C-Class Catamaran Daggerboard: Analysis and Optimization

Sara Filipa Felizardo Santos Silvasara.felizardo@tecnico.ulisboa.pt

Instituto Superior Tecnico, Lisboa, Portugal

October 2014

Abstract

The main purpose of this study is to improve the lift velocity performance of a Portuguese C-ClassCatamaran by upgrading the design of the hydrofoil structure, known as daggerboard. Daggerboardsare hydrofoils placed on the middle of the hulls that allow the catamaran to lift up and gain velocity.Since there is less resistance in the air than underwater, the sooner the boat rises in the water, the earlierit increases its velocity, making the daggerboards the main components responsible for improving thecatamaran’s performance on race. In this work, a two-dimensional hydrofoil geometry was generatedusing an interface between Xfoil software and a program developed in this work. The Class-Shape-Transformation and Differential Evolution methods were implemented to generate a smooth geometryimproving the lift of the catamaran at a velocity equal to 10 m/s without cavitation effects. The three-dimensional model was then created. The profile was developed and improved until the displacementresults were under the ones of the original daggerboard. The daggerboard improvement included amodification on the blade structure and on the depth panel thickness.Keywords: C-Class, Hydrofoil, Class-Shape-Transformation, Differential Evolution, Daggerboard

1. Introduction

The International C-Class Catamaran Champi-onship (ICCCC) is a high speed boat race wherecompetitors can show their creativity and engi-neering skills. The rules for this competition aredescribed in [1]. Last year, team Cascais, bornfrom a partnership between Tony Castro and Opti-mal Structural Solutions company, participated inthis championship with the first portuguese C-Classcatamaran totally made in Portugal.

In the last race, the Catamaran’s daggerboardwas built with a National Advisory Committee forAeronautics (NACA) 2412 profile for the hydrofoilsection and an S profile for the structure of thedaggerboard. Besides this profile, Optimal has alsodeveloped an L daggerboard with NACA 5412 sec-tion. The main goal of this work is to upgrade thedaggerboard by improving the hydrofoil section.

The work developed is divided in four parts:methodology, literature research, hydrofoil design,and daggerboard design. In the methodology sec-tion, the steps taken and the project conditions aredefined, as well as the main goals of this work. Inthe literature research section, a description of thedaggerboard components is made. The hydrofoilcharacteristics and cavitation are also presented anddescribed in this section. Cavitation was one of theidentified problems at the last race and the conceptwas used in this work in order to generate a hydro-

foil section which is able to avoid this effect.

In the hydrofoil design section, a two-dimensionalprofile was designed using the Class-Shape-Transformation (CST) method, and improved us-ing Differential Evolution (DE) method. The finalhydrofoil shape was called CST. The main goal ofthis design was to increase the lift-drag ratio andthe minimum pressure coefficient when comparedwith the profiles used by team Cascais.

In the daggerboard design section, a structuralanalysis was performed using Ansys software. Inorder to achieve the main goal of this work, a dag-gerboard structure was designed with the new hy-drofoil section (CST) and compared to the hydro-foil sections (NACA 2412 and NACA 5412) usedby team Cascais at the last race. A pre-study ofthe blade of the daggerboard was done before thefull structure analysis. The new daggerboard struc-ture was improved until the maximum displacementvalue of the structure was below the one of the dag-gerboard used by team Cascais at the last race.

2. Literature research

In C-Class, the daggerboard has been a constantsubject of investigation, aiming to find and exper-iment with different daggerboard profiles in orderto get less drag at higher speeds. For better under-standing, a brief demonstration of the several partsfrom a catamaran is provided in Fig.1.

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Figure 1: Catamaran Components

2.1. Hydrofoil characteristics

A hydrofoil is similar to an airfoil. The main differ-ence lies on the fact that when choosing the profileit should be considered cavitation effects and thepressure distribution over the upper surface. Thereare several airfoil profiles designed for several spec-ifications. For example, for aircraft wings it is com-mon to use NACA profiles because of their goodbehavior in lift generation. But NACA’s profilesare made without any cavitation effects concerns.For the vessels, it is prudent to choose a geometrythat avoids cavitation, like an Eppler hydrofoil.

