# Numerical Investigation of the Hydrodynamics of Changing Fin Positions within a 4-Fin Surfboard Configuration

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

- In comparison to the 3-fin configuration, the 4-fin configuration is expected to be slower because more drag force is produced, and the surfer will have to apply more work to perform a turn maneuver because of the higher lift force.
- Fluid flow velocity will have a significant influence on the lift and drag forces and will therefore have an immense effect on the surfer.
- The various positions of the rear fins along the axial and transverse direction of the surfboard potentially have a massive impact on the lift and drag forces of the entire configuration, on the stall point, and consequently on the surfing.

## 2. Materials and Methods

#### 2.1. Surfing Hydrodynamics

**Front fins:**the front fins in prevailing surfing direction.**Rear fins:**the fins next to the tail of the surfboard.**Outside fins (OF):**the fins next to the shore.**Inside fins (IF):**the fins next to the wave crest.

#### 2.2. Geometric Dimensions of the Fins

^{2}. The initial distances in the transverse direction between the front fins are 330 mm and for the rear fins 170 mm; see Figure 6a. The axial distance of the rear fins to the front fins is 150 mm. The entire fin configuration is rotated around an axis to realize the AoA between the fins and the fluid flow, which is located precisely at the center point between the leading edges of the front fins; see Figure 6b.

- Both front fins are fixed.
- Just symmetric fin configurations are realized.
- The displacement of the rear fins is oriented on the positions of the intersection point board and leading edge.

#### 2.3. Simulation Region and Boundary Conditions

#### 2.4. Mesh Generation and Independence Study

^{+}between 30 and 60. Furthermore, a wake refinement of the grid downstream of the fins was performed in STAR-CCM+ to accurately resolve the increasing recirculation with increasing AoA; see Figure 9a.

_{i}) with an increasing density to identify the required grid resolution. Figure 9b shows the drag coefficient as a function of the number of cells. The deviation of the drag coefficient for meshes M

_{5}and M

_{6}ranges between 0.3% and 0.4% compared to M

_{4}. We chose that M

_{4}with approximately 1.2 million cells is the proper grid configuration to perform the 4-fin parameter study.

#### 2.5. Numerical Methods

^{+}> 30 [43]. The high-y+ model is based on the standard logarithmic law of the wall model. Depending on the AoA, the y

^{+}values for the first grid layer at the surfaces of the fins were specified to be in the range of 30 to 90. The target value for y

^{+}is 60. Therein, the first cell centers next to the wall are located in the log-layer of the turbulent boundary, which describes y

^{+}as logarithmic function of u

^{+}within the standard log-law [44]. For the transient URANS simulations, the time step size was adapted for each AoA to satisfy a Courant–Friedrichs–Lewy (CFL) condition $\mathrm{CFL}=1.0\pm 0.5$ at the fins surfaces where the mesh is finest [45]. As a time discretization scheme, a 2nd order implicit unsteady method was applied, which works for CFL > 1 [46].

_{L}and F

_{D}) itself, and the forces as non-dimensional lift and drag coefficients, as defined by [16], were determined:

## 3. Results

#### 3.1. Lift and Drag of the Commercial 4-Fin Configuration (fin Position T:80/A:150)

**C**Figure 10a shows the lift coefficient as a mean of all four fins and separately for the single fins, averaged over time, as a function of the AoA for the commercial 4-fin configuration. Regarding the IFs, the lift coefficients of both increase for increasing AoA until their maximum values at AoA = 20°. After the maximum, stall occurs and the lift coefficients decrease for both fins. As a result of this, the lift of the front IF is always larger than the lift of the rear IF due to the non-interrupted fluid flow at the front fins.

_{L:}**C**Figure 10b shows the drag coefficient versus the AoA as a mean value and separately for the single fins. The drag coefficients of both IF and the front OF increase steadily over the whole range of AoAs. In contrast, the rear OF shows a decrease in the AoA range between 20°–30°, with a subsequent increase up to a maximum located at an AoA of 45°. This drag decrease of the rear OF is forced by its position in the wake of the front OF accordingly to the lift forces. In this AoA range, the front OF shadows the rear OF from the main flow, which reduces C

_{D}:_{D}to nearly zero.

