# A New Systematic Series of Foil Sections with Parallel Sides

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Method

#### 2.1. Equations

#### 2.2. Boundaries

#### 2.3. Discretization

#### 2.4. Grid Settings

## 3. Verification

_{i}is the linear cell size ratio between a specific grid (i) and the finest one, U is the numerical uncertainty, and U (%C

_{L}) is the uncertainty in percent of the coefficient.

## 4. Validation Case

## 5. Systematic Investigation

_{L}= 0.4) is set instead, so the angle of attack is slightly different between the cases. It is the given side force from the sails that should be balanced, and this will be achieved at different leeway angles for the different sections. The authors choose a lift coefficient equal to 0.4 as a target value after conducting a study of typical leeway angles of the 470 and Optimist dinghies using a velocity prediction program (VPP). For 420 and 470, the 0° case may not be relevant, since the centerboard is hoisted while sailing downwind. For the Optimist, the centerboard is usually only partly hoisted during the downwind leg, so the 0° case can be useful for this type of dinghy. Furthermore, the 0° case for this type of sections is of interest in other fields than sailing, where the angle of attack is often 0°.

## 6. Results

_{L}= 0.4 case.

_{L}= 0.4 are considered. In order to explore the physics, field plots of the axial velocity are presented, together with line plots of the pressure or friction variation along the section on the two sides. To identify regions of separation, the lower limit of the axial velocity is set to zero, such that negative velocities, i.e., separations, are indicated by a white color.

#### 6.1. Zero Angle of Attack

#### 6.2. Lift Coefficient 0.4

## 7. Conclusions

- The thinner sections have considerably lower drag than the thick ones.
- The length of the leading edge is unimportant for thin sections. For thick sections, it is important in some cases, but not all. The reasons are explained above.
- The trailing edge should be as long as possible.
- The nose radius has a very small effect on the drag.
- The trailing edge angle should be very small for the thin sections. For the thick sections, this is also true, with one exception that is explained above.
- The best thin parallel-sided sections have, on average, 13% higher drag than the four-digit NACA section with the same thickness ratio. For thick sections, the average drag increase is 30%.

- The thin and thick sections have surprisingly equal drag. Exceptions are sections with a short trailing edge at medium, and high Reynolds numbers where the thick sections have higher drag. This effect is explained above.
- Generally, sections with long leading and trailing edges are the best, while sections with short leading and trailing edges are the worst. The performance of other combinations depends on the Reynolds number.
- The trailing edge angle has a very small effect on the drag.
- A rather small (but not too small) leading edge radius is the best for the thin sections. The same is true for the thick sections. There is however an exception at high Reynolds number and short leading edge.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

Symbol | Definition |

C_{D} | Drag Coefficient |

C_{L} | Lift Coefficient |

α | Angle of Attack |

Hi | Refinement Ratio |

U | Numerical Uncertainty |

C(t) | Parametric Curve |

B(t) | Bernstein polynomials |

k(t) | Curvature |

w | Weight of a Control Point |

T | Thickness |

LE | Leading Edge Length |

TE | Trailing Edge Length |

R | Nose Radius |

A | Trailing Edge Angle |

A_{REF} | Trailing Edge Angle (Reference) |

## Appendix A. Parametric Section Design

_{i}are the coordinates of n + 1 control points, w

_{i}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):

${\mathit{B}}_{\mathit{i},\mathbf{3}}\left(\mathit{t}\right)$ | ${\mathit{B}}_{\mathit{i},\mathbf{4}}\left(\mathit{t}\right)$ |

${B}_{0,3}\left(t\right)={\left(1-t\right)}^{3}$ | ${B}_{0,4}\left(t\right)={\left(1-t\right)}^{4}$ |

${B}_{1,3}\left(t\right)=3\text{}t\text{}{\left(1-t\right)}^{2}$ | ${B}_{1,4}\left(t\right)=4\text{}t\text{}{\left(1-t\right)}^{3}$ |

${B}_{2,3}\left(t\right)=3\text{}{t}^{2}\text{}\left(1-t\right)$ | ${B}_{2,4}\left(t\right)=6\text{}{t}^{2}\text{}{\left(1-t\right)}^{2}$ |

${B}_{3,3}\left(t\right)={t}^{3}$ | ${B}_{3,4}\left(t\right)=4\text{}{t}^{3}\text{}\left(1-t\right)$ |

${B}_{4,4}\left(t\right)={t}^{4}$ |

_{1}of the control point P

_{1}, according to Equation (A4).

