C-Class Catamaran Daggerboard: Analysis and Optimization
Sara Filipa Felizardo Santos Silva
[email protected]
Instituto Superior Técnico, Lisboa, Portugal
October 2014
Abstract
The main purpose of this study is to improve the lift velocity performance of a Portuguese C-Class
Catamaran by upgrading the design of the hydrofoil structure, known as daggerboard. Daggerboards
are 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 earlier
it increases its velocity, making the daggerboards the main components responsible for improving the
catamaran’s performance on race. In this work, a two-dimensional hydrofoil geometry was generated
using an interface between Xfoil software and a program developed in this work. The Class-ShapeTransformation and Differential Evolution methods were implemented to generate a smooth geometry
improving the lift of the catamaran at a velocity equal to 10 m/s without cavitation effects. The threedimensional model was then created. The profile was developed and improved until the displacement
results were under the ones of the original daggerboard. The daggerboard improvement included a
modification on the blade structure and on the depth panel thickness.
Keywords: C-Class, Hydrofoil, Class-Shape-Transformation, Differential Evolution, Daggerboard
1. Introduction
foil section which is able to avoid this effect.
In the hydrofoil design section, a two-dimensional
profile was designed using the Class-ShapeTransformation (CST) method, and improved using Differential Evolution (DE) method. The final
hydrofoil shape was called CST. The main goal of
this design was to increase the lift-drag ratio and
the minimum pressure coefficient when compared
with the profiles used by team Cascais.
In the daggerboard design section, a structural
analysis was performed using Ansys software. In
order to achieve the main goal of this work, a daggerboard structure was designed with the new hydrofoil section (CST) and compared to the hydrofoil sections (NACA 2412 and NACA 5412) used
by team Cascais at the last race. A pre-study of
the blade of the daggerboard was done before the
full structure analysis. The new daggerboard structure was improved until the maximum displacement
value of the structure was below the one of the daggerboard used by team Cascais at the last race.
The International C-Class Catamaran Championship (ICCCC) is a high speed boat race where
competitors can show their creativity and engineering skills. The rules for this competition are
described in [1]. Last year, team Cascais, born
from a partnership between Tony Castro and Optimal Structural Solutions company, participated in
this championship with the first portuguese C-Class
catamaran totally made in Portugal.
In the last race, the Catamaran’s daggerboard
was built with a National Advisory Committee for
Aeronautics (NACA) 2412 profile for the hydrofoil
section and an S profile for the structure of the
daggerboard. Besides this profile, Optimal has also
developed an L daggerboard with NACA 5412 section. The main goal of this work is to upgrade the
daggerboard by improving the hydrofoil section.
The work developed is divided in four parts:
methodology, literature research, hydrofoil design,
and daggerboard design. In the methodology section, the steps taken and the project conditions are
defined, as well as the main goals of this work. In
the literature research section, a description of the
daggerboard components is made. The hydrofoil
characteristics and cavitation are also presented and
described in this section. Cavitation was one of the
identified problems at the last race and the concept
was used in this work in order to generate a hydro-
2. Literature research
In C-Class, the daggerboard has been a constant
subject of investigation, aiming to find and experiment with different daggerboard profiles in order
to get less drag at higher speeds. For better understanding, a brief demonstration of the several parts
from a catamaran is provided in Fig.1.
1
the fluid density, ρ, and the hydrofoil chord, c. Both
lift (CL ) and drag (CD ) coefficients are dependent
of the hydrofoil shape.
CL =
CD =
L
1
2
2 ρcv
D
1
2
2 ρcv
(1a)
(1b)
The mean camber line defines the hydrofoil curvature, as shown in Fig. 3. Due to the changing of
the curvature, the lift and drag forces values change
as well.
Figure 1: Catamaran Components
2.1. Hydrofoil characteristics
A hydrofoil is similar to an airfoil. The main difference lies on the fact that when choosing the profile
it should be considered cavitation effects and the
pressure distribution over the upper surface. There
are several airfoil profiles designed for several specifications. For example, for aircraft wings it is common to use NACA profiles because of their good
behavior in lift generation. But NACA’s profiles
are made without any cavitation effects concerns.
