Sailing boat performance prediction

Sailing boat performance prediction Roux et al June 22, 2004

1

Introduction

The mathematical prediction of sailing boat velocities probably traces back to the middle of the nineteenth century when the first modern races were organised. The desire to compare the crew performances rather than that of very different boats yields the necessity for a way to estimate these performances on an objective basis. The primitive formula used were based on the simple, but reasonable idea that the velocity of a given boat could be correctly approximated as an algebraic combination of favourable factors such as length or sails area, and less favourable ones as width or weight. They were rapidly improved Leechter (1974) by including less obvious measurements as for example the depth at particular points of the boats, yielding such well known rating formula as the 12 meter America’s cupper or the IOR (International Offshore Rule) in more recent years. All these formula were fully determined so that the designer could work out his boat in order to optimise the velocity by combining factors which were not explicitly accounted for with others which were assumed more favourable than the weight they account for in the formula. The game was to obtain a boat which was apparently slow for the rule and actually fast. A very important step forward Kerwin (1978) was accomplished by substituting to this a priori known formula a rating which was obtained as the result of a statistical analysis based on the comparison of a large set of boat representative dimensions with their actual measured performances. It becomes therefore more and more difficult to fool the rule and this successful model is still in application today as the IMS rule (International Measurement System http://www.orc.org/imsrule.htm). Although it is much more efficient than any previous one, it is clear that the basis of such a model is almost the same Schlageter and Teeters (1999) : a mathematical formula derived from as much experiments as possible, just the computer being partially substituted to men for the synthesis of the large amount of informations collected. An other result of this new rule could be the renewal of the definition of the wining boat. If it is assumed that the boat maximum velocity is correctly predicted by the rules whatever its design could be, one very important issue should be its capability to reach this maximum velocity easily in various condition. That can makes the design problem very complex and bring the need for a new generation of design tools which can be nothing but CFD based. During the last forty years, the introduction of computers and numerical methods in ship hydrodynamics allowed for the development of a new class of models which were based on the solution of the fluid mechanics equation rather than the analysis of the performances of existing boats. This was the way to the building of actual predictive models and some of these models are today 1

currently used as an aid to ship design. However, almost all these models have been derived for large ships with mechanical propulsion for which the velocity is part of the design problem rather than a result. This is clearly unsuited to the case of a sailing yacht for which the velocity is the result of the combination of complex factors such as the wind or the sea state. Even in calm water, the prediction of sail boat velocities necessitates the balance of aero and hydrodynamic forces, that is the matching of hydro and aerodynamic models Larsson (1990). In this paper, a first step in this direction will be presented. In this work, it was considered that for calm water, the boat velocity remains almost constant so that the variation around the equilibrium point remains small. In contrast, small heel angle will result in a large variation at the top of the mast so that a non-linear model was necessary for the sail power computation. Although a direct coupling was possible, an intermediate modelling was used to synthesize the hydrodynamics data. This was made possible by only considering a reduced set of data and unknowns : the forward velocity, the leeway and the heel angle for the data, the two horizontal components of the hydrodynamic forces and the momentum with respect to the boat axis for the unknowns. Therefore, equilibrium points were constrained to remain on a subset which was parameterized by a finite number of computed points prior to the aerodynamic calculations. Hereafter, this method will be described step by step. First section will be devoted to the hydrodynamics model and section two to the sails performance computation. In section three, the coupling procedure will be presented. Finally, the application of the method to the case of a First Class 8 monotype will be used to analyse and illustrate the behaviour of the method.

2

Hydrodynamics

The hydrodynamic forces were computed by solving an approximated problem based on two main assumptions : the fluid is inviscid with the possible exception of wake generation, and the perturbation due to the boat motion remains small so that a linear model can be used for the free surface. These assumptions yield a linearised problem which can further reduce to the sum of a seakeeping problem describing the motion of the boat when the sea surface is perturbed by periodic sinusoidal waves and a wave resistance problem for the case of a boat with a forward speed and an initially plane surface. Only this last case will be considered in this work. This linearised wave resistance problem is probably one of the very first hydrodynamic problems to have received attention and many solutions have been proposed. The most currently used is based on a boundary integral formulation leading to the definition of an ad hoc Green function. The numerical computation of this Green function is the main difficulty of the problem and satisfactory solutions were not obtained until very recent years, Ba et al. (2003), Brument and Delhommeau (1997), Chen and Noblesse (1998). For the steady irrotational flow of an inviscid fluid, the velocity field can be expressed as the gradient of a potential Φ and the equations to be solved read : ∆Φ = 0 ∂Φ =0 ∂n 2 ∂Φ 2 ∂ Φ U∞ +g =0 2 ∂x ∂z 2

