# Experimental Study of Body-Fin Interaction and Vortex Dynamics Generated by a Two Degree-Of-Freedom Fish Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model

#### 2.2. Kinematics

#### 2.3. Experimental Methods

## 3. Results

#### 3.1. Effect of the Body-Generated Vortices

#### 3.2. Leading Edge Vortex Formation

#### 3.3. Circulation Production in the Wake

#### 3.4. Effect of Nonsinusoidal Trailing Edge Kinematics

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

LE | Leading edge |

LEV | Leading edge vortex |

TE | Trailing edge |

TEV | Trailing edge vortex |

2D2C | Two-dimensional two-component |

2D3C | Two-dimensional three-component |

## References

- Lighthill, M.J. Hydromechanics of Aquatic Animal Propulsion. Annu. Rev. Fluid Mech.
**1969**, 1, 413–446. [Google Scholar] [CrossRef] - Magnuson, J.J. Locomotion by Scombrid Fishes: Hydromechanics, Morphology, and Behavior. In Fish Physiology; Hoar, W.S., Randall, D.J., Eds.; Locomotion: Shildon, UK; Academic Press: Cambridge, MA, USA, 1978; Volume 7, pp. 239–313. [Google Scholar]
- Bernal, D.; Dickson, K.A.; Shadwick, R.E.; Graham, J.B. Review: Analysis of the evolutionary convergence for high performance swimming in lamnid sharks and tunas. Comp. Biochem. Physiol. Part A Mol. Integr. Physiol.
**2001**, 129, 695–726. [Google Scholar] [CrossRef] - Koochesfahani, M.M. Vortical patterns in the wake of an oscillating airfoil. AIAA J.
**1989**, 27, 1200–1205. [Google Scholar] [CrossRef] - Triantafyllou, G.S.; Triantafyllou, M.S.; Grosenbaugh, M.A. Optimal Thrust Development in Oscillating Foils with Application to Fish Propulsion. J. Fluids Struct.
**1993**, 7, 205–224. [Google Scholar] [CrossRef] - Read, D.A.; Hover, F.S.; Triantafyllou, M.S. Forces on oscillating foils for propulsion and maneuvering. J. Fluids Struct.
**2003**, 17, 163–183. [Google Scholar] [CrossRef] - Buchholz, J.H.J.; Smits, A.J. The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel. J. Fluid Mech.
**2008**, 603, 331–365. [Google Scholar] [CrossRef] [PubMed][Green Version] - Green, M.A.; Rowley, C.W.; Smits, A.J. The unsteady three-dimensional wake produced by a trapezoidal pitching panel. J. Fluid Mech.
**2011**, 685, 117–145. [Google Scholar] [CrossRef][Green Version] - King, J.T.; Kumar, R.; Green, M.A. Experimental observations of the three-dimensional wake structures and dynamics generated by a rigid, bioinspired pitching panel. Phys. Rev. Fluids
**2018**, 3, 034701. [Google Scholar] [CrossRef] - Floryan, D.; Van Buren, T.; Rowley, C.W.; Smits, A.J. Scaling the propulsive performance of heaving and pitching foils. J. Fluid Mech.
**2017**, 822, 386–397. [Google Scholar] [CrossRef][Green Version] - Moored, K.W.; Quinn, D.B. Inviscid Scaling Laws of a Self-Propelled Pitching Airfoil. AIAA J.
**2018**, 1–15. [Google Scholar] [CrossRef] - Van Buren, T.; Floryan, D.; Wei, N.; Smits, A.J. Flow speed has little impact on propulsive characteristics of oscillating foils. Phys. Rev. Fluids
**2018**, 3, 013103. [Google Scholar] [CrossRef] - Ayancik, F.; Zhong, Q.; Quinn, D.B.; Brandes, A.; Bart-Smith, H.; Moored, K.W. Scaling laws for the propulsive performance of three-dimensional pitching propulsors. J. Fluid Mech.