For the vessels, it is prudent to choose a geometry
that avoids cavitation, like an Eppler hydrofoil.
The hydrofoil creates a lift force, perpendicular
to the flow direction, and a drag force, which has
the same direction of the flow (Fig. 2). Since the
purpose of this work was to make the boat fly over
the water, the main goal was to increase the lift
generated by the hydrofoil.
Figure 3: Hydrofoil Nomenclature
The upper surface is where the velocity reaches
high speeds and the static pressure reaches low values. The static pressure in the lower surface is
higher than in the upper surface. The pressure gradient between both surfaces generates the lift force.
The distribution of pressure over a hydrofoil
is usually expressed by the pressure coefficient,
Eq. (2).
cp =
p − p∞
1
2
2 ρv
(2)
2.2. Cavitation
In this study, the effect on the flow past hydrofoils
was a matter of concern, since it was one of the identified problems in last race. Using the description
of cavitation by [6], If the pressure above the liquid
is reduced by any means, evaporation recommences
Figure 2: Airfoil’ forces [5]
until a new balance is reached. If the pressure is
sufficiently lowered, the liquid boils when bubbles
The angle of attack is the angle between the flow of vapour are formed in the fluid and rise to the
direction and the hydrofoil chord. If the angle of surface, producing large volumes of vapour. In hyattack increases, the lift force also increases, while draulic engineering the vapour pressure of a liquid
the drag force decreases. The main problem of these is of importance, for there may be places of low lodaggerboards is the variation of the angle of attack, cal pressure, particularly, when the liquid is flowing
with the changes of the sea currents and the occur- over a solid surface. If, in one of these places, the
rence of waves. It is very dangerous for the sailors pressure is reduced until the liquid boils, then buband for the vessel if the waves impact directly. At bles suddenly collapse. There, very rapid collapsing
the beginning, with reduced velocities, it is neces- motions cause high impact pressures if they occur
sary to have big angles of attack on the foils, in against portions of the solid surface, and may evenorder to increase speed and, consequently, lift.
tually cause a local mechanical failure by fatigue of
Lift (L) and drag (D) forces can be calculated by the solid surface.
Eq. (1), respectively. Both equations have similar
The cavitation number is commonly represented
constants, like the flow velocity, represented by v, by σcavit and it is defined by Eq. (3).
2
σcavit =
p − pv
,
1
2
2 ρv
3. Methodology and project conditions
(3)
The design approach implemented on this work was
performed by setting the mass and speed of the boat
[4]. Since the total mass is the sum of the boat mass
and two regular persons, the equilibrium of the boat
is very sensitive to the position of both sailors. This
speed-weight combination must provide the maximum lift-drag ratio relation with minimum cavitation effects. Besides these fixed variables, there
were two global design variables defined for the current project:
The p and pv are the ambient and vapour pressure, ρ represents the fluid density, v is the velocity of upstream flow which corresponds to our boat
speed.
The static pressure has a minimum value somewhere at the surface of the foil. The corresponding
reduction of the static pressure is indicated by the
pressure coefficient, Eq. (4).
cp,min =
pmin − p
1
2
2 ρv
• Depth of the structure, d - since there were already two models developed, an L and S profiles, the depth of these structures was used as
starting point of the new structure design.
(4)
For the critical condition of pmin = pv , it is possible to conclude that the critical cavitation number
is represented by Eq. (5).
σcavit = −cp,min = |cp,min |
• Aspect ratio, A - defined as a range of values
and leading to the daggerboard’s dimensions.
(5)
The design methodology can be described as folThis equation represents the beginning of cavita- lows:
tion. It can also be concluded by Eq. (3) that when
1. By selecting a depth, the cavitation number
velocity increases, the cavitation number decreases,
can be calculated setting the minimum preswhich reduces the range of pressure coefficient alsure coefficient of the structure before the oclowed to avoid cavitation.
currence of cavitation effect (Eq. (6)).
