ ⎯ ⎯→ ⎯⎯ ⎭G,⎯→ b yb , z b ⎧

(18)

2.2.5. Hydrodynamic interaction forces In the system-based approach, it is assumed that the free surface is not perturbed by the yacht's hull. This assumption allows the mathematical model to be simplified and faster to implement. But this assumption also involves that some physics are missing. Then, in order to improve the accuracy of the simulations, a fluid/hull interaction model is used. This model is a simplified 3DOF maneuvering model in surge, sway and yaw whose coefficients are identified from captive model tests in calm water. In some cases, CFD can also be used to identify the hydrodynamic derivatives. The surge force, sway force and yaw moment are expressed as Taylor series expansions and their expressions are given in equations (19)–(21).

Fig. 6. Advance (AD) and tactical diameter (TD) plotted against speed.

XHD = Xu u +

YHD = Yv v + NHD = Nv v +

1 1 Xuu u2 + Xvv v 2 2 2

1 Yvvv v 3 6 1 Nvvv v 3 6

(19) (20) (21)

This 3DOF model is used since the transition phase between displacement hull and foiling hull mainly happens while performing straight line sailing. The linear coefficients Xu , Yv , Nv and nonlinear coefficients Xuu , Xvv , Yvvv , Nvvv are identified from experiments. In first

Fig. 7. Relative elevation of the yacht. 111

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Table 3 Phase plane of the roll motion. a) bare hull, b) L-foils, c) V-foils. a)

b)

(continued on next page)

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Table 3 (continued) c)

Table 4 Summary of the outcomes (++ very efficient, + efficient, - not efficient).

Bare hull L-foil V-foil

Maneuverability

Stability

– ++ +

+ – –

approximation, since the main objective of this work is to show the ability of the system-based method to model the behavior of the ship under foil's action, experimental results from the Bulgarian Ship Hydrodynamics Center (BSHC) presented in Broglia et al. (2015) on Delft 372 Catamaran were used. The hydrodynamic derivatives of a demi-hull are identified by applying a 2nd order polynomial regression to the experimental values presented in Fig. 3. The transition between displacement mode and foiling mode is taken into account by assuming that the hydrodynamic interaction forces can be weighted by the ratio between the static wetted surface Sw0 and the instantaneous wetted surface Sw. This assumption is inspired by previous work from ITTC75 recommendations, ITTC (2008) and Huetz et al. (2013). Then, the components of the hydrodynamic

Fig. 10. Time series of the rudder deflection δ and the yaw angle ψ

forces can be expressed as follows:

⎯⎯⎯⎯→ XHD 0 ⎫ ⎧ FHD ⎫ Sw ⎧ Y 0 = . HD →⎬ ⎨ ⎯⎯⎯⎯⎯⎯ ⎬ Sw 0 ⎨ ⎯x ,⎯→ ⎯ ⎯→ ⎯ ⎩ MHD ⎭G ⎩ 0 NHD ⎭G,⎯→ 1 y1 , z1

(22)

It has to be noticed that the expression 22 is only applicable to the 113

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hull is equipped with two main foils located close to the center of gravity of the yacht. The L-foil is assumed to be composed by a vertical part, the shaft and a horizontal part, the tip. The V-foil is only made of a unique part. The force model of the foil is established with the same potential theory than for the rudder forces. It means that knowing the lift and drag coefficients (CL, CD) associated with the foil's shape, the effects of the L-foils on the yacht behavior can be modelled with the following expression:

⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯→ ⎯⎯⎯→ ⎧ Fshaft ⎫ ⎧ Ffoil ⎫ ⎧ Ftip ⎫ + = → → → ⎨ ⎯⎯⎯⎯⎯⎯⎯ ⎨ ⎯⎯⎯⎯⎯ ⎨ ⎯⎯⎯⎯⎯ M ⎬ Mshaft ⎬ M ⎬ ⎭G ⎩ tip ⎭G ⎩ foil ⎭G ⎩

Fig. 11. Phase planes of the relative elevation of the CoG compared to the heading of the yacht.