The hydrofoil creates a lift force, perpendicularto the flow direction, and a drag force, which hasthe same direction of the flow (Fig. 2). Since thepurpose of this work was to make the boat fly overthe water, the main goal was to increase the liftgenerated by the hydrofoil.

Figure 2: Airfoil’ forces [5]

The angle of attack is the angle between the flowdirection and the hydrofoil chord. If the angle ofattack increases, the lift force also increases, whilethe drag force decreases. The main problem of thesedaggerboards is the variation of the angle of attack,with the changes of the sea currents and the occur-rence of waves. It is very dangerous for the sailorsand for the vessel if the waves impact directly. Atthe beginning, with reduced velocities, it is neces-sary to have big angles of attack on the foils, inorder to increase speed and, consequently, lift.

Lift (L) and drag (D) forces can be calculated byEq. (1), respectively. Both equations have similarconstants, like the flow velocity, represented by v,

the fluid density, ρ, and the hydrofoil chord, c. Bothlift (CL) and drag (CD) coefficients are dependentof the hydrofoil shape.

CL =L

12ρcv

2(1a)

CD =D

12ρcv

2(1b)

The mean camber line defines the hydrofoil cur-vature, as shown in Fig. 3. Due to the changing ofthe curvature, the lift and drag forces values changeas well.

Figure 3: Hydrofoil Nomenclature

The upper surface is where the velocity reacheshigh speeds and the static pressure reaches low val-ues. The static pressure in the lower surface ishigher than in the upper surface. The pressure gra-dient between both surfaces generates the lift force.

The distribution of pressure over a hydrofoilis usually expressed by the pressure coefficient,Eq. (2).

cp =p− p∞

12ρv

2(2)

2.2. Cavitation

In this study, the effect on the flow past hydrofoilswas a matter of concern, since it was one of the iden-tified problems in last race. Using the descriptionof cavitation by [6], If the pressure above the liquidis reduced by any means, evaporation recommencesuntil a new balance is reached. If the pressure issufficiently lowered, the liquid boils when bubblesof vapour are formed in the fluid and rise to thesurface, producing large volumes of vapour. In hy-draulic engineering the vapour pressure of a liquidis of importance, for there may be places of low lo-cal pressure, particularly, when the liquid is flowingover a solid surface. If, in one of these places, thepressure is reduced until the liquid boils, then bub-bles suddenly collapse. There, very rapid collapsingmotions cause high impact pressures if they occuragainst portions of the solid surface, and may even-tually cause a local mechanical failure by fatigue ofthe solid surface.

The cavitation number is commonly representedby σcavit and it is defined by Eq. (3).

2

σcavit =p− pv12ρv

2, (3)

The p and pv are the ambient and vapour pres-sure, ρ represents the fluid density, v is the veloc-ity of upstream flow which corresponds to our boatspeed.

The static pressure has a minimum value some-where at the surface of the foil. The correspondingreduction of the static pressure is indicated by thepressure coefficient, Eq. (4).

cp,min =pmin − p

12ρv

2(4)

For the critical condition of pmin = pv , it is pos-sible to conclude that the critical cavitation numberis represented by Eq. (5).

σcavit = −cp,min = |cp,min| (5)

This equation represents the beginning of cavita-tion. It can also be concluded by Eq. (3) that whenvelocity increases, the cavitation number decreases,which reduces the range of pressure coefficient al-lowed to avoid cavitation.

In cambered sections, there is an optimum lift co-efficient, at which the streamlines meet the sectionnose smoothly [3]. If a thinner profile was chosenwith a non-smooth nose it would have had a higherpressure coefficient (cp) contributing to higher cav-itation and, consequently, increasing the drag. InFig. 4 it is possible to see an example of cp evo-lution with velocity for different depths from freesurface.

Figure 4: Pressure Coefficient vs Velocity

The only way to avoid cavitation, although it willnever be totally avoided, is to generate balance be-tween the angle of attack, the velocity of the boat,and the profile pressure coefficient. In [8], severalstudies were already made in order to create a ”for-mula” to avoid cavitation problems but reliable so-lutions were not found yet.