In cambered sections, there is an optimum lift coefficient, at which the streamlines meet the section
patm + ρgd − pv
nose smoothly [3]. If a thinner profile was chosen
= −cp,min
(6)
σcavit =
1
2
2 ρv
with a non-smooth nose it would have had a higher
pressure coefficient (cp ) contributing to higher cav2. With the minimum pressure coefficient defined,
itation and, consequently, increasing the drag. In
it was possible to generate a hydrofoil profile
Fig. 4 it is possible to see an example of cp evoand optimize the geometry until the maximum
lution with velocity for different depths from free
lift-drag ratio was achieved.
surface.
3. The corresponding CL , produced by the optimized geometry, can be used to calculate the
section area of the hydrofoil (Eq. (7)).
A=
mg
1
2
2 ρv CL
(7)
4. By defining the aspect ratio, the span b of the
hydrofoil was calculated (Eq. (8)).
A=
b2
A
(8)
5. Using the original dimensions and a simpler geometry for the blade, a daggerboard was designed and static and modal analysis were performed. The analysis results were compared to
the daggerboards with NACA 2412 and NACA
5412 profile sections.
Figure 4: Pressure Coefficient vs Velocity
The only way to avoid cavitation, although it will
never be totally avoided, is to generate balance between the angle of attack, the velocity of the boat,
and the profile pressure coefficient. In [8], several
studies were already made in order to create a ”formula” to avoid cavitation problems but reliable solutions were not found yet.
6. A redesign of the structure was made in order
to improve the displacement results maintaining a safety factor above the required by the
Optimal company.
3
To summarize, the main project goals and project
conditions are:
near the leading and trailing edges, where the radius
of curvature is smaller. A frequently used method
for dividing the chord into panels with larger density near edges is the Full Cosine method. In this
method the x coordinate is obtained from Eq. (9).
• A new hydrofoil geometry was required.
• The lift-velocity was 10 m/s.
c
(1 − cos β)
(9)
2
• There are no constrains for the hydrofoil diThe chord is represented by c and, for n chordwise
mensions - depth, span and chord.
panels needed, β is given by Eq. (10), where i is
from 1 to n+1.
• The cavitation number has to be controlled in
π
order to avoid cavitation effects.
(10)
β = (i − 1)
n
• The safety factor of the structure must be
The CST method was developed for aerodynamic
above 5.
design optimization by [7], and it can be used to
generate two and three-dimensional shapes. For
this work, it was only used for the two-dimensional
generation. Any geometry can be represented by
this method. The class function defines which type
of geometry it will produce. Since it was defined
to generate an airfoil or hydrofoil, the only thing
that differentiates one shape from another is a set
of control coefficients that is built into the defining
shape equations. These control coefficients allow
the local modification of the shape of the curvature
until the desired shape is achieved. This method is
based on Bezier curves with an added Class function. The non-dimensional coordinates are defined
in Eq. (11).
Figure 5: Daggerboard dimensions nomenclature
x
(11a)
ψ=
c
y
4. Hydrofoil design
ζ=
(11b)
c
In this work, the main goal was to create a hydroThe upper and lower surface defining equations
foil with a good lift-drag ratio, to get the maximum
are
represented as follows,
amount of lift while producing low drag, and main• The lift angle of attack is equal to 3.5 degrees.
x=
tain a constant load distribution over the hydrofoil
surface by performing a constant pressure distribution.
The design approach for this work consisted on
choosing an existing hydrofoil, that was already
studied and analysed for similar projects, whose
goals coincide with the present work goals. The
main advantage of this approach is that there is
test data available making the prediction of the hydrofoil behaviour easier in similar conditions. The
approximation of the known geometry to a new one
was done by using the Least Squares Error (LSE)
method between them, generating the new hydrofoil coordinates by the CST method. Then, an optimization method, differential evolution, was applied.