in D, on S on Sf s

(1)

where D is the fluid domain partly delimited by the free surface represented by the plane z=0, S is the wetted part of the boat and Sf s is the part of plane z = 0 out of the boat . The last condition is the linearised free surface condition obtained by considering that the waves induced by the boat motion remain small compared to their wave length and to the boat dimensions. Moreover, the solution has to satisfy two additional conditions :

1. a radiation condition which restricts the generated wave propagation to the downstream portion of the half space delimited by the free surface. 2. a Kutta-Joukovski condition at the trailing edges of lifting bodies, which in our case reduced to the boat keel.

In order to obtain a boundary integral formulation, an integral representation for the potential was used. It is based on the definition of source and doublet distributions respectively defined on the hull surface and on the skeleton of the lifting parts. Note that such a definition possibly leads to the use of both source σ and doublet µ for the modelisation of thick lifting appendices. The sources are distributed on the body surface. In the present calculation, the lifting part is split into two different parts. The first one is the keel skeleton S1 which is modelized by means of a doublet distribution. The hull was also considered as a lifting part so this lifting surface could be modelized using doublet distribution only on its skeleton denoted S2 . As a result of the Kutta-Joukovski condition, this lifting surface generates a vortical wake Σ which has been represented using a doublet distribution. The Kutta-Joukovski expressed in term of this doublet distribution yields the simple equation : dµ =0 dt along the trailing edge. The application of the linearisation assumption to the wake allows the approximation of Σ as the ruled surface generated at the trailing edge by U∞ . Thanks to these equations, the following integral representation for the potential derivative was obtained : ∀x ∈ S ZZ ∂Φ σ (x) 1 ∂G (x, x0 ) = − σ0 dsx0 ∂n 2 4π ∂nx S Z ZZ 2 0 U∞ ∂ 2 G (x, x0 ) 1 0 ∂G (x, x ) 0 + σ n · x dlx0 − µ0 dsx0 4πg ∂nx 4π ∂nx ∂nx0 S1 ∪S2 ∪Σ

(2)

C

where the prime indicates that the function is evaluated at x0 , n is either the outward unit normal or the normal to one selected side of the wake surface and C is the waterline. For the case of a boat with constant velocity, there exists a steady solution which is build up by using the steady Green function. It is obtained by taking the limit for the frequency tending to zero of the ship motion Green function given by : G x, x0 = G0 x, x0 + Gw x, x0 3

    Zπ/2   −2 0 Gw x, x = < Ks g1 (Ks ξ) dθ    πLo  −π/2

"

G0 where: ξ=

1 1 1 p −p x, x0 = 0 2 Lo ¯ 0 )2 (x − x ) (x − x

# (3)

1 z + z 0 + i (x − x0 ) cos θ + (y − y 0 ) sin θ L0

G0 is the Rankine part and Gw , the Froude dependant part and x ¯0 is the symmetric of x0 with respect to the horizontal plane z = 0. L0 is a reference length. In equation (3), g1 is a modified complex integral function defined by: g1 (ξ) = eξ ε1 (ξ), with ε1 (ξ) = E1 (ξ) if = (ξ) ≥ 0; ε1 (ξ) = E1 (ξ) − 2iπ if = (ξ) < 0. The complex exponential integral function of order 1 is defined by:  Z ∞ −t e   dt if − π < arg(ξ) < π  t Zξ ∞ −ξt E1 (ξ) = (4)  e   i dt if < (ξ) > 0 t 1 The pole Ks is defined by Ks = 1/F 2 cos2 θ. < and = represent real and imaginary parts. The Rankine part of the Green function has been extensively studied in the Rankine methods and so, will not be further discussed here. Let’s point out that G and φ satisfy the same boundary condition on the free surface and both are harmonic functions almost everywhere. Beside this, G does not satisfy any boundary condition on the body. The body surface S and mean surface S1 + S2 are divided into plane quadrilateral panels where the source and doublet intensities are assumed to be constant. Consequently the waterline is divided into segments where the source intensity is also assumed to be constant and taken equal to the value of the closest panel. The wake is divided into a set of semi-infinite strips. As previously mentioned, the linearization assumption yields a ruled surface approximation for the wake. In order to avoid a possible geometrical singularity at the trailing edge, a continuous smooth matching surface was build and the ruled surface approximation was only used at a small distance of the trailing edge. The body condition was written through equation (2) at each centre of the hull panels. The KuttaJoukovski condition was written using the same equation (2), slightly downstream of the trailing edge of the lifting parts of the hull. The doublet intensity was assumed to be constant on the semi-infinite strips extending downstream and equal to the value at the trailing edge. Equation (2) contains integrals of the Green function and its derivatives which have to be computed on panels, waterline segments or semi-infinite strips. These were performed after having interchanged the boundary and Fourier integrations. The first ones were performed analytically, using a Stokes theorem: ZZ Is = S