Leading Edge | |||
---|---|---|---|

Control Point | x | y | Weight |

P0 | 0 | 0 | 1 |

P1 | 0 | $\frac{\mathrm{T}}{2}$ | $\sqrt{\frac{2\text{}R\text{}{x}_{P2}\text{}{w}_{P0}\text{}{w}_{P2}}{3\text{}{y}_{P2}{}^{2}}}$ |

P2 | $\frac{2\text{}\mathrm{LE}}{3}$ | $\frac{\mathrm{T}}{2}$ | 1 |

P3 | $\mathrm{LE}$ | $\frac{\mathrm{T}}{2}$ | 1 |

Trailing Edge | |||
---|---|---|---|

Control Point | x | y | Weight |

P4 | 0 | $\frac{\mathrm{T}}{2}$ | 1 |

P5 | $\frac{{x}_{P6}}{2}$ | $\frac{\mathrm{T}}{2}$ | 1 |

P6 | $\frac{\mathrm{T}\mathrm{E}}{2.5}$ | $\frac{\mathrm{T}}{2}$ | 1 |

P7 | $\frac{2\text{}\mathrm{T}\mathrm{E}}{3}$ | $\left({x}_{P8}-{x}_{P7}\right)\text{}\mathrm{tan}\left(A\right)$ | 1 |

P8 | $\mathrm{T}\mathrm{E}$ | 0 | 1 |

_{REF}, defined by Equation (A5) for ${y}_{P7}=\frac{\mathrm{T}}{2}$, i.e., it may be computed as

_{REF}. 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.

## Appendix B. Selection of the Best Shape

_{REF}. 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.

Re 300,000 − AoA = 0.0° | Re 300,000 − CL = 0.4 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

T | LE | TE | R | A | C_{D} × 10^{3} | T | LE | TE | R | A | C_{D} × 10^{3} |

4 | 10 | 20 | 20 | $\frac{1}{5}{A}_{REF}$ | 8.46 | 4 | 10 | 20 | 5 | $\frac{4}{5}{A}_{REF}$ | 12.06 |

4 | 10 | 40 | 20 | $\frac{1}{5}{A}_{REF}$ | 6.18 | 4 | 10 | 40 | 5 | $\frac{\mathbf{1}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 11.85 |

4 | 20 | 20 | 2 | $\frac{2}{5}{A}_{REF}$ | 8.38 | 4 | 20 | 20 | 10 | $\frac{1}{5}{A}_{REF}$ | 12.28 |

4 | 20 | 40 | 10 | $\frac{\mathbf{1}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 6.05 | 4 | 20 | 40 | 10 | ${A}_{REF}$ | 13.21 |

8 | 10 | 20 | 20 | $\frac{3}{5}{A}_{REF}$ | 16.18 | 8 | 10 | 20 | 10 | $\frac{2}{5}{A}_{REF}$ | 17.78 |

8 | 10 | 40 | 20 | $\frac{\mathbf{1}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 13.52 | 8 | 10 | 40 | 10 | $\frac{4}{5}{A}_{REF}$ | 15.89 |

8 | 20 | 20 | 10 | $\frac{3}{5}{A}_{REF}$ | 20.21 | 8 | 20 | 20 | 2 | $\frac{2}{5}{A}_{REF}$ | 14.01 |

8 | 20 | 40 | 2 | $\frac{1}{5}{A}_{REF}$ | 10.64 | 8 | 20 | 40 | 5 | $\frac{\mathbf{1}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 12.29 |

Re 900,000 − AoA = 0.0° | Re 900,000 − CL = 0.4 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

T | LE | TE | R | A | C_{D} × 10^{3} | T | LE | TE | R | A | C_{D} × 10^{3} |

4 | 10 | 20 | 5 | $\frac{2}{5}{A}_{REF}$ | 5.47 | 4 | 10 | 20 | 10 | $\frac{1}{5}{A}_{REF}$ | 9.30 |

4 | 10 | 40 | 5 | $\frac{2}{5}{A}_{REF}$ | 3.71 | 4 | 10 | 40 | 10 | $\frac{\mathbf{1}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 8.36 |

4 | 20 | 20 | 2 | $\frac{2}{5}{A}_{REF}$ | 5.43 | 4 | 20 | 20 | 10 | $\frac{4}{5}{A}_{REF}$ | 10.05 |

4 | 20 | 40 | 2 | $\frac{\mathbf{2}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 3.69 | 4 | 20 | 40 | 5 | $\frac{3}{5}{A}_{REF}$ | 9.13 |