N1
ζ U (ψ) = CN
(ψ) S U (ψ) + ψ ∆ζU
2
(12a)
N1
ζL (ψ) = CN
(ψ) SL (ψ) + ψ. 4 ζL
2
(12b)
Eq. (13) represents the class function where, for
a round-nose hydrofoil, the parameters N1 and N2
must be equal to 0.5 and 1, respectively.
N1
CN
(ψ) = ψ N1 (1 − ψ)N2
2
(13)
As mentioned before, the CST method allows
the representation of a hydrofoil only by defining
the class function. In order to achieve the desired
shape, it was necessary to define the shape function,
S U (ψ) =
4.1. Class-Shape-Transformation method
NU
X
AU
i Si (ψ)
(14a)
AL
i Si (ψ)
(14b)
i=0
Before using the CST method, it is necessary to define the x coordinates of the hydrofoil. In most of
the cases involving airfoils, a denser paneling is used
S L (ψ) =
NL
X
i=0
4
where NU and NL are the order of Bernstein polynomial for upper and lower surface, respectively. In
this work NU = NL = N and they are equal to one
less than the number of curvature coefficients (AU
and AL ) used. S is the component shape function
and it was represented by
Si (ψ) = KiN ψ i (1 − ψ)N −1
stage is called crossover. If the result of the objective function is reduced with this new vector, the
vector remains and it is used in the next generation
(iteration). If the result of the objective function
is superior than the target vector, the vector is not
replaced - selection operation.
Mutation
(15)
For each target vector xi,G , a mutant vector is genwhere KiN is the binomial coefficient, that is related
erated according to,
to the order of the Bernstein polynomials used. It
was defined as follows
vi,G+1 = xr1 ,G +F (xr2 ,G −xr3 ,G ) with i = 1, . . . , Np
N!
N
(18)
(16)
Ki =
i!(N − i)!
with random indexes r1 , r2 , ... ∈ 1, 2, ..., Np ,
The complete equations, for upper and lower sur- which are chosen to be different from the running
faces by CST method, are presented in Eq. (17a) index i, so that Np must be greater or equal to four
and Eq. (17b), respectively. The last term, ψ. 4 ζ, to allow for this condition. F controls the amplification of (xr2 ,G − xr3 ,G ) and F > 0.
represents the tail thickness.
Given that the control coefficients, AU and AL , Crossover
are the only unknown terms, it was used the inverse
method to obtain them and create a smooth shape The third vector is presented as,
to begin the study.
ui,G+1 = (u1i,G+1 , u2i,G+1 , ..., uDi,G+1 )
ζU (ψ) = ψ 0.5 (1 − ψ)1.0 KiN ψ i (1 − ψ)NU −1 ] + ψ. 4 ζU
(17a)
ζL (ψ) = ψ 0.5 (1 − ψ)1.0 KiN ψ i (1 − ψ)NL −1 ] + ψ. 4 ζL
(17b)
(19)
The crossover operation crosses two vectors, xi,G
and vi,G+1 and generates the third vector, ui,G+1 .
For each vector component, it generates a random
number in range U [0, 1], randj . Cut off, CR, parameter is introduced and it is between zero and
one. If randj < CR,
4.2. Differential Evolution
Since the maximization of the lift-drag ratio was one
of the goals to achieve, and it was granted by finding
the control coefficients of the shape, it was used
a method that optimizes these control coefficients
independently and in parallel, minimizing the time
of these calculations.
DE is an optimization method to minimize the
function value, by the definition of a range of values
for every single variable of the function. DE uses a
number of parameters, Np , in vectors of dimension
D 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 it
should cover the entire domain of research. This
space has inferior and superior limits, which should
be defined, and it corresponds to the project parameters. In the present work, it represents each
control coefficient of the hydrofoil shape.
For each generation, a new population is born
using three stages: mutation, crossing, and selection. DE generates new vectors with parameters by
adding a weighted difference between the two previous vectors to a third vector of the same population
- mutation operation. The mutated vector’s parameters are then mixed with the parameters of the target vector, to yield the third vector. This mixing
ui,G+1 = vi,G+1
(20)
ui,G+1 = xi,G
(21)
Otherwise,
In order to guarantee the existence of at least one
crossover, a ui,G+1 is randomly chosen to be part of
vector vi,G+1 .