3. Methodology and project conditions

The design approach implemented on this work wasperformed by setting the mass and speed of the boat[4]. Since the total mass is the sum of the boat massand two regular persons, the equilibrium of the boatis very sensitive to the position of both sailors. Thisspeed-weight combination must provide the max-imum lift-drag ratio relation with minimum cavi-tation effects. Besides these fixed variables, therewere two global design variables defined for the cur-rent project:

• Depth of the structure, d - since there were al-ready two models developed, an L and S pro-files, the depth of these structures was used asstarting point of the new structure design.

• Aspect ratio, A - defined as a range of valuesand leading to the daggerboard’s dimensions.

The design methodology can be described as fol-lows:

1. By selecting a depth, the cavitation numbercan be calculated setting the minimum pres-sure coefficient of the structure before the oc-currence of cavitation effect (Eq. (6)).

σcavit =patm + ρgd− pv

12ρv

2= −cp,min (6)

2. With the minimum pressure coefficient defined,it was possible to generate a hydrofoil profileand optimize the geometry until the maximumlift-drag ratio was achieved.

3. The corresponding CL, produced by the opti-mized geometry, can be used to calculate thesection area of the hydrofoil (Eq. (7)).

A =mg

12ρv

2CL

(7)

4. By defining the aspect ratio, the span b of thehydrofoil was calculated (Eq. (8)).

A =b2

A(8)

5. Using the original dimensions and a simpler ge-ometry for the blade, a daggerboard was de-signed and static and modal analysis were per-formed. The analysis results were compared tothe daggerboards with NACA 2412 and NACA5412 profile sections.

6. A redesign of the structure was made in orderto improve the displacement results maintain-ing a safety factor above the required by theOptimal company.

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To summarize, the main project goals and projectconditions are:

• A new hydrofoil geometry was required.

• The lift-velocity was 10 m/s.

• The lift angle of attack is equal to 3.5 degrees.

• There are no constrains for the hydrofoil di-mensions - depth, span and chord.

• The cavitation number has to be controlled inorder to avoid cavitation effects.

• The safety factor of the structure must beabove 5.

Figure 5: Daggerboard dimensions nomenclature

4. Hydrofoil design

In this work, the main goal was to create a hydro-foil with a good lift-drag ratio, to get the maximumamount of lift while producing low drag, and main-tain a constant load distribution over the hydrofoilsurface by performing a constant pressure distribu-tion.

The design approach for this work consisted onchoosing an existing hydrofoil, that was alreadystudied and analysed for similar projects, whosegoals coincide with the present work goals. Themain advantage of this approach is that there istest data available making the prediction of the hy-drofoil behaviour easier in similar conditions. Theapproximation of the known geometry to a new onewas done by using the Least Squares Error (LSE)method between them, generating the new hydro-foil coordinates by the CST method. Then, an op-timization method, differential evolution, was ap-plied.

4.1. Class-Shape-Transformation method

Before using the CST method, it is necessary to de-fine the x coordinates of the hydrofoil. In most ofthe cases involving airfoils, a denser paneling is used

near the leading and trailing edges, where the radiusof curvature is smaller. A frequently used methodfor dividing the chord into panels with larger den-sity near edges is the Full Cosine method. In thismethod the x coordinate is obtained from Eq. (9).

x =c

2(1− cosβ) (9)

The chord is represented by c and, for n chordwisepanels needed, β is given by Eq. (10), where i isfrom 1 to n+1.

β = (i− 1)π

n(10)

The CST method was developed for aerodynamicdesign optimization by [7], and it can be used togenerate two and three-dimensional shapes. Forthis work, it was only used for the two-dimensionalgeneration. Any geometry can be represented bythis method. The class function defines which typeof geometry it will produce. Since it was definedto generate an airfoil or hydrofoil, the only thingthat differentiates one shape from another is a setof control coefficients that is built into the definingshape equations. These control coefficients allowthe local modification of the shape of the curvatureuntil the desired shape is achieved. This method isbased on Bezier curves with an added Class func-tion. The non-dimensional coordinates are definedin Eq. (11).

