In this chapter, it is specified how a two-dimensional hydrofoil shape optimization was made and which tools were used to achieve the final shape desired. Since a hydrofoil is similar to an airfoil, there are some methods in the literature that were already used to design a new shape and that can be applied to this work. However, cavitation must be considered and avoided in order to get less drag force and improve the boat’s lift velocity. It means that a special attention was given to the pressure drop on the upper and lower surfaces of the hydrofoil: it had to be under control in the optimization process in order to guarantee the best performance in race conditions.
It was used a hydrofoil shape already known to start the process. In order to get a smooth geometry, a Class-Shape-Transformation (CST) method was applied. It defines a basic shape, with the class function, and modifies the geometry with the shape function.
Differential Evolution (DE) is a population based optimization algorithm developed to optimize real parameters and value functions. It is an easy method where it is possible to control the minimum and maximum value for each parameter in order to get the best set of parameters which minimize the objective function.
In order to perform the hydrofoil shape study in a viscous environment, Xfoil software was included on the main program. Xfoil is a program for the design of airfoil shapes. The geometry’s coordinates and the Reynolds number must be specified in order to get pressure distribution in upper and lower surfaces, as have to be the lift and drag coefficients for a given angle of attack, or a set of them.
The final hydrofoil geometry is presented, as well as the main characteristics. The improvement process and results are compared to NACA 2412, the daggerboard section used by team Cascais in last race.
23
24 CHAPTER 4. DAGGERBOARD DESIGN
4.1 Hydrofoil design
The main goal of hydrofoil design proceeds from a knowledge of the boundary layer properties and the relation between geometry and pressure distribution. On this work, the main goal was to create a hydrofoil with a good lift-drag ratio, to get a maximum amount of lift while producing low drag, and maintain a constant pressure distribution over the hydrofoil surface.
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 Differences (LSD) method between them, generating the new hydrofoil coordinates by the CST method. Then, an optimization method, differential evolution, was applied.
4.1.1 Class-Shape-Transformation Method
Before using the CST method, it was necessary to define thex coordinates of the hydrofoil. In most of the cases involving airfoils, a denser panelling is used 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. With this method the x coordinate was obtained from Eq. (4.1).
x = c
2(1−cosβ) (4.1)
The chord is represented by c and, for n chordwise panels needed,β is given by Eq. (4.2), wherei is from 1 to n+1.
β = (i−1)π
n (4.2)
The CST method was developed for aerodynamic design optimization by [20], 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 coordi- nates are defined in Eq. (4.3).
4.1. HYDROFOIL DESIGN 25
ψ= x
c (4.3a)
ζ=y
c (4.3b)
The upper and lower surface defining equations are represented as follows,
ζU(ψ) =CNN1
2(ψ)SU(ψ) +ψ∆ζU (4.4a)
ζL(ψ) =CNN1
2(ψ)SL(ψ) +ψ.4ζL (4.4b)
Eq. (4.5) represents the class function where, for a round-nose hydrofoil, the parametersN1andN2
must be equal to 0.5 and 1, respectively.
CNN1
2(ψ) =ψN1(1−ψ)N2 (4.5)
As mentioned before, CST method allows to represent a hydrofoil only by defining the class function.
In order to achieve the desired shape, it is necessary to define the shape function,
SU(ψ) =
NU
X
i=0
AUi Si(ψ) (4.6a)
SL(ψ) =
NL
X
i=0
ALi Si(ψ) (4.6b)
whereNUandNLare the order of Bernstein polynomial for upper and lower surface, respectively. In this workNU=NL=Nand they are equal to one less than the number of curvature coefficients (AUandAL) used.Sis the component shape function and it is represented by
Si(ψ) =KiNψi(1−ψ)N−1 (4.7)
whereKiN is the binomial coefficient, that is related to the order of the Bernstein polynomials used. It is defined as follows
KiN= N!
i!(N−i)! (4.8)
In Fig.4.1, it is represented a series of Bernstein polynomials in the form of Pascal’s triangle.
The complete equations, for upper and lower surfaces by CST method, are presented in Eq. (4.9a)
26 CHAPTER 4. DAGGERBOARD DESIGN
Figure 4.1: Bernstein polynomial decomposition [20].
and Eq. (4.9b), respectively. The last term,ψ.4ζ, represents the tail thickness.
ζU(ψ) =ψ0.5(1−ψ)1.0KiNψi(1−ψ)NU−1] +ψ.4ζU (4.9a) ζL(ψ) =ψ0.5(1−ψ)1.0KiNψi(1−ψ)NL−1] +ψ.4ζL (4.9b)
Given that the control coefficientsAUandALwere the only unknown terms, it was used an approx- imation method to a known geometry to obtain the respective control coefficients, creating a smooth shape to begin the study.
4.1.2 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 coeffi- cients independently and in parallel, minimizing the time of these calculations.