Selection
In order to decide whether or not it should become a
member of generation G+1, the third vector ui,G+1
is compared to the target vector xi,G using the
greedy criterion. If vector ui,G+1 yields a smaller
cost function value than xi,G then xi,G+1 is set to
ui,G+1 . Otherwise, the old value is retained, xi,G .
4.3. CST geometry
A hydrofoil with desirable characteristics, such as
low pressure coefficient (in order to avoid cavitation) in a viscous environment and a good lift-drag
ratio, was chosen. Hydrofoil Eppler 818 (E818) was
a good hydrofoil to start our approximation, since it
had a constant pressure value in both surfaces with
a small area between them. However, Eppler 836
(E836) was also a good starting geometry as it can
be seen by comparison between the already used
profiles, NACA 2412 and NACA 5412, and E817
5
and E836, in Fig. 6. Despite the fact that the E818
has a higher lift coefficient than the E836 (Fig. 7),
this can be improved on the objective function.
shape generation.
With the known geometry and the CST method,
it was possible to generate a CST shape and approximate it to the E836 shape. The LSE method was
used in order to minimize the error between both
curves. By finding the E836 control coefficients it
was defined the first set of control coefficients to begin the hydrofoil shape optimization. The structure
of the optimization program can be consulted in [2].
Xfoil software performs analysis to the airfoils
in viscous conditions by introducing the Reynolds
number (Re). The Re was calculated with the formula represented in Eq. (22). The original chord
has 0.23 m of length. For the initial analysis it was
used a chord of 0.25 m. The lift velocity was equal
to 10 m/s and a density of 1025 kg/m3 was considered.
ρcv
(22)
Re =
µ
The new geometry had to achieve the lift-drag
ratio and a cp,min value as its highest reference, in
order to avoid cavitation. The main goals to achieve
are described as follows.
Figure 6: cp vs x/c - NACA and Eppler profiles
• (cp,min )CST ≥ (cp,min )NACA 2412 ;
• (L/D)CST ≥ (L/D)NACA 2412 ;
• The pressure distribution should be the most
flat possible.
Since the daggerboard used in the last competition was an S profile with NACA 2412, the first set
of analysis was performed using these geometry values as reference. The lift-drag ratio of this profile,
for the angle of attack defined, is equal to 114 and
the cp,min to -1.1154.
It was used the LSE to approximate both shapes.
In Fig. 8, the E836 shape (black line) and the initial
geometry (red line) are presented. The control coefficients which correspond to the initial geometry
are presented in Eq. (23).The main goal of this approximation was to reduce the error between them
until the initial geometry became equal to E836’s
shape and thereby achieved the correspondent control coefficients.
Figure 7: cl vs alpha - NACA and Eppler profiles
In Fig. (6), it is possible to visualize the pressure distribution only on the upper surface where
AU = [1, 1, 1, 1, 1]
(23a)
the load of the vessel is distributed. The more constant the pressure distribution, the more constant
AL = [1, 1, 1, 1, 1]
(23b)
the load distribution. It means that the daggerboard will suffer less oscillation in race, while the
The coefficients obtained for E836 geometry are
vessel lifts. The hydrofoil E836 has an inferior value
represented
as follows (Eq. (24)).
for the minimum cp value than E818, which provides less cavitation effect. In spite of having a
AU = [0.1082, 0.1519, 0.1592, 0.1478, 0.3136] (24a)
lower drag coefficient when compared with NACA
profiles, it also has a lower lift coefficient which can
AL = [0.1448, 0.1416, 0.1487, 0.1528, 0.3148] (24b)
be improved by the optimization method. For this
reason, the E836 profile was chosen to begin the
6
number is 2.285. This value provided a limit for the
pressure coefficient, Eq. (26), and it allowed a better control on the pressure coefficient obtained from
each iteration. The pressure value allowed must respect Eq. (26).