**2019**, 871, 1117–1138. [Google Scholar] [CrossRef][Green Version] - Paraz, F.; Schouveiler, L.; Eloy, C. Thrust generation by a heaving flexible foil: Resonance, nonlinearities, and optimality. Phys. Fluids
**2016**, 28, 011903. [Google Scholar] [CrossRef][Green Version] - Quinn, D.B.; Lauder, G.V.; Smits, A.J. Scaling the propulsive performance of heaving flexible panels. J. Fluid Mech.
**2014**, 738, 250–267. [Google Scholar] [CrossRef] - Rosic, M.-L.N.; Thornycroft, P.J.M.; Feilich, K.L.; Lucas, K.N.; Lauder, G.V. Performance variation due to stiffness in a tuna-inspired flexible foil model. Bioinspir. Biomim.
**2017**, 12, 016011. [Google Scholar] [CrossRef] [PubMed] - Ay, M.; Korkmaz, D.; Ozmen Koca, G.; Bal, C.; Akpolat, Z.; Bingol, M. Mechatronic Design and Manufacturing of the Intelligent Robotic Fish for Bio-Inspired Swimming Modes. Electronics
**2018**, 7, 118. [Google Scholar] [CrossRef] - Coral, W.; Rossi, C.; Curet, O.M.; Castro, D. Design and assessment of a flexible fish robot actuated by shape memory alloys. Bioinspir. Biomim.
**2018**, 13, 056009. [Google Scholar] [CrossRef] - Mignano, A.; Kadapa, S.; Tangorra, J.; Lauder, G. Passing the Wake: Using Multiple Fins to Shape Forces for Swimming. Biomimetics
**2019**, 4, 23. [Google Scholar] [CrossRef] - Zhou, K.; Liu, J.; Chen, W. Study on the Hydrodynamic Performance of Typical Underwater Bionic Foils with Spanwise Flexibility. Appl. Sci.
**2017**, 7, 1120. [Google Scholar] [CrossRef] - Li, N.; Liu, H.; Su, Y. Numerical study on the hydrodynamics of thunniform bio-inspired swimming under self-propulsion. PLoS ONE
**2017**, 12, e0174740. [Google Scholar] [CrossRef] - Liu, G.; Ren, Y.; Dong, H.; Akanyeti, O.; Liao, J.C.; Lauder, G.V. Computational analysis of vortex dynamics and performance enhancement due to body–fin and fin–fin interactions in fish-like locomotion. J. Fluid Mech.
**2017**, 829, 65–88. [Google Scholar] [CrossRef] - Zhu, Q.; Wolfgang, M.J.; Yue, D.K.P.; Triantafyllou, M.S. Three-dimensional flow structures and vorticity control in fish-like swimming. J. Fluid Mech.
**2002**, 468, 1–28. [Google Scholar] [CrossRef][Green Version] - Borazjani, I.; Daghooghi, M. The fish tail motion forms an attached leading edge vortex. Proc. R. Soc. B Biol. Sci.
**2013**, 280, 20122071. [Google Scholar] [CrossRef] [PubMed][Green Version] - Maia, A.; Lauder, G.V.; Wilga, C.D. Hydrodynamic function of dorsal fins in spiny dogfish and bamboo sharks during steady swimming. J. Exp. Biol.
**2017**, 220, 3967–3975. [Google Scholar] [CrossRef] [PubMed] - Drucker, E.G.; Lauder, G.V. Locomotor function of the dorsal fin in teleost fishes: Experimental analysis of wake forces in sunfish. J. Exp. Biol.
**2001**, 204, 2943–2958. [Google Scholar] [PubMed] - Drucker, E.G.; Lauder, G.V. Locomotor function of the dorsal fin in rainbow trout: Kinematic patterns and hydrodynamic forces. J. Exp. Biol.
**2005**, 208, 4479–4494. [Google Scholar] [CrossRef] [PubMed] - Tytell, E.D. Median fin function in bluegill sunfish Lepomis macrochirus: Streamwise vortex structure during steady swimming. J. Exp. Biol.
**2006**, 209, 1516–1534. [Google Scholar] [CrossRef] [PubMed] - Akhtar, I.; Mittal, R.; Lauder, G.V.; Drucker, E. Hydrodynamics of a biologically inspired tandem flapping foil configuration. Theor. Comput. Fluid Dyn.
**2007**, 21, 155–170. [Google Scholar] [CrossRef] - Muscutt, L.E.; Weymouth, G.D.; Ganapathisubramani, B. Performance augmentation mechanism of in-line tandem flapping foils. J. Fluid Mech. Camb.