8 | 10 | 20 | 40 | $\frac{2}{5}{A}_{REF}$ | 12.14 | 8 | 10 | 20 | 10 | $\frac{3}{5}{A}_{REF}$ | 13.79 |

8 | 10 | 40 | 40 | $\frac{1}{5}{A}_{REF}$ | 10.74 | 8 | 10 | 40 | 40 | $\frac{1}{5}{A}_{REF}$ | 10.22 |

8 | 20 | 20 | 20 | $\frac{3}{5}{A}_{REF}$ | 14.28 | 8 | 20 | 20 | 10 | $\frac{3}{5}{A}_{REF}$ | 12.55 |

8 | 20 | 40 | 2 | $\frac{\mathbf{2}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 5.01 | 8 | 20 | 40 | 5 | $\frac{\mathbf{2}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 8.48 |

Re 1,500,000 − AoA = 0.0° | Re 1,500,000 − CL = 0.4 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

T | LE | TE | R | A | C_{D} × 10^{3} | T | LE | TE | R | A | C_{D} × 10^{3} |

4 | 10 | 20 | 5 | $\frac{2}{5}{A}_{REF}$ | 3.80 | 4 | 10 | 20 | 5 | $\frac{1}{5}{A}_{REF}$ | 8.71 |

4 | 10 | 40 | 2 | $\frac{3}{5}{A}_{REF}$ | 2.94 | 4 | 10 | 40 | 5 | $\frac{\mathbf{1}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 8.06 |

4 | 20 | 20 | 10 | $\frac{2}{5}{A}_{REF}$ | 3.78 | 4 | 20 | 20 | 10 | $\frac{1}{5}{A}_{REF}$ | 8.83 |

4 | 20 | 40 | 2 | $\frac{\mathbf{3}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 2.93 | 4 | 20 | 40 | 10 | $\frac{1}{5}{A}_{REF}$ | 8.17 |

8 | 10 | 20 | 40 | $\frac{2}{5}{A}_{REF}$ | 10.81 | 8 | 10 | 20 | 20 | $\frac{2}{5}{A}_{REF}$ | 10.47 |

8 | 10 | 40 | 40 | $\frac{1}{5}{A}_{REF}$ | 9.68 | 8 | 10 | 40 | 20 | $\frac{2}{5}{A}_{REF}$ | 8.50 |

8 | 20 | 20 | 20 | $\frac{2}{5}{A}_{REF}$ | 10.15 | 8 | 20 | 20 | 10 | $\frac{1}{5}{A}_{REF}$ | 10.12 |

8 | 20 | 40 | 2 | $\frac{\mathbf{3}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 3.79 | 8 | 20 | 40 | 10 | $\frac{\mathbf{2}}{\mathbf{5}}{\mathit{A}}_{\mathit{R}\mathit{E}\mathit{F}}$ | 7.48 |

_{L}= 0.4), the best computed configuration is: T = 4, LE = 10, TE = 20, R = 5, A = $\frac{4}{5}{A}_{REF}$. For the downwind condition (AoA = 0.0), the best computed configuration is: T = 4, LE = 20, TE = 20, R = 2, A = $\frac{2}{5}{A}_{REF}$. 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.

## References

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**Figure 2.**View of the mesh surrounding the test case section, and close-ups of leading and trailing edges. For the test case, the angle of attack is achieved by rotation of the foil, while for the systematic computations the direction of the inflow is varied to obtain the requested lift coefficient (0.4).

**Figure 4.**Comparison of drag coefficients for the coarse grid and a finer one. The independent variable is the trailing edge angle (see below).

**Figure 6.**Drag coefficient (C

_{D}× 10

^{3}) for Reynolds number 300,000. Independent variable: trailing edge angle (

**a**,

**b**); nose radius (

**c**,

**d**).

**Figure 7.**Drag coefficient (C

_{D}× 10

^{3}) for Reynolds number 900,000. Independent variable: trailing edge angle (

**a**,

**b**); nose radius (

**c**,

**d**).

**Figure 8.**Drag coefficient (C

_{D}× 10

^{3}) for Reynolds number 1,500,000. Independent variable: trailing edge angle (

**a**,

**b**); nose radius (

**c**,

**d**).

**Figure 9.**Plots of axial velocity (upper) and pressure distribution (lower) for a thin and a thick section.

**Figure 10.**Velocity plot and skin friction coefficient of short leading edge length (

**left**) and long leading edge length (

**right**).

**Figure 11.**Velocity plot and pressure plot of short trailing edge (

**left**) and long trailing edge (

**right**) sections.

**Figure 12.**Five different radii are compared. x-axis trailing edge angle, y-axis drag coefficient times 10

^{3}.