σcavit ≥ |cp,min |CST
(26)
Several iterations were made by changing the
weight of the objective function. In the first iterations it was used a βOF = 0.5 which was incrementally increased until βOF = 0.9. Eq. (27) clarifies
the 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 and
the cp,min to -0.80157. These were acceptable values
since the modulus of the pressure coefficient stayed
Figure 8: Hydrofoil Eppler 836 (black line) and
below the cavitation number defined as 2.2846. The
initial shape (red line)
lift-drag ratio had an increase of 38 % and the minimum pressure coefficient had an increase of 39 %
when compared to NACA 2412 lift-drag ratio and
pressure coefficient values, respectively, which fulfills one of the goals of this work.
The CST final shape is presented in Fig. (??).
The
CST hydrofoil has a flat pressure distribution
Figure 9: Eppler 836 and CST shape
which
allows a flat load distribution over the hydroapproximation
foil surface that avoids structural oscillations. In
Tab. (1), the characteristics of the final shape of
In Fig. (9) both shapes are presented. As can the hydrofoil can be seen.
be seen, there is a slight difference between both
shapes, 1.36% minimum error.
When the CST shape control coefficients were
found, it was constructed an objective function
based on the main goals to be achieved. It was
added some thickness in the trailing edge in order
Figure 10: CST final shape
to prepare the geometry for structural analysis.
The objective function is defined in Eq. (25).
Table 1: CST Final Geometry Characteristics
L
ObjFunc = − βOF cp,min − (βOF − 1)
λ
D
(25)
v [m/s]
CL
CD
L/D
cp,min
The optimization program minimizes the objective function. In this case, Eq. (25) is negative in
10
0.6423 0.00405 158 -0.80157
order to be maximized. The lift-drag ratio has to be
multiplied by a constant λ to have the same order
magnitude of cp,min . It was used λ = 0.01.
Considering that the cavitation effect is one of
the responsible parameters for the drag increase, 5. Daggerboard design
it was established to give a biggest importance to The structural analysis was performed using Anthe cp,min control by giving it a higher weight in sys software. The element chosen for the analysis
the objective function formula along the iteration was SOLID185 which is used for the modelling of
process.
solid structures and is defined by eight nodes having
Using the Eq. (6), and defining a depth of 1.8 m, three degrees of freedom (DOF) at each node. The
and a velocity equal to 10 m/s, the real cavitation material simulated was T800, a composite used by
7
5.2. Daggerboard’s analysis and results
Optimal company. The mesh refinement was elaborated for the three profiles in order to find the best
mesh length for the analysis with less computational
time waste. The final mesh (Fig. 11) has a length
of 0.004 m.
The profile depth considered is equal to 1.8 m, the
initial span length is around 0.5 m and the chord
equal to 0.25 m. The material and element types
were maintained, such as the mesh refinement. The
constrains were applied on the top of the daggerboard, simulating the hull fitting. The pressure
distribution was applied only over the blade. The
CSTinitial daggerboard has higher stress distribution over all the surface. The NACA 5412 daggerboard has lower stress value distribution over
the depth panel due to its higher thickness which
allows lower deformation in this zone. The critical stress point is common in all structures. This
is a curved zone where stress tends to concentrate
and it can lead to critical situations like fatigue(see
Fig. 13). The results from this analysis are presented in Tab. 2.
Figure 11: Mesh detail- CSTinitial .
Table 2: Daggerboard maximum displacements
and stresses
5.1. Pre-study
The three profiles have the same dimensions: chord
of 0.25 m and span equal to 0.5 m. The pressure
distribution was obtained by the Xfoil software and
distributed over the blade surface. Since the prestudy was only to understand the main differences
between the three different profiles, they are fully
constrained on one side. A static analysis was performed and stress distribution results are illustrated
in Fig. 12. Since the CST profile is thinner than the
NACA profiles, it was expected a higher displacement of the CST profile blade. The CST profile
presents an increase in displacement of 133% and
69% when compared with NACA 2412 and 5412,
respectively, which is not the a goal of this work.