#### 3.2. Lift and Drag for Four Different u_{in} (5 m/s, 7.5 m/s, 10 m/s, and 18 m/s) at the Commercial 4-Fin Configuration

_{L}and lift force: The mean lift coefficients for the four velocities, as displayed in Figure 12a, follow the trend of the mean C

_{L}shown in Figure 10a: after an increase up to its maximum at AoA = 20°, C

_{L}first decreases until an AoA of 35°. Subsequently, it rises again up to an AoA = 45°. Between AoA = 25° and 35°, the lift coefficients for the two largest inflow velocities 10 and 18 m/s decrease faster than for the two lower velocities. The reason is that total flow separation occurs more rapidly for higher u

_{in}, owing to a larger adverse pressure gradient $\frac{dp}{dx}$ at the suction side of the fins [47].

_{in}, considering AoA = const.

_{D}and drag force: Figure 13 presents the mean drag coefficients and the corresponding forces as a function of the AoA for the four velocities. Regarding the drag coefficients in Figure 13a, its trend is also similar to the mean drag coefficient shown in Figure 10b, and they reach their maximum at AoA = 45°. Corresponding to the lift coefficients, the drag coefficient for u

_{in}= 10 and 18 m/s deviates at AoA = 25° and 30° from the two lower u

_{in}.

#### 3.3. Rear Fin Position Study: Axial vs. Transverse Shift of the Rear Fins

_{L}: Figure 14a presents the mean lift coefficients of the 4-fin configurations versus the AoA for different transverse positions of the rear fins (T:0–120) at a fixed axial position A = 150 mm. For each configuration, the primary trend increases with increasing AoA and decreases after stall occurs similar to Figure 10a. Therefore, the reached maximum lift coefficient for T:0 and T:20 is 20% and 15% larger than that for the other configurations. In the further progress up to AoA = 45°, C

_{L}either increases again (T:50–100), becomes almost constant (T:0–20), or further decreases (T:120). Furthermore, for T:0–50, the maximum lift coefficient is higher at larger AoAs than for other configurations T:80–120. The significant increase in C

_{L}is generated by the in-line position of the front and rear fins serving as one larger fin.

_{L}at AoA = 25°. Furthermore, the absolute value of the maximum C

_{L}is also very similar for all configurations.

_{D}: Figure 15 presents the mean drag coefficients C

_{D}of the entire 4-fin configurations versus the AoA for different transverse positions of the rear fins (Figure 15a) and different axial positions (Figure 15b). In general, C

_{D}increases with increasing AoA similar to the mean drag coefficient of the commercial configuration, as shown in Figure 10b.

_{D}deviates for different transverse positions. Therefore, the lowest drag is generated by the configurations T:80–120. This can be explained by the rear OFs being in the wake of the front fins at the AoA range of 25°–40°.

_{D}devolutions differ from each other, with the lowest drag seen in A:50–150. However, the configuration A:150 highly increases between AoA = 35° and 45°, whereas the configurations A:50–100 still produce the lowest drag. The reason for these low drag coefficient values is that the rear OF fin is, again, in the wake of the front fin so that the main inflow is shadowed by the front fins.

**Lift-to-drag:**Figure 16 presents the lift-to-drag ratio of the 4-fin configurations versus the AoA for different transverse (Figure 16a) and axial positions of the rear fins (Figure 16b).

## 4. Discussion

#### 4.1. Lift and Drag of the Investigated Commercial 4-Fin (T:80/A:150) Configuration

#### 4.2. Comparison of 3- and 4-Fin Configurations Using FCS Accelerator Fins from This Study and Falk et al. 2019, and FCS k2.1 Fins from Gudimetla et al. 2009

#### 4.2.1. Comparing Both 4-Fin Configurations

#### 4.2.2. Comparing the 3- and 4-Fin Configurations

#### 4.2.3. Summary of the Comparisons

#### 4.3. Lift and Drag for Four Different Inflow Velocities (5 m/s, 7.5 m/s, 10 m/s, and 18 m/s) at the Investigated 4-Fin

#### 4.4. Rear Fin Position Study: Axial vs. Transverse Shift of the Rear Fins

- The stall point is moving from an AoA of 30° (T:0) to 20° (T:120),
- the maximum of the lift coefficients decreases significantly from 0.87 (T:0) to approx. 0.7 (T:50–120), and
- the lift coefficients also decrease in the range of high AoA of 30°–45°.

- The stall point is moving from an AoA of 25° (A:0) to 20° (A:50–200),
- the maximum lift coefficient decreases slightly from 0.72 (A:0) to 0.69 (A:50–200), and
- the lift coefficients start to increase again at an AoA = 35° for the positions A:150 and A:200.