**Figure 13.**Velocity plot and skin friction coefficient of large (

**left**), medium (

**center**) and small (

**right**) trailing edge angle sections.

**Figure 15.**Velocity plot and skin friction coefficient of short leading edge and long trailing edge (

**left**) and long leading edge and short trailing edge (

**right**) sections.

**Figure 17.**Five different trailing edge angles are compared, x-axis nose radius, y-axis drag coefficient times 10

^{3}.

**Figure 18.**Velocity plot and pressure plot of short leading edge (

**left**) and long leading edge (

**right**) sections.

**Table 1.**Numerical uncertainty of lift and drag coefficient for the test case at different grid densities.

Nr. Cells | H_{i} | C_{L} | U (C_{L}) | U (%C_{L}) | C_{D} | U | U (%C_{D}) |
---|---|---|---|---|---|---|---|

6.5 M | 1 | 0.4354 | 0.00187 | 0.5 | 0.00850 | 0.000102 | 1.2 |

3.6 M | 1.34 | 0.4337 | 0.00411 | 0.9 | 0.00849 | 0.000060 | 0.7 |

1.6 M | 2 | 0.4346 | 0.00237 | 0.5 | 0.00848 | 0.000106 | 1.3 |

0.4 M | 4.03 | 0.4342 | 0.00606 | 1.4 | 0.00862 | 0.000301 | 3.5 |

Angle of Attack | Test Case | Abbott |
---|---|---|

α | C_{D} | C_{D} |

0.0 | 0.00344 | 0.0039 |

1.0 | 0.00348 | 0.0040 |

1.5 | 0.00352 | 0.0050 |

2.0 | 0.00362 | 0.0060 |

2.5 | 0.00640 | 0.0062 |

3.0 | 0.00670 | 0.0063 |

4.0 | 0.00770 | 0.0069 |

**Table 3.**Ranges of Reynolds numbers, thicknesses, leading edge lengths, trailing edges lengths of 470, 420 and Optimist dinghies.

Boat | Reynolds Number | Thickness (% Chord) | Leading Edge (% Chord) | Trailing Edge (% Chord) |
---|---|---|---|---|

Optimist | 280,000 | 5.52 | 21.43 | 21.43 |

290,000 | 7.14 | 20.69 | 20.69 | |

420 | 830,000 | 3.76 | 25.30 | 25.30 |

850,000 | 4.82 | 24.71 | 24.71 | |

470 | 910,000 | 4.26 | 18.08 | 35.49 |

1,500,000 | 7.89 | 11.70 | 22.97 |

Reynolds Number | Sailing Condition | Thickness (% Chord) | Leading Edge (% Chord) | Trailing Edge (% Chord) | Nose Radius (% Thickn.) | Trailing Edge Angle [°] |
---|---|---|---|---|---|---|

300,000 | AoA = 0.0° | T1 = 4% | LE1 = 10% | TE1 = 20% | R1 = 40% | A1 = A_{REF} |

900,000 | C_{L} = 0.4 | T2 = 8% | LE2 = 20% | TE2 = 40% | R2 = 20% | A2 = $\frac{4}{5}$A_{REF} |

1,500,000 | - | - | - | - | R3 = 10% | A3 = $\frac{3}{5}$A_{REF} |

- | - | - | - | - | R4 = 5% | A4 = $\frac{2}{5}$A_{REF} |

- | - | - | - | - | R5 = 2% | A5 = $\frac{1}{5}$A_{REF} |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Saporito, A.; Persson, A.; Larsson, L.; Mancuso, A.
A New Systematic Series of Foil Sections with Parallel Sides. *J. Mar. Sci. Eng.* **2020**, *8*, 677.
https://doi.org/10.3390/jmse8090677

**AMA Style**

Saporito A, Persson A, Larsson L, Mancuso A.
A New Systematic Series of Foil Sections with Parallel Sides. *Journal of Marine Science and Engineering*. 2020; 8(9):677.
https://doi.org/10.3390/jmse8090677

**Chicago/Turabian Style**

Saporito, Antonio, Adam Persson, Lars Larsson, and Antonio Mancuso.
2020. "A New Systematic Series of Foil Sections with Parallel Sides" *Journal of Marine Science and Engineering* 8, no. 9: 677.
https://doi.org/10.3390/jmse8090677