m

X f (ζk+1 ) − f (ζk ) d2 0 f (ζ)ds = Ck dζ 2 ζk+1 − ζk

(5)

k=1

where m is the number of panel vertices. For example, thanks to equation (5), the boundary integration on a panel for the Froude number dependent part of the Green function can be written 4

as:

Z ZZ

−2 Gwds = < π 0

(Z

π/2

m X

Ck Dk dθ

−π/2 k=1

,

Dk =

χk+1

Z dξ

)

g1 (ξ)dξ χk

χk+1 − χk

(6)

where χk = {Ks /L0 )[z + z 0 k +i [(x − x0 k ) cos θ + (y − y 0 k ) sin θ]}. The results of the integration for the diffraction-radiation with forward speed Green function have been extensively detailed in Boin et al. (2003) for the diffraction-radiation with forward speed Green function and the process used in the present work is very similar. The present results have p been checked by comparisons with those obtained in the unsteady case with ωe → 0 (ω˜e = ωe L0 /g = 0.01). The same method is also applied to the integration on the waterline segments. The panel method also involves integrals of the second order derivatives of the Green function in the wake strips extending from an element of the trailing edge. For the boat studied hereafter, a grid of 960 panels for the hull and a grid of 10 strips of 20 panels for the keel (the lifting part) were used. To help convergence of the results for the Fourier integrals, the body has been slightly submerged to z=- 0.05 m. In order to calculate the total resistance, an additional term accounting for the viscous resistance was also computed according to the ITTC57 formula : 1 0.075 U∞ L 2 Rf = ρw U∞ , RL = Sm Cf , Cf = (7) 2 2 ν (log RL − 2) where Rf is the viscous resistance, Sm the total wetted surface and Cf is the friction coefficient calculated using the Reynolds number RL with υ=1.1610−6 m2 /s. ρw is the water density.

3

Aerodynamics

As mentioned in the introduction, the flow around the sails was computed by means of a nonlinear solver. The non-linearity cannot be fully justified in the present case since only stationary equilibrium was considered. However, the use of a non-linear solver would be required for the study of a boat in periodic waves which is our next target. Moreover, the coupling procedure used in this work is based on a pseudo-unsteady scheme for which the non-linearity of the aerodynamic solver will help convergence. At least, it must be considered that the linearised case is included in the non-linear one so that we do not actually have any consistency problem. Our aerodynamic solver is based on a boundary integral formulation for the sails representation and a particle model for their wake. This is also an inviscid flow approximation excepted for the wake generation at the trailing edges of the sails. Due to the small disturbances approximation, the sea surface is no more than a symmetry plane and the wave generated wind perturbation will be neglected. Therefore, the unperturbed flow reduces to a uniform velocity field with possible vertical variation in order to account for the wind gradient. In this last case, the incoming flow is rotational and the distortion of the vortex filament by the sail should be accounted for. However, these phenomena will be neglected in the sequel which obviously reduces the validity domain of our approach. Nevertheless we believe that only the wake geometry is concerned where it is partly 5

accounted for through the wake deformation. Thus it can be expected that neglecting this distortion should be consistent with any linearised model : the non-linear model used hereafter is expected to work slightly better at least. Denoting ω the vorticity within the wake, Uω the velocity component induced by this vorticity field, U∞ the undisturbed incoming flow and φ an additional potential component, the equations to be solved read : in D,