It also developed a higher stress distribution over
the blade with an increase of 91% and 25% when
compared with NACA 2412 and 5412, respectively.
The maximum stress verified in the CST profile provides a safety factor of 24 which is higher than the
corresponding project’s requirement.
Profile
NACA 2412
NACA 5412
CSTinitial
Max. Displacement
[m]
Max. Stress
[MPa]
0.055
0.072
0.118
85.6
137
158
The CSTinitial profile has a higher displacement
due to its thickness. To prevent this fact, one of
the possible solutions is to increase the thickness in
the depth panel since it is where the structure has
the higher stresses. The CSTinitial structure has a
security factor of almost 17.
The static analysis was just a first step to perform
a daggerboard behaviour study. A modal analysis allowed to obtain the natural frequencies of the
structure and respective modes of vibration. Initially ten modes were chosen for the analysis but
taking into account the results, a refinement was
done and only five modes were considered since the
structure begins to deform in a non-sense way, the
number of modes was reduced to five. This range
of frequencies deforms the daggerboard in different
directions. For a better understanding of the structure’s deformation, a node was picked in the critical displacement zone which was coincident on the
three profiles.
The static and modal analysis results demonstrated the CSTinitial behaviour when the pressure
and the natural frequencies were applied on the
blade. In this section, the geometric dimensions
were modified in order to get a daggerboard structure similar to the one constructed by Optimal com-
Figure 12: Stress distribution - CST
8
pany, in which the blade has a trapezoidal geometry
instead of a rectangular one.
Structural modifications were done in order to
decrease the displacement of the blade to avoid vibrations when the boat increases the velocity. So,
in order to obtain an optimized daggerboard calculations were performed to designed a new blade
structure and compared it to the CSTinitial daggerboard. The methodology described was applied on
this section.
The mass considered for the boat is around
300 kg, where 150 kg corresponds to the vessel and
two sailors with 75 kg each. The boat is supported
by two daggerboards and two rudders. It was considered that the rudders supports around 400 N
each, according to Optimal company information.
From this point, it was also considered a conservative approach. When the catamaran changes direction, it tends to lift up one side of the boat and the
weight is placed totally on one daggerboard. For
this reason, the weight considered was equal to the
total weight of the boat, around 2543 N without
counting on rudders.
With the weight and the section area defined,
Eq. (28) the same aspect ratio (A) of original daggerboards is used, which is equal to 3. The Eq. (29)
calculates the CSTimproved span.
A=
mg
1
2
2 ρv CL
⇔A=
P
1
2
2 ρv CL
With this new configuration, the maximum displacement decreased to 0.115 m which was not a
significant modification. However, the curved zone
of the daggerboard was not a critical zone anymore,
which prevents fracture. The factor of safety of the
improved daggerboard remained at 17.
By performing a modal analysis of this structure, the range of natural frequencies increases
when compared to the CSTinitial daggerboard which
means that this configuration has a higher capability to avoid oscillations and, consequently, to prevent fractures due to fatigue.
The movements of the same point on both structures were compared, CSTinitial , and CSTimproved
daggerboard. It was notorious that there were almost no modifications in the amplitude of movements. In fact, in the three directions the amplitude
of movements higher.
Since the displacement values were not satisfied,
it was decided to improve the depth panel by increasing the thickness. This new configuration is
named as CSTfinal . The maximum displacement
decreased drastically from 11 cm to 3 cm. The maximum stress verified was around 116 MPa which
leads 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 compared to CSTinitial . The amplitude of movements,
considering the same time of analysis, decreased,
which was the main goal of this structural improvement.
The evolution of modal and structural results are
summarized in Tab.4 and Tab.5, respectively. The
CSTfinal daggerboard stress distributions is illustrated in Fig. 13. The final configuration is the
one with best structural behaviour for the project
conditions presented.
with g = 9.81 m/s2
(28)
A=
b2
with A = 3
A
(29)
Since the sectional area of the original structures
is a trapezium, it was defined that the major chord
was twice the minor chord, i.e cminor = 0.5Cmajor ,
which leads to Eq. (30).