ψ =x

c(11a)

ζ =y

c(11b)

The upper and lower surface defining equationsare represented as follows,

ζU (ψ) = CN1

N2(ψ)SU (ψ) + ψ∆ζU (12a)

ζL(ψ) = CN1

N2(ψ)SL(ψ) + ψ.4 ζL (12b)

Eq. (13) represents the class function where, fora round-nose hydrofoil, the parameters N1 and N2

must be equal to 0.5 and 1, respectively.

CN1

N2(ψ) = ψN1(1− ψ)N2 (13)

As mentioned before, the CST method allowsthe representation of a hydrofoil only by definingthe class function. In order to achieve the desiredshape, it was necessary to define the shape function,

SU (ψ) =

NU∑i=0

AUi Si(ψ) (14a)

SL(ψ) =

NL∑i=0

ALi Si(ψ) (14b)

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where NU and NL are the order of Bernstein poly-nomial for upper and lower surface, respectively. Inthis work NU = NL = N and they are equal to oneless than the number of curvature coefficients (AU

and AL) used. S is the component shape functionand it was represented by

Si(ψ) = KNi ψi (1− ψ)N−1 (15)

where KNi is the binomial coefficient, that is related

to the order of the Bernstein polynomials used. Itwas defined as follows

KNi =

N !

i!(N − i)!(16)

The complete equations, for upper and lower sur-faces by CST method, are presented in Eq. (17a)and Eq. (17b), respectively. The last term, ψ.4 ζ,represents the tail thickness.

Given that the control coefficients, AU and AL,are the only unknown terms, it was used the inversemethod to obtain them and create a smooth shapeto begin the study.

ζU (ψ) = ψ0.5(1−ψ)1.0KNi ψ

i(1−ψ)NU−1]+ψ.4ζU(17a)

ζL(ψ) = ψ0.5(1−ψ)1.0KNi ψ

i(1−ψ)NL−1] +ψ.4 ζL(17b)

4.2. Differential Evolution

Since the maximization of the lift-drag ratio was oneof the goals to achieve, and it was granted by findingthe control coefficients of the shape, it was useda method that optimizes these control coefficientsindependently and in parallel, minimizing the timeof these calculations.

DE is an optimization method to minimize thefunction value, by the definition of a range of valuesfor every single variable of the function. DE uses anumber of parameters, Np, in vectors of dimensionD to optimize as population of each generation, G,i.e. for each iteration of the minimization process,[9]. The number of optimization parameters, Np,does not change during the minimization process.The initial population is randomly chosen and itshould cover the entire domain of research. Thisspace has inferior and superior limits, which shouldbe defined, and it corresponds to the project pa-rameters. In the present work, it represents eachcontrol coefficient of the hydrofoil shape.

For each generation, a new population is bornusing three stages: mutation, crossing, and selec-tion. DE generates new vectors with parameters byadding a weighted difference between the two previ-ous vectors to a third vector of the same population- mutation operation. The mutated vector’s param-eters are then mixed with the parameters of the tar-get vector, to yield the third vector. This mixing

stage is called crossover. If the result of the objec-tive function is reduced with this new vector, thevector remains and it is used in the next generation(iteration). If the result of the objective functionis superior than the target vector, the vector is notreplaced - selection operation.

Mutation

For each target vector xi,G, a mutant vector is gen-erated according to,

vi,G+1 = xr1,G+F (xr2,G−xr3,G) with i = 1, . . . , Np

(18)with random indexes r1, r2, ... ∈ 1, 2, ..., Np,

which are chosen to be different from the runningindex i, so that Np must be greater or equal to fourto allow for this condition. F controls the amplifi-cation of (xr2,G − xr3,G) and F > 0.

Crossover

The third vector is presented as,

ui,G+1 = (u1i,G+1, u2i,G+1, ..., uDi,G+1) (19)

The crossover operation crosses two vectors, xi,Gand vi,G+1 and generates the third vector, ui,G+1.For each vector component, it generates a randomnumber in range U [0, 1], randj . Cut off, CR, pa-rameter is introduced and it is between zero andone. If randj < CR,

ui,G+1 = vi,G+1 (20)

Otherwise,ui,G+1 = xi,G (21)

In order to guarantee the existence of at least onecrossover, a ui,G+1 is randomly chosen to be part ofvector vi,G+1.