∆φ = 0. ∂φ = − (Uω + U∞ ) · n ∂n lim φ ≈| x |−1

on ∂D0 , in D,

|x|→∞

∂ω + div (U ⊗ ω) = div (ω ⊗ U) ∂t U = Uω + U∞ + gradφ ZZZ 1 ω 0 ∧ (x − x0 ) 0 Uω (x) = 0 3 dx 4π D |x−x |

in D, in D, (8)

where D is the computational domain delimited by the sea surfaces and the sails surfaces denoted ∂D0 . It can be observed that this problem naturally split into two coupled problems for the determination of φ and ω. The first three equations are solved using a boundary integral formulation and the last three by means of a particle method. This model has been extensively used for sail flow in previous works, Charvet and Huberson (1992), Charvet et al. (1996), and will be considered as fully validated in the present study. The integral representation of gradφ on ∂D reads : ZZ 1 (x − x0 ) gradφ(x) = (9) gradx0 µ0 ∧ ds(x0 ) 2π | x − x0 |3 ∂D Note that φ automatically satisfies the first and third equation so that only the second one need to be verified. Using the previous integral expression in this equation yields : ZZ 1 (x − x0 ) 0 0 gradx0 µ ∧ (10) ds(x ) · n = − (U∞ (x) + Uω (x)) · n 2π | x − x0 |3 ∂D which is an integral equation for the doublet distribution µ on the sails surface. The discretisation of this equation is readily obtained by using a polyhedral approximation of ∂D and a piecewise constant approximation of µ0 . The wake was discretized using vortex carrying particles. There are many possible definitions for such particle. We use an integral one which consists in attributing to each particle the quantities : Z Z Z ZZZ ZZZ Xi = xdx dx , Ωi = ω(x)dx (11) Pi

Pi

Pi

which characterize respectively the particle location and the transported vorticity. The next step consists in writing the vorticity transport equation using lagrangian coordinates. Using the previous particles definitions yields an approximation in the form of two sets of non-linear differential equations : (Xi − Xj ) Ωj ∧ Ωi dΩi 3 X = Ωi · Ωj ∧ (Xj − Xi ) + 5 5 dt 4π | Xj − Xi | + | Xj − Xi |3 +3 j6=i

6

+ Ωi · grad U∞ + gradφ (Xi ) dXi 1 X Ωj ∧ (Xi − Xj ) = + gradφh (Xj ) + Uw dt 4π | Xj − Xi |3 +3

(12)

j6=i

It must be observed that these discrete equations do not strictly correspond to equations 8 and it is well established now that a set of regularised equation has to be used. The parameter is a direct measure of this regularisation process. In these equations, we also used φh which stand for the discrete approximation of φ obtained by substituting piecewise constant functions σk and µk to σ(x) and µ(x) on the sail surfaces. The two discrete functions Ωi and µk are connected through three different mechanisms : 1. through the velocity field which is used to express the integral equation for µh and the differential equation for Xi . The complete expression of the velocity is a function of both µk and Ωi : 1 X Ωj ∧ (Xi − Xj ) U= (13) + gradφh (Xj ) + Uw 4π | Xj − Xi |3 +3 j6=i

2. through the velocity gradient which is derived from the previous expression and used to compute the deformation of the vortex filament expressed in the form of a differential equation for Ωi , 3. through a specific boundary condition which relates the wake to the doublet distribution on each sail. This condition is similar to the usual Kutta-Joukovski condition for two dimensional flows. It was extended to the case of three dimensional flows by means of heuristic arguments yielding the following form for the vorticity flux at the trailing edge : Φω = (U · n) gradµ

(14)

where n is a unit vector of the plane tangent to the sail at the trailing edge and normal to the trailing edge, and gradµ stand for the vorticity distribution on the sails surfaces. The discretisation of this condition result in the creation, at each time steps, of new particles along the trailing edge defined by δt U+ + U− Xi = Xbf + 2 2 + − U +U Ωi = δt gradµh 2

(15)

where µh is the piecewise constant approximation of µ taking the value µk on the segment ”k”. The aerodynamic forces on the sails are derived from the pressure field which can be computed in different ways. In the present work, these forces are simply obtained by a straightforward application of the Lagally’s theorem yielding :

F(x) = ρU ∧ gradµh 7

(16)

Note that due to the slip condition U · n = 0 on the sail, F is always normal to the sail which is consistent with the assumption that the aerodynamic forces are actually pressure forces for inviscid flows. To achieve a complete aerodynamic model, the resultant and momentum of the aerodynamic torsor have to be computed. They are readily obtained through a summation on the panels for all sails.