#### 4.5. Shortcomings

## 5. Conclusions

- The surfer has to apply much more work to perform a turn maneuver with a 4-fin configuration due to the higher lift forces and is losing more velocity during maneuvers because of the higher drag forces compared to a 3-fin configuration.
- The investigated surfing velocity has nearly no effect on the lift and drag coefficients, but the lift and drag forces increase nonlinearly with increasing velocity.
- At the 4-fin configuration, the front and rear fins of each side act like two large fins when the OFs and RFs are nearly aligned (T:0 A:150 and T:20 A:150) and therefore produce the highest lift forces with a later onset of force fluctuations.
- The maximum of the lift-to-drag ratio increases while shifting the rear fins in the transverse direction toward the longitudinal axis of the surfboard with a fixed axial position.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Mc Cagh, S. The Surfboard Book: How Design Drives Performance; McCagh O’Neill Pty Ltd.: Palm Beach, Australia, 2013. [Google Scholar]
- Lavery, N.; Foster, G.; Carswell, D.; Brown, S. CFD modelling of the effect of fillets on fin drag. Reef J.
**2009**, 1, 93–111. [Google Scholar] - WSL History. Available online: http://www.worldsurfleague.com/pages/history (accessed on 28 November 2018).
- Warshaw, M. The History of Surfing, 1st ed.; Chronicle Books: San Francisco, CA, USA, 2010. [Google Scholar]
- Pallis, J.; Mehta, R. Aerodynamics and hydrodynamics in sports. Eng. Sport
**2002**, 4, 31–39. [Google Scholar] - Hanna, R.K. CFD in Sport—A Retrospective; 1992–2012. Procedia Eng.
**2012**, 34, 622–627. [Google Scholar] [CrossRef][Green Version] - Azcueta, R.; Rousselon, N. CFD applied to super and mega yacht design. In Proceedings of the Design, Construction and Operation of Super and Mega Yachts Conference, Genova, Italy, 1–2 April 2009. [Google Scholar]
- Rosen, B.S.; Laiosa, J.P.; Davis, W.H. CFD design studies for America’s Cup 2000. In Proceedings of the 18th Applied Aerodynamics Conference, Denver, CO, USA, 14–17 August 2000; p. 4339. [Google Scholar]
- Paton, J. Computational Fluid Dynamics and Fluid Structure Interaction of Yacht Sails. Ph.D. Thesis, University of Nottingham, Nottingham, UK, 2011. [Google Scholar]
- Bienz, C.; Larsson, T.; Sato, T.; Ullbrand, B. In Front of the Grid—CFD at SAUBER PETRONAS F1 Leading the Aerodynamic Development. In Proceedings of the 1st European Automotive CFD Conference, Bingen, Germany, 25–26 June 2003; pp. 51–60. [Google Scholar]
- Falk, S.; Kniesburges, S.; Janka, R.; Grosso, R.; Becker, S.; Semmler, M.; Döllinger, M. Computational hydrodynamics of a typical 3-fin surfboard setup. J. Fluids Struct.
**2019**, 90, 297–314. [Google Scholar] [CrossRef] - Carswell, D.; Lavery, N.; Brown, S. Computational modelling of surfboard fins for enhanced performance. In The Engineering of Sport 6; Springer: New York, NY, USA, 2006. [Google Scholar]
- Gudimetla, P.; Kelson, N.; El-Atm, B. Analysis of the hydrodynamic performance of three- and four-fin surfboards using computational fluid dynamics. Aust. J. Mech. Eng.
**2009**, 7, 61–67. [Google Scholar] [CrossRef] - Oggiano, L.; Pierella, F. CFD for Surfboards: Comparison between Three Different Designs in Static and Maneuvering Conditions. In Proceedings of the 12th Conference of the International Sports Engineering Association, Brisbane, Australia, 26–29 March 2018; Volume 2, p. 309. [Google Scholar]
- Oggiano, L. Numerical Comparison between a Modern Surfboard and an Alaia Board using Computational Fluid Dynamics (CFD). In Proceedings of the 5th International Congress on Sport Sciences Research and Technology Suppor (icSports), Funchal, Portugal, 30–31 October 2017; pp. 75–82. [Google Scholar]
- Durst, F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows; Springer: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Argyropoulos, C.D.; Markatos, N.C. Recent advances on the numerical modelling of turbulent flows. Appl. Math. Model.
**2015**, 39, 693–732. [Google Scholar] [CrossRef] - Menter, F.R. Zonal Two Equation k-w Turbulence Models for Aerodynamic Flows. In Proceedings of the 23rd Fluid Dynamics, Plasmadynamics, and Lasers Conference, Orlando, FL, USA, 6–9 July 1993. [Google Scholar]
- Markatos, N.C. The mathematical modelling of turbulent flows. Appl. Math. Model.