This formulation is a quasi-static two-dimensional formulation, where the effect of the flow in the spanwise direction is not taken into account. The foil is considered with a discrete geometry and each element has its own reference frame. This gives the opportunity to study complex foil geometry and also to consider some future developments on fluid/structure interactions.

bare hull without appendages. The models used in the modelling of the effect of the appendages (rudder, foils, daggerboards …) are presented in the next sections. 2.2.6. Control forces The control forces tend to modify the yacht orientation and heading while performing maneuvers. Two kinds of modelling are applicable. If experimental data are available, then, an Abkowitz (1964) formulation can be used. In this study, inspired by Yoshimura (2005) work, an MMG based formulation is used. Such a model is defined from the potential theory by taking into account the interactions between the hull and the appendix. In first approximation, the shapes of the rudder and the foils are symmetrical NACA profiles.

2.2.6.3. Free surface effects. When performing steady sailing, the effects of the free surface on the foils are first evaluated for a 2D flow. Faltinsen (2005) proposed to define a submergence Froude number Fnh as follows:

Fnh =

L=

1 ρ 2

1 ρ 2

∫ Vr2 CD dSr

(23)

∫ Vr2 CL dSr

(24)

U gh

(27)

The definition of such a Froude number allows to express the lift coefficient CL as a function of the non-dimensional submergence h/c, where c is the chord of the profile.

2.2.6.1. Rudder. The rudder forces are modelled as external forces that ⎯z and the angle of attack α . depend on the rudder deflection δ around ⎯→ b r The formulations of the drag force D and lift force L derives from the potential theory and take into account the relative resulting velocity Vr experienced by the rudder.

D=

(26)

1

h

c

⎧ CL ( c = ∞)[1 + 16 ( h )2]when Fnh → 0 ⎪ h 1 h CL ( ) = 1 + ( )2 ⎨CL ( h = ∞)[ 16 c ]when Fnh > 10/ h/ c c 1 h c ⎪ 2 + ( )2 16 c ⎩

(28)

Thus, after having taken into account the previous-mentioned considerations, the total control forces can be expressed as the sum of the rudder and foil forces.

⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯→ ⎧ Ffoils ⎫ ⎧ Frudder ⎫ ⎧ FCtrl ⎫ = + →⎬ ⎨ ⎯⎯⎯⎯⎯⎯ ⎨ ⎯⎯⎯⎯⎯⎯⎯⎯⎯→ ⎬ ⎨ ⎯⎯⎯⎯⎯⎯→ ⎬ ⎩ MCtrl ⎭G ⎩ Mrudder ⎭G ⎩ Mfoils ⎭G

⎧⎛ cos δ 0 −sin δ ⎞ ⎛ cos αr 0 sin αr ⎞ ⎛ D ⎞⎫ ⎯⎯⎯⎯⎯⎯⎯⎯→ ⎪ 0 1 0 ⎟⎜ 0 1 0 ⎟ ⎜ L ⎟⎪ ⎧ Frudder ⎫ ⎜ → ⎬ = ⎨⎝ sin δ 0 cos δ ⎠ ⎝−sin αr 0 cos αr ⎠ ⎝ 0 ⎠⎬ ⎨ ⎯⎯⎯⎯⎯⎯⎯⎯⎯ ⎩ Mrudder ⎭G ⎪ ⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯→ ⎪ Or G ∧ FCtrl ⎯⎯x ,⎯→ ⎯ ⎯→ ⎯⎯ ⎩ ⎭G,⎯→ b yb , z b

(29)

2.2.6.4. Autopilot models. In order to perform course-keeping tests, a differential controller is used to control the rudder deflection. The simple expression of such an autopilot is given by equation (30) where ψc is the desired course.

(25) When the rudder is equipped with a T-foil, a vertical lifting component is added in equation (25).