Selection

In order to decide whether or not it should become amember of generation G+1, the third vector ui,G+1

is compared to the target vector xi,G using thegreedy criterion. If vector ui,G+1 yields a smallercost function value than xi,G then xi,G+1 is set toui,G+1. Otherwise, the old value is retained, xi,G.

4.3. CST geometry

A hydrofoil with desirable characteristics, such aslow pressure coefficient (in order to avoid cavita-tion) in a viscous environment and a good lift-dragratio, was chosen. Hydrofoil Eppler 818 (E818) wasa good hydrofoil to start our approximation, since ithad a constant pressure value in both surfaces witha small area between them. However, Eppler 836(E836) was also a good starting geometry as it canbe seen by comparison between the already usedprofiles, NACA 2412 and NACA 5412, and E817

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and E836, in Fig. 6. Despite the fact that the E818has a higher lift coefficient than the E836 (Fig. 7),this can be improved on the objective function.

Figure 6: cp vs x/c - NACA and Eppler profiles

Figure 7: cl vs alpha - NACA and Eppler profiles

In Fig. (6), it is possible to visualize the pres-sure distribution only on the upper surface wherethe load of the vessel is distributed. The more con-stant the pressure distribution, the more constantthe load distribution. It means that the dagger-board will suffer less oscillation in race, while thevessel lifts. The hydrofoil E836 has an inferior valuefor the minimum cp value than E818, which pro-vides less cavitation effect. In spite of having alower drag coefficient when compared with NACAprofiles, it also has a lower lift coefficient which canbe improved by the optimization method. For thisreason, the E836 profile was chosen to begin the

shape generation.With the known geometry and the CST method,

it was possible to generate a CST shape and approx-imate it to the E836 shape. The LSE method wasused in order to minimize the error between bothcurves. By finding the E836 control coefficients itwas defined the first set of control coefficients to be-gin the hydrofoil shape optimization. The structureof the optimization program can be consulted in [2].

Xfoil software performs analysis to the airfoilsin viscous conditions by introducing the Reynoldsnumber (Re). The Re was calculated with the for-mula represented in Eq. (22). The original chordhas 0.23 m of length. For the initial analysis it wasused a chord of 0.25 m. The lift velocity was equalto 10 m/s and a density of 1025 kg/m3 was consid-ered.

Re =ρcv

µ(22)

The new geometry had to achieve the lift-dragratio and a cp,min value as its highest reference, inorder to avoid cavitation. The main goals to achieveare described as follows.

• (cp,min)CST ≥ (cp,min)NACA 2412;

• (L/D)CST ≥ (L/D)NACA 2412;

• The pressure distribution should be the mostflat possible.

Since the daggerboard used in the last competi-tion was an S profile with NACA 2412, the first setof analysis was performed using these geometry val-ues as reference. The lift-drag ratio of this profile,for the angle of attack defined, is equal to 114 andthe cp,min to -1.1154.

It was used the LSE to approximate both shapes.In Fig. 8, the E836 shape (black line) and the initialgeometry (red line) are presented. The control co-efficients which correspond to the initial geometryare presented in Eq. (23).The main goal of this ap-proximation was to reduce the error between themuntil the initial geometry became equal to E836’sshape and thereby achieved the correspondent con-trol coefficients.

AU = [1, 1, 1, 1, 1] (23a)

AL = [1, 1, 1, 1, 1] (23b)

The coefficients obtained for E836 geometry arerepresented as follows (Eq. (24)).

AU = [0.1082, 0.1519, 0.1592, 0.1478, 0.3136] (24a)

AL = [0.1448, 0.1416, 0.1487, 0.1528, 0.3148] (24b)

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Figure 8: Hydrofoil Eppler 836 (black line) andinitial shape (red line)

Figure 9: Eppler 836 and CST shapeapproximation

In Fig. (9) both shapes are presented. As canbe seen, there is a slight difference between bothshapes, 1.36% minimum error.

When the CST shape control coefficients werefound, it was constructed an objective functionbased on the main goals to be achieved. It wasadded some thickness in the trailing edge in orderto prepare the geometry for structural analysis.

The objective function is defined in Eq. (25).