A=
Cmajor + cminor
2A
b ⇔ Cmajor =
2
1.5b
Table 4: Natural frequencies - CST profile
evolution
(30)
Profile
f1
[Hz]
f2
[Hz]
f3
[Hz]
f4
[Hz]
f5
[Hz]
NACA2412
NACA5412
CSTinitial
CSTfinal
7.8
8.4
6.3
15.5
38.3
39.0
29.9
57.3
45.6
49.0
36.8
73.3
88.8
88.5
79.2
104.8
101.1
107.3
80.8
168.1
The dimensions for the final blade are presented
in Tab. 3.
Table 3: CST blade improvement
Weight
Span
P
[N]
Section
area
A
[m2 ]
b
[m]
Major
chord
Cmajor
[m]
Minor
chord
cminor
[m]
2543
0.077
0.48
0.21
0.11
6. Conclusions and Future Work
The main goal of this work consisted in creating
a new hydrofoil shape that could decrease the lift
velocity of a C-Class Catamaran. To achieve the
9
Table 5: Daggerboard results
Profile
NACA2412
NACA5412
CSTinitial
CSTfinal
Displac.
[m]
Stress
[MPa]
Saf.Factor
0.055
0.072
0.118
0.030
86
137
158
116
31
19
17
23
variation of thickness. The safety factor of the final daggerboard configuration increased from 17 to
23, which fulfills the safety factor requirement. Additionally to the improvement of the displacement
values, the natural frequency range is also higher
than the NACA’s daggerboard, meaning a better
fatigue damage tolerance due to a higher resonance
resistance.
Future studies and researches can be made considering the free surface effects. While doing this,
the constant pressure line, simulating the atmosphere pressure between the underwater daggerboard and the hull, can be considered. This case
can be simulated using computational fluid dynamics (CFD) study in order to understand the real
forces generated by the flow over the structure.
The method used for the hydrofoil generation creates a smooth geometry which allows us to modify
the hydrofoil shape locally. In order to achieve better design, in future research the thickness parameter improvement should be included in the objective
function.
References
[1] Championship
http://www.restronguetsc.org.
24 February 2014.
Figure 13: CST final daggerboard
defined goals, a new CST hydrofoil shape was designed that is able to lift the catamaran at a velocity of 10 m/s without cavitation effects. The liftdrag ratio increased in 39% and the minimum pressure decreased in 72%, when compared to NACA
2412 values for the same conditions which fulfills
the goals of this work. For a profile depth defined
as 1.8 m the cavitation appears at a pressure coefficient equal to 2.285. Since the CST hydrofoil has
a minimum pressure coefficient of 0.8016 it means
that the boat is able to increase the velocity up to
16 m/s without crossing the cavitation limit. The
pressure load also has a flatter distribution over the
new hydrofoil, thus reducing the structure oscillations preventing damage due to fatigue.
A pre-study of the blade was performed for the
three sections: NACA 2412, NACA 5412, and CST.
Structural analysis demonstrated a higher displacement for the CST blade when compared to the other
two profiles.
The three-dimensional daggerboard was designed
with the CST section. The improvement of the daggerboard structure was developed until the maximum displacement verified became lower than
the daggerboard’s configurations with NACA’s sections. The final CST daggerboard has an L configuration, a trapezoidal blade and a depth panel with
10
rules.
Accessed
[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 energy absorption by two-body heaving wecs with
latching 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 for
fast ships. 1998.
[5] H. J. Fluid Mechanics slides. Instituto Supeior
Tecnico, 2013.
[6] F. J.R.D. A textbook of fluid mechanics. Edward
Arnold, 1971.
[7] B. M. and Kulfan. Universal parametric geometry representation method. 2008.
[8] S. M., Z. P., and M. M. Cfd analysis of cavitation erosion potential in hydraulic machinery.
2009.
[9] S. R. and P. K. Differencial evolution - a simple and efficient heuristic for global optimization
over continuous spaces. 1997.