δ = −Kr (ψ − ψc ) − Kr TD r

2.2.6.2. Foils. In this paper, two kinds of foils are studied: the L-foil and the V-foil with symmetrical NACA profiles. As presented in Fig. 4, the

(30)

The rake angles of the foils are also controlled using a similar

a)

b)

Fig. 12. Time series of rudder deflection δ and yaw angle ψ. a) L-foil without control, b) L-foil with control. 114

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Wind

W

av

es

Fig. 13. Catamaran with L-foil in regular waves.

differential controller whose inputs are the yacht elevation z and the vertical velocity w. In equation (31), zc is the desired elevation of the center of gravity above the free surface of the water.

u ⎛ cos α w sin α w 0 ⎞ ⎛ ⎯⎯⎯⎯⎯⎯→ ⎯⎯⎯⎯⎯⎯⎯→ AWS = ⎜− sin α w cos α w 0 ⎟ TWS − ⎜⎛ v ⎟⎞ − GOsail ∧ ⎜ w ⎝ ⎠ 0 0 1⎠ ⎝ ⎝

αfoil = −Kf (z − z c ) − Kf TDf w

Based on previous work from Harris (1980), Flay and Vuletich (1995) expressed the true wind speed TWS profile with a log-law formulation as a function of the height z, the friction velocity U∗ and the roughness length z0.

(31)

When the rudders are equipped with T-foils, the dynamical control of their deflection αTfoil is made through equation (32) whose inputs are the pitch angle θ and the pitch rate q. The value of the desired trim θc is manually adjusted by the crew while sailing.

αTfoil = −KTf (θ − θc ) − KTf TDTf q

TWS 1 z = ln ( ) z0 U∗ k

(32)

p ⎛ ⎞⎞ ⎜q ⎟⎟ ⎝ r ⎠⎠

(33)

(34)

with:

For each controller, the gains are adjusted from a bench of semicaptive simulations. Then, the gains are chosen according to optimal criteria for speed, precision and stability. For the particular zigzag 1010 test case presented in this paper, the values of the gains are given in Table 1. It should be reminded that the values given in Table 1 might be different for different foil geometries and can have a significant effect on the outcomes of the maneuvers.

k = 0.4

(35)

z 0 = 10exp(−0.4/ K )

(36)

K = 10−4 (7.5 + 0.67U10)

(37)

In the 6DOF model presented in this paper, Asail is the sail area, CDsail is the drag coefficient and CLsail is the lift coefficient defined from lifting line calculations. As mentioned by Heppel (2015), such a simple model is useful for gaining insight, and is adequate for the sail plan or a submerged T-foil but is too simple to represent the nonlinear behavior of a surface-piercing foil.

2.2.7. Propulsive forces In first approximation, a simple model for sail forces is implemented. Since the formulation of the propulsive force is a function of the true wind conditions (speed and angle) and also a function of the lift and drag coefficients of the sail plan, it can be assumed that the yacht is actually sail-powered. When expressing these forces at the center of gravity of the yacht, the heeling, yaw and pitch moments due to the sails can be known. The simplified model is inspired by previous works from Otto Scherer (1974) where the resulting sail forces and moments are expressed as functions of the height of center of effort Osail. As shown in Fig. 5, further article from Fossati and Muggiasca (2010) mentioned the use of a “dynamic resultant wind speed Vris” and a “dynamic apparent wind angle βdin” evaluated at the center of effort. Then, the lift and drag coefficients of the sails are expressed as functions of these dynamic parameters. For the C-class or AC45 yachts, the total propulsive force is generated via a wing sail. Knowing the wing deflection αw at each time step, the dynamic apparent wind speed denoted AWS is computed as follows:

⎯→ ⎯ ⎧ CDsail 0 ⎫ ⎧ FP ⎫ 1 = ρa AWS 2Asail CLsail 0 ⎯ ⎯⎯ → ⎨ ⎬ ⎨M ⎬ 2 ⎩ P ⎭Osail 0 ⎭Osail ⎩ 0

(38)

The control of the wing deflection is done via a proportional controller whose inputs are the yacht speed and roll angle. Also, the accuracy of the predictions could be improved by adding some damping and added mass contributions of the rig. 2.2.8. Wave forces When the yacht is sailing in a seaway, the incident waves are modelled using a 2nd order Stokes theory. The forces are computed from the Froude-Krylov assumption where the yacht does not affect the velocity field of the flow. Also, the flow is assumed to be irrotational and incompressible and the pressure is expressed as a function of the incident wave potential Φi . 115