ObjFunc = −[βOF cp,min − (βOF − 1)

(L

D

)λ

](25)

The optimization program minimizes the objec-tive function. In this case, Eq. (25) is negative inorder to be maximized. The lift-drag ratio has to bemultiplied by a constant λ to have the same ordermagnitude of cp,min. It was used λ = 0.01.

Considering that the cavitation effect is one ofthe responsible parameters for the drag increase,it was established to give a biggest importance tothe cp,min control by giving it a higher weight inthe objective function formula along the iterationprocess.

Using the Eq. (6), and defining a depth of 1.8 m,and a velocity equal to 10 m/s, the real cavitation

number is 2.285. This value provided a limit for thepressure coefficient, Eq. (26), and it allowed a bet-ter control on the pressure coefficient obtained fromeach iteration. The pressure value allowed must re-spect Eq. (26).

σcavit ≥ |cp,min|CST (26)

Several iterations were made by changing theweight of the objective function. In the first itera-tions it was used a βOF = 0.5 which was incremen-tally increased until βOF = 0.9. Eq. (27) clarifiesthe final control coefficients for the CST final shape.

AU = [0.1093, 0.2710, 0.0286, 0.5432, 0.0150] (27a)

AL = [0.0603, 0.0186, 0.0079, 0.0003, 0.0124] (27b)

The lift-drag ratio values converged to 158 andthe cp,min to -0.80157. These were acceptable valuessince the modulus of the pressure coefficient stayedbelow the cavitation number defined as 2.2846. Thelift-drag ratio had an increase of 38 % and the min-imum pressure coefficient had an increase of 39 %when compared to NACA 2412 lift-drag ratio andpressure coefficient values, respectively, which ful-fills one of the goals of this work.

The CST final shape is presented in Fig. (??).The CST hydrofoil has a flat pressure distributionwhich allows a flat load distribution over the hydro-foil surface that avoids structural oscillations. InTab. (1), the characteristics of the final shape ofthe hydrofoil can be seen.

Figure 10: CST final shape

Table 1: CST Final Geometry Characteristics

v [m/s] CL CD L/D cp,min

10 0.6423 0.00405 158 -0.80157

5. Daggerboard design

The structural analysis was performed using An-sys software. The element chosen for the analysiswas SOLID185 which is used for the modelling ofsolid structures and is defined by eight nodes havingthree degrees of freedom (DOF) at each node. Thematerial simulated was T800, a composite used by

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Optimal company. The mesh refinement was elabo-rated for the three profiles in order to find the bestmesh length for the analysis with less computationaltime waste. The final mesh (Fig. 11) has a lengthof 0.004 m.

Figure 11: Mesh detail- CSTinitial.

5.1. Pre-study

The three profiles have the same dimensions: chordof 0.25 m and span equal to 0.5 m. The pressuredistribution was obtained by the Xfoil software anddistributed over the blade surface. Since the pre-study was only to understand the main differencesbetween the three different profiles, they are fullyconstrained on one side. A static analysis was per-formed and stress distribution results are illustratedin Fig. 12. Since the CST profile is thinner than theNACA profiles, it was expected a higher displace-ment of the CST profile blade. The CST profilepresents an increase in displacement of 133% and69% when compared with NACA 2412 and 5412,respectively, which is not the a goal of this work.It also developed a higher stress distribution overthe blade with an increase of 91% and 25% whencompared with NACA 2412 and 5412, respectively.The maximum stress verified in the CST profile pro-vides a safety factor of 24 which is higher than thecorresponding project’s requirement.

Figure 12: Stress distribution - CST

5.2. Daggerboard’s analysis and results

The profile depth considered is equal to 1.8 m, theinitial span length is around 0.5 m and the chordequal to 0.25 m. The material and element typeswere maintained, such as the mesh refinement. Theconstrains were applied on the top of the dagger-board, simulating the hull fitting. The pressuredistribution was applied only over the blade. TheCSTinitial daggerboard has higher stress distribu-tion over all the surface. The NACA 5412 dag-gerboard has lower stress value distribution overthe depth panel due to its higher thickness whichallows lower deformation in this zone. The criti-cal stress point is common in all structures. Thisis a curved zone where stress tends to concentrateand it can lead to critical situations like fatigue(seeFig. 13). The results from this analysis are pre-sented in Tab. 2.