**2017**, 827, 484–505. [Google Scholar] [CrossRef][Green Version] - Broering, T.M.; Lian, Y.-S. The effect of phase angle and wing spacing on tandem flapping wings. Acta Mech Sin
**2012**, 28, 1557–1571. [Google Scholar] [CrossRef] - Dewey, P.A.; Quinn, D.B.; Boschitsch, B.M.; Smits, A.J. Propulsive performance of unsteady tandem hydrofoils in a side-by-side configuration. Phys. Fluids
**2014**, 26, 041903. [Google Scholar] [CrossRef] - Wu, T.Y.-T. Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid. J. Fluid Mech.
**1971**, 46, 337–355. [Google Scholar] [CrossRef][Green Version] - Katz, J.; Weihs, D. Hydrodynamic propulsion by large amplitude oscillation of an airfoil with chordwise flexibility. J. Fluid Mech.
**1978**, 88, 485–497. [Google Scholar] [CrossRef] - Wolfgang, M.J.; Anderson, J.M.; Grosenbaugh, M.A.; Yue, D.K.; Triantafyllou, M.S. Near-body flow dynamics in swimming fish. J. Exp. Biol.
**1999**, 202, 2303–2327. [Google Scholar] [PubMed] - Schouveiler, L.; Hover, F.S.; Triantafyllou, M.S. Performance of flapping foil propulsion. J. Fluids Struct.
**2005**, 20, 949–959. [Google Scholar] [CrossRef] - Kaya, M.; Tuncer, I.H. Nonsinusoidal Path Optimization of a Flapping Airfoil. AIAA J.
**2007**, 45, 2075–2082. [Google Scholar] [CrossRef] - Van Buren, T.; Floryan, D.; Quinn, D.; Smits, A.J. Nonsinusoidal gaits for unsteady propulsion. Phys. Rev. Fluids
**2017**, 2, 053101. [Google Scholar] [CrossRef] - Das, A.; Shukla, R.K.; Govardhan, R.N. Foil locomotion through non-sinusoidal pitching motion. J. Fluids Struct.
**2019**, 89, 191–202. [Google Scholar] [CrossRef] - Qi, Z.; Zhai, J.; Li, G.; Peng, J. Effects of non-sinusoidal pitching motion on the propulsion performance of an oscillating foil. PLoS ONE
**2019**, 14, e0218832. [Google Scholar] [CrossRef] - Idyll, C.P.; Sylva, D.D. Synopsis of Biological Data on Yellowfin Tuna Thunnus Albacares (Bonnaterre) 1788 (Western Atlantic); Species Synopsis No. 26; FAO Fisheries Biology Synopsis; FAO: Rome, Italy, 1963; pp. 771–777. [Google Scholar]
- Triantafyllou, M.S.; Triantafyllou, G.S.; Gopalkrishnan, R. Wake mechanics for thrust generation in oscillating foils. Phys. Fluids A Fluid Dyn.
**1991**, 3, 2835–2837. [Google Scholar] [CrossRef] - Van Buren, T.; Floryan, D.; Smits, A.J. Scaling and Performance of Simultaneously Heaving and Pitching Foils. AIAA J.
**2018**, 1–12. [Google Scholar] [CrossRef] - Hunt, J.C.R.; Wray, A.A.; Moin, P. Eddies, Streams, and Convergence Zones in Turbulent Flows; Center for Turbulent Research: Stanford, CA, USA, 1988. [Google Scholar]
- Buchholz, J.H.J.; Green, M.A.; Smits, A.J. Scaling the circulation shed by a pitching panel. J. Fluid Mech.
**2011**, 688, 591–601. [Google Scholar] [CrossRef][Green Version] - DeVoria, A.C.; Ringuette, M.J. Vortex formation and saturation for low-aspect-ratio rotating flat-plate fins. Exp. Fluids
**2012**, 52, 441–462. [Google Scholar] [CrossRef] - Young, J.; Lai, J.C. Mechanisms Influencing the Efficiency of Oscillating Airfoil Propulsion. AIAA J.
**2007**, 45, 1695–1702. [Google Scholar] [CrossRef] - Lu, K.; Xie, Y.H.; Zhang, D. Numerical study of large amplitude, nonsinusoidal motion and camber effects on pitching airfoil propulsion. J. Fluids Struct.
**2013**, 36, 184–194. [Google Scholar] [CrossRef]