4

The coupling algorithm

In this section, the coupling method is described. As it was mentioned in the introduction, this is a pseudo-unsteady procedure in the sense that the hydrodynamic solver is a linearised stationary solver whereas the aerodynamic solver is a non-linear unsteady one. Therefore, the time is an actual time for the last one and a pseudo-time for the first one. Thanks to this remark, it was considered that a real-time approach, that is the in-line calculation of the hydrodynamic solution would be useless and time consuming. Instead, the results of this solver were synthesized in the form of a subset of the six-dimensional space used to describe the hydrodynamic state of the boat. This surface is approximated by a spline interpolation from a set of grid points computed before the calculations. Each point in this space has three geometrical coordinates representing respectively : the wave resistance possibly completed by a viscous drag derived from the well known ITTC57 approximation, the drift angle and the heel angle and three dynamical coordinates representing the drift force and the wave resistance of the hull and the hydrodynamic force momentum with respect to the boat axis. The equilibrium of the hull was computed once for all at the beginning of the calculations and is expected to endorse only small variations, consistently with the linearization hypothesis. For this reason, the modification of the sinkage was not further considered. Each one of the three dynamic components was related to the geometric components by determining a surface in R6 . This surface was approximated by using cubic splines to described the variation of any couples of the dynamic variables. Let Fsx , Fsy denote the drift and propulsion components of the aerodynamic force, Msx the moment of this force with respect to the boat axis and Fhx , Fhy and Mhx the similar force components and moment for the hydrodynamic forces. The three following equations are satisfied at the equilibrium point : Fsx = Fhx Fsy = Fhy Msx + Mhx + Mgx = 0

(17)

where Mgx is the moment of the gravity forces. However, these very simple equations synthesised the results of two complex problems. Moreover, although the hydrodynamic solver is a linear one, this is a non linear problem since the actual geometry variations were accounted for when the hypersurface simulating the hydrodynamic behaviour of the boat was computed. An additional non-linearity was obviously introduced with the aerodynamic solver. Therefore, a non-linear solver

8

was used in the form of a pseudo unsteady coupling resulting in the following set of equations. Mt γx = Fsx + Fhx Mt γy = Fsy + Fhy Msx + Mhx + OGk ∧ Mk gez + OGc ∧ Mc ez = 0

(18)

where γx and γy are the two components of the pseudo-acceleration, Mk and Gk the keel mass and the keel mass centre respectively, Mc and Gc the same quantities relatively to the crew and Mt the total mass of the boat. The first two equations are pseudo-dynamic equations whereas the last one is a static equilibrium equation. Moreover, only the keel and the crew masses are considered in this equation, the hull mass effect being comparatively very small. More important factors are probably those of the mast and sails which were also neglected. Two reasons can be invoked to justify this simplification. First, we have to keep in mind that the equilibrium of the boat was fixed once for all at the beginning of the calculations and only small variations were expected in the convergence phase, consistently with the linearization assumption. Second, the method was applied to a First Class 8 monotype which is a boat equipped with a retractable keel so that the crew and the keel mass centres are adjustable parameters. The different forces are represented on the figure 1. In the previous equations, the three unknowns are the two components of the velocity which can be determined by integration according to : dU =γ dt and the heel angle which is a straightforward result of the last equation. These calculations define a new point on the hypersurface which is used to compute the new forces components and moments. These differential equations were discretised using an Adam-Bashford scheme. It must be pointed out that an alternative exists when dealing with the equilibrium equation. As already mentioned, the crew mass is to some extent an adjustable quantity. Actually, the crew members can rest at any distance from the boat axis so that the contribution of their own mass can be adjusted to obtain any desired heel angle lying within a given range. Therefore, an alternative procedure can be used to reach an equilibrium point. It consists in selecting a heel angle and computing the location of the crew mass centre or the part of the crew mass which as to be located at the more outward possible location. This last procedure as been retained hereafter. A last particular point of this procedure concerns the extension of the subset of the hypersurface considered as accessible to the equilibrium point. Due to the somewhat artificial nature of the convergence procedure, the computed equilibrium point can found itself far enough from the initial point, thus violating the linearization assumption. Therefore, it was set to remain within a bounded domain which was expected to be large enough to include the final converged state. If the equilibrium reaches one of the boundaries, it runs along this boundary as long as necessary to eventually move back inside the domain under the current conditions. Only inner points are therefore considered as valuable solutions.