Table 2: Daggerboard maximum displacementsand stresses

Profile Max. Displacement Max. Stress[m] [MPa]

NACA 2412 0.055 85.6NACA 5412 0.072 137CSTinitial 0.118 158

The CSTinitial profile has a higher displacementdue to its thickness. To prevent this fact, one ofthe possible solutions is to increase the thickness inthe depth panel since it is where the structure hasthe higher stresses. The CSTinitial structure has asecurity factor of almost 17.

The static analysis was just a first step to performa daggerboard behaviour study. A modal analy-sis allowed to obtain the natural frequencies of thestructure and respective modes of vibration. Ini-tially ten modes were chosen for the analysis buttaking into account the results, a refinement wasdone and only five modes were considered since thestructure begins to deform in a non-sense way, thenumber of modes was reduced to five. This rangeof frequencies deforms the daggerboard in differentdirections. For a better understanding of the struc-ture’s deformation, a node was picked in the criti-cal displacement zone which was coincident on thethree profiles.

The static and modal analysis results demon-strated the CSTinitial behaviour when the pressureand the natural frequencies were applied on theblade. In this section, the geometric dimensionswere modified in order to get a daggerboard struc-ture similar to the one constructed by Optimal com-

8

pany, in which the blade has a trapezoidal geometryinstead of a rectangular one.

Structural modifications were done in order todecrease the displacement of the blade to avoid vi-brations when the boat increases the velocity. So,in order to obtain an optimized daggerboard cal-culations were performed to designed a new bladestructure and compared it to the CSTinitial dagger-board. The methodology described was applied onthis section.

The mass considered for the boat is around300 kg, where 150 kg corresponds to the vessel andtwo sailors with 75 kg each. The boat is supportedby two daggerboards and two rudders. It was con-sidered that the rudders supports around 400 Neach, according to Optimal company information.From this point, it was also considered a conserva-tive approach. When the catamaran changes direc-tion, it tends to lift up one side of the boat and theweight is placed totally on one daggerboard. Forthis reason, the weight considered was equal to thetotal weight of the boat, around 2543 N withoutcounting on rudders.

With the weight and the section area defined,Eq. (28) the same aspect ratio (A) of original dag-gerboards is used, which is equal to 3. The Eq. (29)calculates the CSTimproved span.

A =mg

12ρv

2CL

⇔ A =P

12ρv

2CL

with g = 9.81 m/s2

(28)

A =b2

Awith A = 3 (29)

Since the sectional area of the original structuresis a trapezium, it was defined that the major chordwas twice the minor chord, i.e cminor = 0.5Cmajor,which leads to Eq. (30).

A =Cmajor + cminor

2b⇔ Cmajor =

2A

1.5b(30)

The dimensions for the final blade are presentedin Tab. 3.

Table 3: CST blade improvement

Weight Section Span Major Minorarea chord chord

P A b Cmajor cminor

[N] [m2] [m] [m] [m]

2543 0.077 0.48 0.21 0.11

With this new configuration, the maximum dis-placement decreased to 0.115 m which was not asignificant modification. However, the curved zoneof the daggerboard was not a critical zone anymore,which prevents fracture. The factor of safety of theimproved daggerboard remained at 17.

By performing a modal analysis of this struc-ture, the range of natural frequencies increaseswhen compared to the CSTinitial daggerboard whichmeans that this configuration has a higher capabil-ity to avoid oscillations and, consequently, to pre-vent fractures due to fatigue.

The movements of the same point on both struc-tures were compared, CSTinitial, and CSTimproved

daggerboard. It was notorious that there were al-most no modifications in the amplitude of move-ments. In fact, in the three directions the amplitudeof movements higher.

Since the displacement values were not satisfied,it was decided to improve the depth panel by in-creasing the thickness. This new configuration isnamed as CSTfinal. The maximum displacementdecreased drastically from 11 cm to 3 cm. The max-imum stress verified was around 116 MPa whichleads to the increase of the safety factor to 23.

The natural frequencies range also increased,meaning a structure more resistant to vibrations.The movements of the node were once again com-pared to CSTinitial. The amplitude of movements,considering the same time of analysis, decreased,which was the main goal of this structural improve-ment.

