**Nomenclature**

#### **Symbol Definition**


#### **Appendix A. Parametric Section Design**

A rational Bézier curve can be defined [21] as shown in Equation (A1):

$$\mathcal{C}(t) = \frac{\sum\_{l=0}^{n} w\_{l} B\_{l,n}(t) P\_{l}}{\sum\_{l=0}^{n} w\_{l} B\_{l,n}(t)}, \quad t = 0, \ldots, 1 \tag{A1}$$

where *n* is the degree of the curve, *Pi* are the coordinates of *n* + 1 control points, *wi* are the weights of the control points and *B*(*i*,*n*) (*t*) are the Bernstein polynomials over the parametric abscissa, *t*, whose definition is given in Equation (A2):

$$B\_{l,n}(t) = \frac{n!}{i!(n-i)!}t^i(1-t)^{n-i}, \quad i = 0, 1, \ldots, n \tag{A2}$$

In the case of third- and fourth-degree curves, Equation (A2) gives:

$$\begin{array}{ll} B\_{\mathbf{i},\mathbf{3}}(t) & B\_{\mathbf{i},\mathbf{4}}(t) \\ B\_{0,3}(t) = \left(1-t\right)^{3} & B\_{0,4}(t) = \left(1-t\right)^{4} \\ B\_{1,3}(t) = 3\left(1-t\right)^{2} & B\_{1,4}(t) = 4\left(1-t\right)^{3} \\ B\_{2,3}(t) = 3\left(t^{2}\left(1-t\right)\right) & B\_{2,4}(t) = 6\left(t^{2}\left(1-t\right)\right) \\ B\_{3,3}(t) = t^{3} & B\_{3,4}(t) = 4\left(t^{3}\left(1-t\right)\right) \\ B\_{4,4}(t) = t^{4} & B\_{4,4}(t) = t^{4} \end{array}$$

A cubic curve (*n* = 3) is selected to define the shape of the leading edge of the profile. There are two reasons leading to this choice: the needs to assure G2 continuity of the profile between the leading edge and the flat zone, and the control of the curvature at the beginning of the profile. The G2 continuity is obtained by assuring that the two neighboring curves have the same tangent line and also the same center of curvature at their common boundary. The control of the curvature at the beginning of the profile is obtained with the following Equation (A3):

$$k(t\_0) = \frac{w\_0 w\_2}{w\_1^2} \frac{n-1}{n} \frac{h}{a^2} \tag{A3}$$

2

where *k*(*t*0) is the curvature, while *h* and *a* are defined in Figure A1.

**Figure A1.** Leading edge designed with a rational cubic Bézier curve.

In this specific case, *k*(*t*0) = <sup>1</sup> *<sup>R</sup>* , *a* = *yP*<sup>1</sup> and *h* = *yP*2. Equation (A3) can be solved to obtain the correct value of the weight *w*<sup>1</sup> of the control point P1, according to Equation (A4).

$$w\_1 = \sqrt{\frac{2 \text{ R } \text{x}\_{P2} \text{ } w\_{P0} \text{ } w\_{P2}}{3 \text{ } y\_{P2}^2}} \tag{A4}$$

Table A1 shows the values of the coordinates of each control point of the leading edge and the corresponding weights.


**Table A1.** Coordinates and weights of leading edge control points.

A fourth-degree (*n* = 4) curve is selected to define the shape of the trailing edge of the profile. In the same way as the leading edge, the G2 continuity is assured. A higher degree is needed to control the trailing edge angle at the end of the profile without violating the G2 continuity. The shape of the trailing edge can be seen in Figure A2, while Equation (A5) shows how to properly obtain the value of the trailing edge angle.

**Figure A2.** Trailing edge designed with a rational Bézier curve.

$$\mathbf{A} = \arctan\left(\frac{y\_{P7}}{x\_{P8} - x\_{P7}}\right) \tag{A5}$$

Table A2 shows the values of the coordinates of each control point of the trailing edge and the corresponding weights.


**Table A2.** Coordinates and weights of trailing edge control points.

In Figure A3, the whole flat section is shown. It is obtained by translating the values of the coordinate points of the trailing edge by a quantity equal to the leading edge length plus the length of the flat zone of the profile.

**Figure A3.** Flat section designed with three rational Bézier curves.

A reference value for the trailing edge angle is *AREF*, defined by Equation (A5) for *yP*<sup>7</sup> = <sup>T</sup> <sup>2</sup> , i.e., it may be computed as

$$A\_{REF} = \arctan\left(\frac{\frac{\text{T}}{2}}{\text{TE} - \frac{2\text{TE}}{\text{J}}}\right) = \arctan\left(1.5\frac{\text{T}}{\text{TE}}\right) \tag{A6}$$

To define the profile of a parallel-sided section, rational Bézier curves are defined and linked to the geometrical variables of the problem: thickness (T), leading edge length (LE), trailing edge length (TE), nose radius (R) and trailing edge angle (A). T, LE, TE are expressed in percent of the chord, and R is expressed in percent of the thickness. A is expressed as a fraction of *AREF*. The shape of the profile is completely parameterized, so it is possible to automatically generate all 200 shapes needed for the present study. Since this technique is very versatile, it can easily be implemented in different cases from the one presented here. A code, developed in Excel VBA, starts from the values of the design variables, processes this information to define the geometry of each curve, and saves the data for each of the sections in a neutral CAD format (e.g., IGES, CSV). Figure A4 shows the geometry of one section. As can be seen, it has been divided into three main zones: leading edge, flat zone, trailing edge. The length of each zone is related to the class rules of each dinghy. The red dots represent the control points of the curve, while the blue ones represent the points of the curves whose internal spacing can be modified to better follow the shape.

**Figure A4.** An example of the shape of a Parallel-Sided section.

#### **Appendix B. Selection of the Best Shape**

Tables A3–A5 show the best shapes for each Reynolds number. These can be useful for designers that can enter the desired values of thickness (T), leading edge length (LE) and trailing edge length (TE) and see what radius (R) and trailing edge angle (A) to use for minimizing the drag coefficient. Note that T, LE, TE are given in percent of the chord length, R is expressed in percent of the thickness and A as a fraction of *AREF*. If there are many possibilities, the tables can be useful to compare all the possible solutions and design the best one for the specific application. The best LE, TE, R and A combination for each combination of thickness, angle of attack and Reynolds number is given in bold.


**Table A3.** Best shapes for Reynolds number 300,000.


**Table A4.** Best shapes for Reynolds number 900,000.

**Table A5.** Best shapes for Reynolds number 1,500,000.


With the information presented in the tables, it is possible to find a good set of parameters for a given dinghy. For example, as seen in Table 3, the Optimist centerboard has a Reynolds number near 300,000, so Table A3 should be used. The maximum trailing edge length is 21.43% of the chord, so TE2 cannot be used. Considering the upwind condition (CL = 0.4), the best computed configuration is: T = 4, LE = 10, TE = 20, R = 5, A = <sup>4</sup> <sup>5</sup>*AREF*. For the downwind condition (AoA = 0.0), the best computed configuration is: T = 4, LE = 20, TE = 20, R = 2, A = <sup>2</sup> <sup>5</sup>*AREF*. It is important to consider the fact that, during the downwind leg of a regatta, it is common practice to partially hoist the centreboard in order to reduce the wetted surface, so the designer should give more importance to the upwind configuration.