9

5

Numerical results

The method was applied to a First Class 8 monotype as already mentioned. This is a 7.85 meter long, 2.49 m wide boat with a total mass of 1 400 kg. The keel mass is 515 kg. The water line main dimensions are 7.10 meters for the length and 2.00 for the width and the maximum draft is 1.75 meter. The keel mass centre can be lowered down to 1 meter and the average crew mass centre is located at a point with coordinates (0, 1.2, 0), the unit length still being the meter. The domain of variation of the equilibrium point was bounded by the following conditions : 0 ≤ Ux ≤ 6m/s,

0 ≤ α ≤ 10

(degrees),

−10 ≤ ψ ≤ 10

(degrees)

The aerodynamic model is able to compute the flow around the sails as long as the incidence angle allows the flow to remain attached. That means that the ratio between the wind velocity and the boat velocty must be limited. In the present calculations, angles up to 70 degrees were allowed although this is widely in excess for our no-separation assumption. It was observed that such a high value was never reached with the possible exception of the early stages when the boat speed is not sufficient to provide an adequate resistance to drift. We start with the hydrodynamic calculations. As already mentioned, these calculations are included in a pre-processor which provides the hydrodynamic forces in the form of a set of parameterised surfaces. The hull was discretised into 960 panels which was checked to be enough to reach a plateau of the accuracy/ panels number relation, cf. Boin (2001). The keel was modelised by means of a 10 × 20 grid thus amounting to a total number of 1160 panels for the boat (figure 2). On figure 3, the non symmetric pressure distribution on the hull is given as well as the free surface elevation for the case of a boat sailing at 4.85 knots with a 10 degrees leeway and 15 degrees heel. The second value has been somewhat exagerated in order to enhance the lifting effect and the non symmetry of the wake. The flow around the sails was computed using the method described in section 3. A 20 × 6 panels were used for the foresail discretisation and 20 × 7 for the main, figure 4. These grids are known to provide good results from the many earlier tests which have been performed using this model, Charvet and Huberson (1992), Charvet et al. (1996). The coupled algorithm was used to simulate the case of a boat sailing windward. We started with a wind velocity of 7 knots and a direction angle of 44 degrees. Although the coupling algorithm does not correspond to an actual simulation of the unsteady hydrodynamic problem, the sequence leading to convergence was found able to reproduce some reality. Indeed, the first step consists in computing the flow around the sails with the boat at rest. This can be observed on figure 5a where the boat speed is set to zero during 1.75 second. The necessity to let the flow around the sail established itself before simulating pseudo navigating conditions can be related to the impossibility to have the sails impulsively set up on an actual boat. A similar problem was encountered for the leeway which had to be set to zero until the velocity reached a sufficient value in order to prevent an excessive drift which would not be compatible with the non-viscous flow assumption, otherwise separation could massively occurs. Therefore, an initial 4 degrees value was selected and maintained during 4.25 seconds. 10

The last results on figure 5.b concerns the crew mass. It was observed in the previous section that this quantity can be considered either as a parameter or as a data in the convergence sequence. This last method correspond to a prescribed heel angle and has been used hereafter. In these calculations, the crew weight variation was expected to remain small compared to the boat weight so that the sinkage of the boat has been neglected. This is coherent with the computation of the boat average position, once for all at the beginning of the calculation, which was used. In the next example, the crew weight was fixed and equal to 391 kg. The four unknowns are the boat velocity and direction, the leeway and the heel angles. The results are given on figure 6. The boat velocity is still initially set to 0 during 1.75 seconds as well as the leeway and the heel angles. Even in this case, the leeway angle reached the maximum allowed value and remained on this value between time 4.4 and 5.3. For an actual boat living its mooring under sails alone, it is well known that the course has to be relaxed downwind in order to help the acceleration until a convenient speed is obtained. Only at this stage do the lift forces on the keel enter into action and the boat can sail upwind with a reduced leeway. Figure 6 reproduced the pressure on the sail computed at five different times. For the first stage, the boat velocity is zero and a large high pressure zone can be observed on the foresail. The extension of this zone reduced furthermore and somewhat abruptly after four seconds (fourth and fifth stages) which corresponds to the release of the constrain on the leeway. A steady solution is then reached approximately after 6.5 second. Our last result concerns the comparison with the prediction of a VPP based on statistics. These last results where provided by the boat designer J.M. Finot. In these calculations, the heel angle did not result of the previous equilibrium equations and was prescribed. Different values where tested in order to obtain a set of curves, figure 7. It must be observed that the agreement is quite good for true wind angle ranging from 50 to 70 degrees and a 12 degrees heel angle. For lower value of the true wind angle, the boat speed rapidly decreases and it is probable that the flow around both keel and hull begin to separate which cannot be reproduced with the present model.