The evolution of modal and structural results aresummarized in Tab.4 and Tab.5, respectively. TheCSTfinal daggerboard stress distributions is illus-trated in Fig. 13. The final configuration is theone with best structural behaviour for the projectconditions presented.

Table 4: Natural frequencies - CST profileevolution

Profile f1 f2 f3 f4 f5

[Hz] [Hz] [Hz] [Hz] [Hz]

NACA2412 7.8 38.3 45.6 88.8 101.1NACA5412 8.4 39.0 49.0 88.5 107.3CSTinitial 6.3 29.9 36.8 79.2 80.8CSTfinal 15.5 57.3 73.3 104.8 168.1

6. Conclusions and Future Work

The main goal of this work consisted in creatinga new hydrofoil shape that could decrease the liftvelocity of a C-Class Catamaran. To achieve the

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Table 5: Daggerboard results

Profile Displac. Stress Saf.Factor[m] [MPa]

NACA2412 0.055 86 31NACA5412 0.072 137 19CSTinitial 0.118 158 17CSTfinal 0.030 116 23

Figure 13: CST final daggerboard

defined goals, a new CST hydrofoil shape was de-signed that is able to lift the catamaran at a veloc-ity of 10 m/s without cavitation effects. The lift-drag ratio increased in 39% and the minimum pres-sure decreased in 72%, when compared to NACA2412 values for the same conditions which fulfillsthe goals of this work. For a profile depth definedas 1.8 m the cavitation appears at a pressure coef-ficient equal to 2.285. Since the CST hydrofoil hasa minimum pressure coefficient of 0.8016 it meansthat the boat is able to increase the velocity up to16 m/s without crossing the cavitation limit. Thepressure load also has a flatter distribution over thenew hydrofoil, thus reducing the structure oscilla-tions preventing damage due to fatigue.

A pre-study of the blade was performed for thethree sections: NACA 2412, NACA 5412, and CST.Structural analysis demonstrated a higher displace-ment for the CST blade when compared to the othertwo profiles.

The three-dimensional daggerboard was designedwith the CST section. The improvement of the dag-gerboard structure was developed until the max-imum displacement verified became lower thanthe daggerboard’s configurations with NACA’s sec-tions. The final CST daggerboard has an L config-uration, a trapezoidal blade and a depth panel with

variation of thickness. The safety factor of the fi-nal daggerboard configuration increased from 17 to23, which fulfills the safety factor requirement. Ad-ditionally to the improvement of the displacementvalues, the natural frequency range is also higherthan the NACA’s daggerboard, meaning a betterfatigue damage tolerance due to a higher resonanceresistance.

Future studies and researches can be made con-sidering the free surface effects. While doing this,the constant pressure line, simulating the atmo-sphere pressure between the underwater dagger-board and the hull, can be considered. This casecan be simulated using computational fluid dynam-ics (CFD) study in order to understand the realforces generated by the flow over the structure.

The method used for the hydrofoil generation cre-ates a smooth geometry which allows us to modifythe hydrofoil shape locally. In order to achieve bet-ter design, in future research the thickness parame-ter improvement should be included in the objectivefunction.

References

[1] Championship rules.http://www.restronguetsc.org. Accessed24 February 2014.

[2] H. J. C. C., L. M. F. P., G. R. P. F., G. L. M.C., and F. ao A. F. O. On the annual wave en-ergy absorption by two-body heaving wecs withlatching control. Renewable Energy, 2012.

[3] H. S. F. Fluid-Dynamic Drag. 1965.

[4] B. J., S. A., T. G., L. K., H. H., H. J., C. H.,and C. T. Hydrofoil design and optimization forfast ships. 1998.

[5] H. J. Fluid Mechanics slides. Instituto SupeiorTecnico, 2013.

[6] F. J.R.D. A textbook of fluid mechanics. EdwardArnold, 1971.

[7] B. M. and Kulfan. Universal parametric geom-etry representation method. 2008.

[8] S. M., Z. P., and M. M. Cfd analysis of cavi-tation erosion potential in hydraulic machinery.2009.

[9] S. R. and P. K. Differencial evolution - a sim-ple and efficient heuristic for global optimizationover continuous spaces. 1997.

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