**1986**, 10, 190–220. [Google Scholar] [CrossRef] - Catalano, P.; Tognaccini, R. RANS analysis of the low-Reynolds number flow around the SD7003 airfoil. Aerosp. Sci. Technol.
**2011**, 15, 615–626. [Google Scholar] [CrossRef] - Sogukpinar, H. Numerical simulation of 4-digit inclined NACA 00XX airfoils to find optimum angle of attach for airplane wing. Uludağ Univ. J. Fac. Eng.
**2017**, 22, 169. [Google Scholar] - Sogukpinar, H.; Bozkurt, I. Implementation of different turbulence model to find proper model to estimate aerodynamic properties of airfoils. In AIP Conference Proceedings; AIP Publishing: Melville, NY, USA, 2018; Volume 1935. [Google Scholar]
- Wang, C.; Xiong, Y.; Wang, G.L.; Guo, H.P. Prediction of hydrodynamic performance of hydrofoil with suction and jet equipment. Appl. Mech. Mater.
**2014**, 444–445, 432–436. [Google Scholar] [CrossRef] - Catalano, P.; Amato, M. An evaluation of RANS turbulence modelling for aerodynamic applications. Aerosp. Sci. Technol.
**2003**, 7, 493–509. [Google Scholar] [CrossRef] - Rezaei, F.; Pasandideh-Fard, M. Stall simulation of flow around an airfoil using LES model and comparison of RANS models at low angles of attack. In Proceedings of the 15th Conference On Fluid Dynamics, Bandar Abbas, Iran, 18–20 December 2013. [Google Scholar]
- Peng, S.-H.; Eliasson, P.; Davidson, L. Examination of the Shear Stress Transport Assumption with a Low-Reynolds Number k-ω Model for Aerodynamic Flows. In Proceedings of the 37th AIAA Fluid Dynamics Conference and Exhibit, Miami, FL, USA, 25–28 June 2007. [Google Scholar]
- Mohamed, M.H.; Ali, A.M.; Hafiz, A.A. CFD analysis for H-rotor Darrieus turbine as a low speed wind energy converter. Eng. Sci. Technol. Int. J.
**2015**, 18, 1–13. [Google Scholar] [CrossRef][Green Version] - Bhargav, M.M.S.R.S.; Ratna Kishore, V.; Laxman, V. Influence of fluctuating wind conditions on vertical axis wind turbine using a three dimensional CFD model. J. Wind Eng. Ind. Aerodyn.
**2016**, 158, 98–108. [Google Scholar] [CrossRef] - Lin, S.; Ma, Y.; Zheng, W.; Zhang, S.; Lei, X.; He, Y. Investigation on Rudder Hydrodynamics for 470 Class Yacht. In Proceedings of the The Conference of the International Sports Engineering Association, Brisbane, Australia, 26–29 March 2018; Volume 2, p. 308. [Google Scholar]
- Ocaña-Blanco, D.; Castañeda-Sabadell, I.; Souto-Iglesias, A. CFD and potential flow assessment of the hydrodynamics of a kitefoil. Ocean. Eng.
**2017**, 146, 388–400. [Google Scholar] [CrossRef] - Paine, M. Hydrodynamics of Surfboards. Bachelor’s Thesis, Sydney University, Sydney, Australia, 1974. [Google Scholar]
- Dally, W. The Maximum Speed of Surfers. J. Coast. Res.
**2001**, 33–40, Special Issue No. 29. [Google Scholar] - Young, I.R. Wind Generated Ocean. Waves, 2nd ed.; Elsevier: Oxford, UK, 1999. [Google Scholar]
- Sandwell, D.T. Physics of Surfing. Energetics of a Surfer. Available online: https://topex.ucsd.edu/ps/energy.pdf (accessed on 13 November 2019).
- Hendricks, T. Surfboard Hydrodynamics Part 4: Speed. Surfer Mag.
**1969**, 10, 34. [Google Scholar] - Guiness World Records: Largest Wave Surfed (Unlimited) Male. Available online: https://www.guinnessworldrecords.com/world-records/78115-largest-wave-surfed-unlimited (accessed on 11 July 2019).
- Carswell, D.J. Hydrodynamics of Surfboard Fins. Ph.D. Thesis, Swansea University, Swansea, UK, 2007. [Google Scholar]
- Scarfe, B.E.; Elwany, M.H.S.; Mead, S.T.; Black, K.P. The Science of Surfing Waves and Surfing Breaks—A Review; Scripps Institution of Oceanography: UC San Diego, CA, USA, 2003. [Google Scholar]
- Beggs-French, R.C. Surfboard Hydrodynamics. UNSW Canberra ADFA J. Undergrad. Eng. Res.
**2009**, 2. [Google Scholar] - Grosso, R. Construction of Topologically Correct and Manifold Isosurfaces. Comput. Graph. Forum
**2016**, 35, 187–196. [Google Scholar] [CrossRef] - El-Atm, B.; Kelson, N.; Gudimetla, P. A Finite Element Analysis of the Hydrodynamic Performance of 3- and 4- Fin Surfboard. In Proceedings of the 9th Global Congress on Manufacturing & Management, Gold Coast, Australia, 12–14 November 2008. [Google Scholar]
- Menter, F.R.; Kuntz, M.; Langtry, R. Ten Years of Industrial Experience with the SST Turbulence Model. Turbul. Heat Mass Transf.
**2003**, 4, 625–632. [Google Scholar] - Siemens PLM Software STAR-CCM+ Documentation Version 11.06; Siemens PLM Software Inc.: Plano, TX, USA, 2016.
- von Karman, T. Mechanical Similitude and Turbulence; Gesellschaft der Wissenschaften zu Göttingen: Göttingen, Germany, 1930. [Google Scholar]
- Lewy, H.; Friedrichs, K.; Courant, R. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann.
**1928**, 100, 32–74. [Google Scholar] - Breuer, M. Direkte Numerische Simulation und Large-Eddy Simulation Turbulenter Strömungen auf Hochleistungsrechnern; Shaker Verlag: Aachen, Germany, 2002. [Google Scholar]
- Anderson, J.D. Fundamentals of Aerodynamics, 5th ed.; McGraw-Hill Education: New York, NY, USA, 2010. [Google Scholar]