6

Conclusion

This work is a first attempt to derive a full sailing boat model by coupling an aerodynamic solver with an hydrodynamic solver. This model has been defined as the minimum basis which can respect the main characteristics of the flow. The crude algorithm used to solve the resulting non-linear coupled problems never failed to converge. It is possible that a substantial part of this property can be attributed to the limitation of the domain where the equilibrium point was allowed to move freely. It must be pointed out that these limitations where derived from observation of real sailing boat. To some extent, the convergence sequence can be expected to produce a quasi-static approximation of the unsteady motion of the boat. One of the more encouraging result in this respect is the similarity between the solution obtained and what can be observed on real boats. Moreover, our results compare reasonably well with the results computed by the boat designer VPP. Next step will be to include actual unsteady features in the model as well as fluid-structure interactions.

11

References BA, M; BOIN, J.P.; DELHOMMEAU, G.; GUILBAUD, M.; MAURY, C. (2003), Comparison between numerical computations and experiments for seakeeping on ship’s models with forward speed, J. Ship Res., 47-4, pp. 347-364 BOIN, J.P.; (2001), Calcul des efforts hydrodynamiques sur un navire soumis ` a une houle régulière : applications d’une méthode de singularités de Kelvin, Thèse de doctorat, Université de Poitiers (France) BOIN, J.P.; GUILBAUD, M.; BA, M. : (2003), On the integration of the diffraction-radiation with forward speed Green function, Ship Techn. Res., 50, pp. 106-124 BRUMENT, A.; DELHOMMEAU, G. : (1997), Evaluation de la fonction de Green de la tenue ` a eme la mer avec vitesse d’avance, 6 Journées Hydrodynamiques, pp. 147-160, Nantes (France) CHARVET, T.; HUBERSON, S. : (1992), Numerical calculation of the flow around sails, Eur. J. Mech. B/Fluids, 5, pp. 599-610. CHARVET, T.; HAUVILLE, F.; HUBERSON, S. : (1996), Numerical simulation of the flow around sails in real sailing conditions, J. of Wind Engin. and Indust. Aerod., 63, pp. 111-129 CHEN, X.B.; NOBLESSE, F.; YANG, C. : (1999), Generic super green functions, Ship Technology Research, Vol 47, No 2, pp. 22-34 KERWIN, J.E. : (1978), A velocity prediction program for ocean racing yachts MIT Ocean Engineering Report No 78-11 LEECHTER, J.S. : (1974), Handicap rules and performances of sailing yachts, Proc. of the 1st Chesapeake Sailing Yachts Symposium, Annapolis (USA) LARSSON, L. : (1990), Scientific methods in yacht design, Ann. Rev. Fluid Mech. 22, pp. 349-385 SCHLAGETER, E.C., TEETERS, J.R. : (1999) Performance prediction software for IACC yachts, Proc. of the 11th Chesapeake Sailing Yachts Symposium, Annapolis (USA), pp. 289-304

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Figure 1: Sailing boat equilibrium.

Figure 2: Example of the mesh used for the First Class 8 : 960 panels have been used for the hull, 200 for the keel.

13

Figure 3: Pressure coefficient and free surface elevation for a First Class 8 at 2.5 m/s (= 4.85 knots)

14

Figure 4: Grid used and pressure distribution on the first class 8 sails.

Figure 5: (a) : evolution of the apparent wind angle (AWA) and boat speed during a 14 seconds sequence, (b) evolution of the leeway and crew mass during the same period. 15

Figure 6: Sequence of snapshot of the pressure distribution on the sails of a First Class 8.

16

Figure 7: Velocity prediction for different true wind angles. The lowest curve was obtained form a empirical VPP by J.M. Finot and the others correspond to heel angles of 12, 8, 4 and 0 degrees going up from the curve next to the first one.

17