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3. Applications 3.1. Yacht Considering the catamaran is wide enough for the two hulls not to interact together and using the Kelvin theory to compute the wake angle, it can be assumed that the wave field generated at the bow of the starboard hull disturbs the wave field of the port hull beyond the aft of the yacht. Then, a hull geometry close to the AC45 geometry has been chosen. Its features are mainly chosen according to the Class AC Test features of Groupama Team France and are given in Table 2. 3.2. Test cases Using the previous mentioned mathematical model, numerical tests in calm water and in waves are performed on a catamaran. The chosen test cases are based on conventional ship maneuvering requirements of the IMO. For the turning circle maneuver, the ability of the yacht to tack with a constant rudder deflection can be evaluated from the values of the advance and the tactical diameter. By performing zigzag tests, the abilities of the yacht to change its course and to control the yaw motion are evaluated. The ability to change the course is evaluated from the response of the yacht after a moderate rudder deflection while the ability to control the yaw motion is evaluated from the analysis of the response of the yacht after a rudder deflection in the opposite direction. These maneuvers are usually performed for conventional ships in order to evaluate their maneuvering abilities but for a foiling yacht, when the standard maneuvers are carried out in 6DOF, they also provide information about coupling effects between maneuverability and heave, roll and pitch stability. First, turning circle tests in calm water are performed for several initial Froude numbers from 0.1 to 0.85 respectively 2.2 and 19 knots. These tests gave quantitative values of the maneuvering abilities of the catamaran without foils, with L-foils and with V-foils as described in Fig. 6. Zigzag tests 20-20 for L-foil shape at 15 knots are also numerically performed in order to evaluate the dynamical response of the yacht by analyzing the values of the first and second overshoot of the yaw motion. The effects of the aerodynamic loads on the wing sail are investigated through the results of a zigzag test 10-10 on the catamaran with L-foil while sailing in a close reach. For the same wind and foil conditions, a straight line course-keeping test in following waves while sailing under active control of the rake angle is carried out. The wavelength to yacht length ratio is equal to 1 and the wave height to wavelength ratio is equal to 0.04. As first configuration for the simulations, and even if it is not the case in reality when sailing on a tack, both foils (port and starboard) are used at the same time. For all the cases, the yacht is free in all six degrees of freedom: heave is free and controlled by the foil rake autopilot; pitch is free and controlled by the rudder rake autopilot; yaw is free and controlled by the rudder angle autopilot; roll is free and controlled by the wing sail autopilot; surge and sway are free and ship velocity is prescribed by the test.

a)

b)

c) Fig. 14. Phase planes: a) roll, b) pitch, c) yaw.

⎯⎯⎯⎯→ N ⎯⎯⎯→ ⎧ ⎫ ∑1 f d FFK ⎧ FW ⎫ = ⎯ ⎯⎯⎯⎯ → ⎯ ⎯⎯⎯ → ⎯ ⎯⎯⎯ → N ⎨ M ⎬ ⎨ f ⎬ ⎩ W ⎭G ⎩∑1 GGf ∧ d FFK ⎭G

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with:

⎯⎯⎯⎯→ ∂Φ 1 → ⎯n d FFK = −ρ [− i − ( ∇ Φi )2]. dSw. ⎯→ f ∂t 2

3.3.1. Turning circle After a stationary phase where the yaw rate of the yacht is equal to zero, the rudder deflection is increased up to 35°. Advance (AD) and tactical diameter (TD) are read and the course stability of the yacht is evaluated through the comparison of these maneuvering parameters with the IMO standard values. The results are presented in Fig. 6. It can be noticed that up to 5 m s−1 the use of L-foils seems to improve the maneuvering abilities of the yacht. Indeed, the values of the

(40)

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B

C

A

a)

b)

Fig. 15. Experimental apparatus: a) T-rudder details, b) semi-captive model test at low speed (A: T-rudder, B: joint for pitch, C: 6DOF dynamometer).