Differential Evolution (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 in vectors of dimensionDto optimize a population of each generation,G, i.e. for each iteration of the minimization process, [33]. 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 represented 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 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.
4.1. HYDROFOIL DESIGN 27 Mutation
For each target vectorxi,G, a mutant vector is generated according to,
vi,G+1=xr1,G+F(xr2,G−xr3,G) withi = 1, ... ,Np (4.10)
with random indexesr1,r2, ...∈1, 2, ...,Np, which are chosen to be different from the running index i, so thatNp must be greater or equal to four to allow for this condition. F controls the amplification of (xr2,G−xr3,G) andF >0. An example is illustrated in Fig.4.2.
Figure 4.2: Two-dimensional cost function showing its contour lines and the process for generating [33].
Crossover
The third vector is presented as,
ui,G+1= (u1i,G+1,u2i,G+1, ...,uDi,G+1) (4.11) The crossover operation crosses two vectors,xi,G andvi,G+1 and generates the third vector,ui,G+1. For each vector component, it generates a random number in rangeU[0, 1],randj. Cut off, CR, parameter is introduced and it is between zero and one. Ifrandj <CR,
ui,G+1=vi,G+1 (4.12)
Otherwise,
ui,G+1=xi,G (4.13)
In order to guarantee the existence of at least one crossover, aui,G+1is randomly chosen to be part of vectorvi,G+1. This operation is illustrated in Fig.4.3.
28 CHAPTER 4. DAGGERBOARD DESIGN
Figure 4.3: D = 7 parameters, crossover example [33].
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 vectorxi,G using the greedy criterion. If vectorui,G+1 yields a smaller cost function value thanxi,Gthenxi,G+1is set toui,G+1. Otherwise, the old value is retained,xi,G.
The study performed by [33], where different optimizing methods are compared to DE, concludes that DE outperformed all the other minimization methods in terms of required number of function evaluations necessary to locate a global minimum of the test functions. DE can be used in this type of problem as it requires few robust control variables.
4.1.3 Hydrofoil Shape Generation
The CST method provides a smooth geometry by the definition of the control coefficients. Just to illustrate the shape generation, it was introduced the following coefficients.
AU= [1, 1, 1, 1] (4.14a)
AL= [1, 1, 1, 1] (4.14b)
These coefficients generated a hydrofoil geometry (Fig.4.4).
This was not the most indicated shape to begin the study, so a known hydrofoil was selected and all the process started from this point.
4.1. HYDROFOIL DESIGN 29
Figure 4.4: Hydrofoil shape - example.
4.1.4 Case of Study
A hydrofoil with desirable characteristics, such as low pressure coefficient (in order to avoid cavita- tion) 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 and E836, in Fig.4.7, Fig.4.5and Fig.4.6.
In Fig.4.7, it is possible to visualize the pressure distribution only on the upper surface where the load of the vessel is distributed. The more constant the pressure distribution, the more constant the load distribution. It means that the daggerboard will suffer less oscillation in race, while the vessel gets out of water. In spite of this, the E818 generates more lift, and produces much more drag than E836 or even than NACA profiles. Even though both generate constant pressure distributions, the E836 has an inferior value for the minimum cp value, which provides less cavitation effect appearance. For this reason, the E836 profile was chosen to begin the shape generation and the optimization process.
4.1.5 CST shape
With the known geometry and the CST method, it was possible to generate a CST shape and ap- proximate 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
30 CHAPTER 4. DAGGERBOARD DESIGN
Figure 4.5: CL vs alpha - NACA and Eppler profiles with Re = 2.5×106.
Figure 4.6: CD vs alpha - NACA and Eppler profiles with Re = 2.5×106.
Figure 4.7: cp vs x/c - NACA and Eppler profiles with Re = 2.5×106andα= 3.5◦.
the hydrofoil shape optimization. The used Differential Evolution code was taken from [25].
4.1. HYDROFOIL DESIGN 31 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. (A.1). The original chord has 0.23 m of length. For 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/m3was considered.
The corresponding lift-drag ratio and cp,min values are 114 and -1.115 for NACA 2412, 169 and - 1.2467 for NACA 5412, respectively. The correspondent cp distribution for both profiles in Fig.4.8and Fig.4.9.
Figure 4.8: NACA 2412 - Re = 2.5×106and Alpha = 3.5◦.
Figure 4.9: NACA 5412 - Re = 2.5×106and Alpha = 3.5◦.
The new geometry, which from now on is called CST shape, had to achieve the lift-drag ratio and hadcp,minvalue as its highest reference, in order to avoid cavitation.
32 CHAPTER 4. DAGGERBOARD DESIGN
4.1.6 Control coefficients
First of all, it was used the LSD method to approximate both shapes. In Fig.4.10, the E836 shape (black line) and the initial geometry (red line) are presented. The main goal of this approximation was to reduce the error between them until the initial geometry becomes equal to E836 shape and thereby achieved the correspondent control coefficients.