**Figure 1.**The velocity of the surfer (dashed lines) and the wave (solid lines) versus the wave height at the breaking point of the wave. The green dots show the simulated inflow velocities of 5, 7.5, 10, and 18 m/s, which were used in this study, and the corresponding wave heights of 0.9, 2.8, 6.0, and 25.6 m. The red dot shows the Guinness World Record of Rodrigo Koxa in 2017 with the confirmed wave height of 24.38 m (80 ft.). The small subfigures show corresponding waves of Hossegor (France), Pipeline (Hawaii), Mavericks (California), and Nazaré (Portugal) in order of the maximum reachable wave height.

**Figure 2.**M. Döllinger surfs a bottom turn in a left-hand wave at Uluwatu (Bali). (1) The Lip: the top of the wave. (2) The Falling Lip: the wave zone is thrown forward as the wave breaks. (3) The Pocket: the optimal energy zone for getting the maximum surfing velocity. (4) The Flat Zone: the zone in front of the broken wave, which will slow down the surfing velocity.

**Figure 3.**(

**a**) Lift ${\mathrm{F}}_{\mathrm{L}}$, drag ${\mathrm{F}}_{\mathrm{D}}$, and resultant forces ${\mathrm{F}}_{\mathrm{R}}$ applying on the fins under an angle of attack (AoA) of $\alpha =10\xb0$ (bottom view, left-hand wave). (

**b**) The surfboard, its longitudinal axis, and the fluid flow with an AoA of $\alpha =10\xb0$ (bottom view, left-hand wave).

**Figure 4.**Exemplary proceeding of CT scanning and post-processing: (

**a**) A 3-fin NSP surfboard positioned in the CT for scanning; (

**b**) post-processed 3D data: surfboard tail cutaway and remaining fins imported in STAR-CCM+.

**Figure 5.**(

**a**) Cant and Toe-In angle of the surfboard fins. (

**b**) The asymmetric and symmetric shapes of the rail fins and the center fin are visible in the bottom view of the surfboard.