The transversal stability of the yacht is first evaluated from the mean values of the roll angle during the stationary phase of the maneuver. In Fig. 8, it can be noticed that with or without foils, the catamaran remains stable during the maneuver and the amplitude of the roll angle does not exceed 6°. The dynamical transversal stability is evaluated from the phase plane of the roll motion given in Fig. 9. Depending on the shape of the attractors, the yacht motion can be defined as stable, or unstable. The phase plane also helps to define criteria and boundaries of dynamical stability. When the attractor is a stable focus, Table 3a, the yacht is stationary stable. When the attractor is a stable limit-cycle, second figure of Table 3. b, then, the motion is stable but is no longer stationary. The third mode that can be identified is the unstable mode, where the attractor is an unstable limit-cycle where the motion is chaotic. This phenomenon can be identified from the second figure of Table 3. c. The modes of stability that have been encountered during the maneuver are shown in Fig. 8. Finally, for the turning circle maneuver, the effects of the foils can be summarized as a function of their impact on the stability and the maneuverability of the yacht. However, in Table 4, the outcomes of the maneuver highly depend on the gains of the dynamic controller of the foils and can significantly change for different gains. Based on systembased models, further studies could be dedicated to the choice of optimal controller parameters for yacht with high maneuvering and stability abilities. Even if the foils seem to have a negative effect on the transversal stability of the yacht at high speed, the mean values of the roll angle and their fluctuations still remain acceptable. For this kind of maneuvers, the advantage of the foils seems to be the improvement of the maneuvering abilities of the yacht. However, for low speeds, the foil's shape seems to have a significant effect on the advance and the tactical diameter.

Fig. 16. Pitch evolution of the catamaran under T-rudder action. (EXP: measured values, NUM: numerical values).

advances and tactical diameters are below the values of the bare hull and remain below the maximal values of the IMO requirements. For the loading conditions defined in Table 2, beyond 5 m s−1 the V-foils seem to improve the maneuvering abilities of the yacht better than the Lfoils. This can be partly explained from Fig. 7, where it can be noticed that the dynamical controller increases the yacht elevation for the V-foil configuration. In this paper, the geometry of the foils is a self-stable configuration. For the turning circle maneuver, the targeted elevation of the hull is the maximum allowed elevation and the dynamic controller of the foil tends to make the yacht sail as high as possible from the free surface. Nevertheless, the L-foils tend to improve the maneuvering abilities of the yacht for all the range of tested speeds. When the drift of the yacht increases during the maneuver, the shaft of the foil acts like a daggerboard and tends to create a sway force pointed towards the trajectory that reduces the tactical diameter. Fig. 7 compares the relative elevation of the hull for several speeds and foil's configurations with the foiling boundary defined as the static draft T of the bare hull. Depending on the foil's area, it can be noticed that the range of speeds where the V-foils are the most efficient seems to be between 4 and 7 m s−1, i.e. 8 and 14 knots, while the range of the Lfoils is between 6 and 8 m s−1. The mode of sailing can also be identified from this figure. When the elevation is below the foiling boundary, then, the yacht is sailing in a displacement mode. Conversely, if the elevation is above this boundary, then, the yacht is sailing in a foiling mode.

3.3.2. Zigzag 20-20 with L-foil shape at 15 knots The zigzag tests 20-20 are performed in order to evaluate the dynamical response of the yacht. The features of the zigzag test under active control of the foil's rake angle are compared with the results of the bare hull. In both cases, the catamaran is equipped with twin rudders and the surge velocity is constant and is equal to 15 knots. After a stationary phase where the yaw rate of the yacht is equal to zero, the rudder deflection is set to 20° on starboard. When the deviation of the yaw angle reaches the value of 20°, then, the rudder deflection is set to 20° on port. Only the first overshoot is taken into 117

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account. From Fig. 10, it comes that the value of the first overshoot of the bare hull is equal to 0.5° and 4.7° for the catamaran with L-foil. Even if the overshoot is higher for the L-foil, the time to reach the desired course is lower. For the zigzag 20-20 maneuver, the dynamic behavior in heave is given in Fig. 11. This relative elevation of the center of gravity CoG compared to the initial stationary elevation is highly coupled with the roll motion. Fig. 11 shows that when foiling above the water, the heave motion follows a stable limit-cycle that makes the behavior of the foiling yacht more unstable than the one of the bare hull. This can be partly explained since in foiling mode, the hull is no longer in the water, meaning that the behavior of the yacht is only governed by the external forces applied on the appendices. Thus, since the hydrodynamic loads on the hull no longer affect the behavior of the yacht, a small change of the deflection of an appendix can lead to a big change on the behavior.