Figure 4.10: Hydrofoil Eppler 836 (black line) and initial shape (red line).
The approximation of the CST shape and E836 was accomplished. The coefficients obtained for E836 geometry are represented as follows.
AU= [0.10825, 0.15195, 0.15925, 0.14782, 0.31366] (4.15a) AL= [0.14485, 0.14163, 0.14873, 0.15288, 0.31482] (4.15b)
In Fig.4.11both shapes are presented. As can be seen, there is a slight difference between both shapes, 1.36% minimum error. Since 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, 0.5 mm, in the trailing edge, as can be seen in Fig. (4.12) inside the red circle, in order to prepare the geometry to perform structural analysis.
4.1. HYDROFOIL DESIGN 33
Figure 4.11: Eppler 836 and CST shape approximation.
Figure 4.12: CST shape with thickness in the trailing edge.
4.1.7 Objective Function
Before defining the objective function it was important to recall the main goals to be achieved.
• (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 theSprofile with NACA 2412, the first set of analysis were performed using only these geometry values as reference.
The objective function is defined in Eq. (4.18). It was used an initialβOF = 0.50 and then this value was increased toβOF = 0.75 andβOF = 0.90.
The first attempt to define an objective function focused on the pressure distribution of the hydrofoil.
Since there was a minimum pressure value defined by the cavitation number, it was decided to design the pretended pressure distribution using only the negative pressure values. For this, it was defined a vector U which included the sum of negative pressure values, sum(cp,neg) and the pressure limit to be achieved, -0.5. The result equation is presented in Eq. (4.16).
U= (cp,neg−0.5)2 (4.16)
Eq. (4.16) is then added to lift-drag ratio, defining the objective function, Eq. (4.17).
ObjFuncpre=−
βOFU−(βOF−1) L
D
λ
(4.17)
34 CHAPTER 4. DAGGERBOARD DESIGN However, this combination was decreasing the minimum pressure but not the lift-drag ratio. For this reason, it was decided to use the minimum pressure of the profile and maximize it, Eq. (4.18). This last decision worked, and lead to an increase in the lift-drag ratio and the minimum pressure.
ObjFunc =−
βOFcp,min−(βOF−1) L
D
λ
(4.18)
The optimization program minimizes the objective function. In this case, Eq. (4.18) is negative in order to be maximized. The lift-drag ratio has to be multiplied by a constantλto have the same order magnitude ofcp,min. It was usedλ= 0.01.
As the boat starts to increase the velocity, it tends to lift up of the water, making the depth of the hydrofoil decrease, until the weight of the vessel equalizes the lifting force provided by the daggerboards.
Considering that the cavitation effect is one of the responsible parameters for the drag increase, it was established to give a biggest importance to thecp,mincontrol by giving it a higher weight in the objective function formula along the iteration process.
Using the Eq. (2.3) described in the previous chapter, and defining a depth of 1.8 m, and a velocity equal to 10 m/s, the real cavitation number is 2.285 (Tab.4.1). This value provides a limit for the pressure coefficient, Eq. (4.19), and it allows a better control on the pressure coefficient obtained from each iteration.
Table 4.1: Scenario of study
velocity [m/s] depth [m] σcavit
10 1.8 2.285
The pressure value allowed must respect Eq. (4.19).
σcavit ≥ |cp,min|CST (4.19)
Several iterations were made by changing the weight of the objective function. In the first iterations it was used aβOF= 0.5 and then it was increased untilβOF = 0.9. The iteration’s evolution is illustrated in Fig.4.13and Fig.4.14.
Eq. (4.20) clarifies the final control coefficients for the CST final shape.
AU= [0.109363, 0.271069, 0.028628, 0.543274, 0.015074] (4.20a) AL= [0.060382, 0.018628, 0.007965, 0.000362, 0.012472] (4.20b)
4.1. HYDROFOIL DESIGN 35
Figure 4.13:L/Dratio evolution - E836. Figure 4.14:cp,minevolution - E836.
The lift-drag ratio values converged to 158 and thecp,minto -0.80157. These were acceptable values since the modulus of the pressure coefficient stayed below the cavitation number defined as 2.2846.
The 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 and its pressure distribution are presented in Fig. 4.15 and Fig. 4.16, cor- respondingly. The CST hydrofoil has a flat pressure distribution when compared to E836 (Fig. 4.17) pressure distribution, considering the same conditions.
Figure 4.15: CST final shape.
For the scenario presented in Tab. (4.1) the final shape has the following hydrofoil characteristics, Tab. (4.2).
Table 4.2: CST Final Geometry Characteristics
Angle [◦] Velocity [m/s] CL CD L/Dratio Cpmin
3.5 10 0.642 0.004 158 -0.802
36 CHAPTER 4. DAGGERBOARD DESIGN
Figure 4.16: CST final shape - pressure distribution.
Figure 4.17: E836 - pressure distribution for the same conditions of CST.