**Figure 6.**(

**a**) Dimensions of the initial commercial 4-fin configuration in the axial and transverse direction with fin depth and base. (

**b**) The position of the axis of rotation to realize the AoA.

**Figure 7.**(

**a**) Definition of the transverse and axial direction for the rear fin displacement, (

**b**) ten positions of the rear fins for each side (inside fins/outside fins (IF/OF)) marked with red dots. The figure represents the configuration with rear fins positioned at a transversal position of 80 mm and an axial position of 200 mm (T:80 A:200).

**Figure 8.**The geometry and boundary conditions of the numerical simulation region with the commercial 4-fins configuration at an AoA of 45°. The surfaces of the fins are also defined as walls.

**Figure 9.**(

**a**) Wake refinement of the fins shown at a mesh plane surface at the bottom wall. (

**b**) Drag coefficient as a function of the number of control cells. Six different meshes with an increasing number of cells and a decreasing target cell volume were included in the mesh independence study.

**Figure 10.**(

**a**) Lift coefficients versus the AoA, and (

**b**) drag coefficients versus the AoA for the commercial fin position (T:80/A:150). The small figures show the flow field of two AoAs.

**Figure 11.**Governing velocity field and streamlines around the fins for (

**a**) AoA = 15°, (

**b**) AoA = 20°, and (

**c**) AoA = 25°.

**Figure 12.**(

**a**) Mean lift coefficient of the four velocities for the entire 4-fin configuration versus the AoA and (

**b**) the mean lift force of the four velocities for the entire 4-fin configuration versus the AoA.

**Figure 13.**(

**a**) Mean drag coefficient of the four velocities for the entire 4-fin configuration versus the AoA and (

**b**) the mean drag force of the four velocities for the entire 4-fin configuration versus the AoA.

**Figure 14.**(

**a**) Mean lift coefficients versus the angle of attack for six entire fin configurations. The rear fins are fixed in one axial position and shifted for six positions of the transversal direction of the surfboard. (

**b**) The mean lift coefficients versus the angle of attack for five entire fin configurations. The rear fins are fixed in one transversal position and shifted for five positions of the axial direction of the surfboard.

**Figure 15.**(

**a**) Mean drag coefficients versus the angle of attack for six entire fin configurations. The rear fins are fixed in one axial position and shifted for six positions of the transversal direction of the surfboard. (

**b**) Mean drag coefficients versus the angle of attack for five entire configurations. The rear fins are fixed in one transversal position and shifted for five positions of the axial direction of the surfboard.

**Figure 16.**(

**a**) Lift-to-drag ratio versus the angle of attack for six entire fin configurations. The rear fins are fixed in one axial position and shifted in six positions of the transversal direction of the surfboard. (

**b**) the lift-to-drag ratio versus the angle of attack for five entire fin configurations. The rear fins are fixed in one transversal position and shifted for five positions of the axial direction of the surfboard.

**Figure 17.**(

**a**) Mean lift coefficient of the entire fin configuration versus the angle of attack. (

**b**) The mean drag coefficient of the entire fin configuration versus the angle of attack. The solid lines present the commercial 3- and 4-fin configurations with the FCS Accelerator fins from this study and [11]. The dashed lines present the results of [13] with 3- and 4-fin configurations with plugged FCS k2.1 fins.

© 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**

Falk, S.; Kniesburges, S.; Janka, R.; O’Keefe, T.; Grosso, R.; Döllinger, M.
Numerical Investigation of the Hydrodynamics of Changing Fin Positions within a 4-Fin Surfboard Configuration. *Appl. Sci.* **2020**, *10*, 816.
https://doi.org/10.3390/app10030816

**AMA Style**

Falk S, Kniesburges S, Janka R, O’Keefe T, Grosso R, Döllinger M.
Numerical Investigation of the Hydrodynamics of Changing Fin Positions within a 4-Fin Surfboard Configuration. *Applied Sciences*. 2020; 10(3):816.
https://doi.org/10.3390/app10030816

**Chicago/Turabian Style**

Falk, Sebastian, Stefan Kniesburges, Rolf Janka, Tom O’Keefe, Roberto Grosso, and Michael Döllinger.
2020. "Numerical Investigation of the Hydrodynamics of Changing Fin Positions within a 4-Fin Surfboard Configuration" *Applied Sciences* 10, no. 3: 816.
https://doi.org/10.3390/app10030816