In order to setup methodologies to validate the DVPP, the results of the previous mentioned mathematical model were compared to the experimental values. The same loading conditions of the catamaran were used between simulations and experiments. Fig. 16 shows the evolution of the induced pitch angle of the catamaran compared to the forward speed for three T-rudder deflections, αT-rudder: 0, 1.5 and 3°. It can be noticed that a good agreement is found between the numerical predictions of the mathematical model and the experiments. Indeed, the pitch angle evolves in a linear + quadratic manner when the speed is increasing. This kind of evolution can be partly explained since the drag and lift forces are increasing with the square of the velocity and creates a quadratic induced pitch moment. 5. Conclusions The results of this study show that the system-based approach is a strong mathematical method that takes into account a large range of existing aero and hydrodynamic models in order to predict the behavior of the yacht. Depending on the test case to study, the assumptions that are made make the calculations quicker. That is the reason why this kind of modelling is used in the DVPP. The presented methodology and the numerical tool that have been developed at the LHEEA lab. can be applied to every kind of yachts from mono to multihulls. However, such a modelling requires experimental data or CFD calculation in order to identify the coefficients of its models and to validate the predictions. In the design stage, the application of such a system-based modelling associated with the experience of the aircraft industry, Bunge (2015), which has over a hundred years of history of flight dynamics could help to established stability criteria and to evaluate the stability boundaries of foiling yachts.

3.3.3. Zigzag 10-10 with L-foil shape and wing sail The aerodynamic loads on the wing are investigated through the results of a zigzag test 10-10. The wing sail deflection is adjusted in time using a proportional controller. The target surge velocity is 20 knots in 30 knots of wind. The true wind angle is 60° and the yacht is sailing on a starboard tack. For the zigzag 10-10, the values of the first and second overshoots are taken into account. From Fig. 12, it can be noticed that the overshoots are higher when sailing under active control of the rake angle. The overshoots are also higher when the yacht is going downwind. For L-foil with active control of the rake angle, the downwind overshoot reaches a maximum value of 2°. 3.3.4. Straight line with L-foil shape, wind and waves The aim of this sub-section is to evaluate the ability of the systembased method to predict the yacht behavior in a seaway. The true wind angle is still 60° and the yacht is sailing on a starboard tack with a forward speed of 20 knots. The course keeping is made using the differential controller presented in paragraph 2.2.6.4. As can be seen in Fig. 13, the waves are regular quartering waves whose wavelength to yacht length ratio is equal to 1 and whose wave height to wavelength ratio is equal to 0.04. In these conditions, the stability of the yacht motion is evaluated through the three phase planes given in Fig. 14. From Fig. 14. b it can be noticed that while sailing in waves, the foils act like dampers that reduce the amplitude of the pitch motion. In this case, the amplitude of the pitch motion of the L-foil catamaran without control, i.e. zero rake angle, is almost twice the amplitude of the catamaran with active control of the rake angle. However, the amplitudes of the roll and yaw motions seem to be higher when the catamaran is sailing with active control of the foil's rake angle. In quartering waves, the instability in yaw observed for the L-foil catamaran in foiling conditions is mainly due to the phase shift between the encounter of the wave on the starboard foil and on the port foil. In foiling mode, the anti-drift plan of the hull is reduced to the wetted surface of the foils and the controller is no longer efficient to keep the initial course.

6. Perspectives In order to validate the predictions of the mathematical model, steady and unsteady semi-captive model tests should be performed on a foiling catamaran in the towing tank of the LHEEA laboratory (ECN/ CNRS). Further studies could also be carried out to improve the accuracy of the simulations and better take into account the interactions and the complex phenomena such as the ventilation. Furthermore, heave and pitch response could also be looked at in straight line sailing, or in the presences of time-varying sail forces, in order to simulate wind speed variations. New tests could also be chosen or developed specifically targeted to the requirements of foiling catamarans. Acknowledgments Investigations on a T-rudder were performed using a model that was built and provided by the students of the Centrale Nantes HydroProject. Experiments were carried out in the towing tank of the LHEEA lab. (ECN/CNRS) thanks to S. Bourdier, A. Levesque and the technical staff of the laboratory. Thanks to French sailor Benoît Marie for his comments on this work. References

4. Experimental investigations on a T-rudder Abkowitz, M.A., 1964. “Lectures on Ship Hydrodynamics - Steering and Maneuvering”, Hydro- and Aerodynamics Laboratory Report Hy-5. Broglia, R., Zaghi, S., Campana, E.F., Visonneau, M., Queutey, P., Dogan, T., SadatHosseini, H., Stern, F., Milanov, E., 2015. CFD validation for DELFT 372 catamaran in static drift conditions, including onset and progression analysis. In: 5th World Maritime Technology Conference - WMTC15, Rhode Island, United States. Bunge, R.A., 2015. Aircraft Flight Dynamics. Stanford University. Faltinsen, O.M., 2005. Hydrodynamics of High-speed Marine Vehicles. Norwegian University of Science and Technology, pp. 197–199. Flay, R.G.J., Vuletich, I.J., 1995. Development of a wind tunnel test facility for yacht aerodynamic studies. J. Wind Eng. 231–258. Fossati, F., Muggiasca, S., 2010. Numerical modelling of sail aerodynamic behavior in

Experimental investigations were performed in the 148 m long towing tank of the LHEEA laboratory (ECN/CNRS). The influence of the T-rudder on the behavior of the catamaran was studied by performing stationary straight line tests in calm water for several forward speeds and T-rudder deflection up to 3°. The model was free in heave and pitch and a 6DOF dynamometer was used to measure the resistance force in surge direction. The time evolutions of the heave and pitch motions were recorded by respectively using a vertical laser and a potentiometer mounted on the joint for pitch as described in Fig. 15a and 15. b. 118

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Towing Tank Conference - Recommended Procedures and Guidelines. ITTC 7.5-02-07-04.5, 2011. Numerical estimation of roll damping. In: International Towing Tank Conference - Recommended Procedures. Otto Scherer, J., 1974. Aerodynamics of high-performance wing sails. In: Chesapeake Sailing Yacht Symposium. The Amateur Yacht Research Society, 1970. Sailing hydrofoils. Hermitage, Newbury, Berkshire, England. White, A.S., Gleeson, P.T., Karamanoglu, M., 2007, “Control of ship capsize in stern quartering sea”, Int. J. Simulat. Syst. Sci. Technol., 8(2) pp. 20-31. ISSN 1473-804X. Yoshimura, Y., 2005. Mathematical model for manoeuvring ship motion (MMG model). In: Workshop on Mathematical Models for Operations Involving Ship-ship Interaction.

dynamic conditions. In: Proceedings of the Second International Conference on Innovation in High Performance Sailing Yachts. Heppel, P., 2015. Flight dynamics of sailing foilers. In: 5th High Performance Yacht Design Conference, Auckland, New-zealand. Horel, B., 2016, “reportPhysical Modelling of Ship's Behaviour in Astern Seas”, (PhD dissertation). Horel, B., Guillerm, P.E., Rousset, J.M., Alessandrini, B., 2013. A method of immersed surface capture for broaching application. In: Proceedings of the ASME 2013 32nd International Conference on Ocean. Offshore and Arctic Engineering. Huetz, L., Guillerm, P.E., 2013. Database building and statistical methods to predict sailing yachts hydrodynamics. In: Proceedings of the Third International Conference on Innovation in High Performance Sailing Yachts. ITTC 7.5-02-02-01, 2008. Testing and data analysis resistance test. In: International

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