**Figure 1.**(

**A**) Exploded view of the model showing the drive system. (

**B**) Front view of model showing dimensions. (

**C**) Side view of model showing dimensions and main components.

**Figure 3.**Tail and caudal fin angle for two representative cases. Circles represent the measured angles; the black curve is a fitted sinusoidal curve for comparison; the dotted line is at zero; and the dot-dashed line is the angular misalignment, $\Delta {\theta}_{T}$ in (

**A**) and $\Delta {\theta}_{C}$ in (

**B**). (

**A**) Tail angle, ${\theta}_{T}$, for case 4. (

**B**) Caudal fin angle, ${\theta}_{C}$, for case 1.

**Figure 4.**(

**A**) Schematic of the water tunnel and experimental setup. (

**B**) Top view of the domain relative to the model. (

**C**) Five planes where data was collected.

**Figure 5.**Five planes of spanwise vorticity data for the second kinematic group (cases 5, 6, 7, and 8). Positive spanwise vorticity is shown in red and negative in blue. The top row is at $t/T=0.32$ where the TE is moving out of the page and highlights the trailing edge vortex. The bottom row is at $t/T=0.80$ where the TE is moving into the page and highlights the leading edge vortex.

**Figure 6.**Three planes for case 8, where ${\theta}_{T,o}=3.63$ and ${\theta}_{C,o}\approx 0$, are shown here. Body (and sting) generated vortices between the body and the caudal fin are visualized using two-dimensional Q-criterion (Q) contours ($Q=[1,5,20,50]{s}^{-2}$). (

**A**) midspan plane (0 mm). (

**B**) $+20$ mm plane. (

**C**) $+40$ mm plane.

**Figure 7.**Two-dimensional Q-criterion (Q) contours ($Q=[1,5,20,50]$ s${}^{-2}$) are used to identify leading edge vortices during the second half-cycle. These plots show the typical life-cycle of a caudal fin LEV for case 7 at the $-40$ mm plane between $t/T=0.48$ and $1.40$.

**Figure 8.**Two-dimensional Q-criterion (Q) contours ($Q=[1,5,20,50]$ s${}^{-2}$) are used to identify leading edge vortices during the second half-cycle. These plots show the typical life-cycle of a caudal fin LEV for case 7 at the $-20$ mm plane between $t/T=0.48$ and $1.40$.

**Figure 9.**Two-dimensional Q-criterion (Q) contours ($Q=[1,5,20,50]$ s${}^{-2}$) are used to identify leading edge vortices during the second half-cycle. Cases 5 through 8 are shown when $t/T=0.88$. The $-40$ mm plane is shown in (

**A**–

**D**) and the $-20$ mm plane is shown in (

**E**–

**H**). The size and strength of the LEV increases with increasing maximum tail amplitude.

**Figure 10.**Rectangular region used for calculating positive circulation for case 5. Vorticity contours of ${\omega}_{z}=\pm [1,4,9,16]$ s${}^{-1}$ where positive values are shown in red and negative values in blue. The time history for this case can be seen in Figure 11B as the solid red curve. (

**A**) $t/T=-0.12$. (

**B**) $t/T=0.32$ (peak positive circulation). (

**C**) $t/T=0.64$.

**Figure 11.**Circulation magnitude of both positive (red) and negative (blue) circulation versus nondimensional time over a pitching half-cycle. (

**A**) Total same-sign circulation shed per half-cycle by kinematic group. (

**B**) Kinematic Group 1 (${\theta}_{T,o}\approx {0}^{\circ}$ and ${\theta}_{C,o}\approx {11}^{\circ}$). (

**C**) Kinematic Group 2 (${\theta}_{T,o}\approx {2}^{\circ}$ and ${\theta}_{C,o}\approx {9}^{\circ}$). (

**D**) Kinematic Group 3 (${\theta}_{T,o}\approx {3}^{\circ}$ and ${\theta}_{C,o}\approx {5}^{\circ}$). (

**E**) Kinematic Group 4 (${\theta}_{T,o}\approx 3.{6}^{\circ}$ and ${\theta}_{C,o}\approx {0}^{\circ}$).

**Figure 12.**Sample sinusoid summation showing the phase offset between the trailing edge excursion (A) and the caudal fin (C) motion: (

**A**) Motion mainly due to the tail (T). (

**B**) Motion mainly due to the caudal fin (C).

**Figure 13.**Motion Profiles for cases 1 through 4 where the solid curve represents the trailing edge amplitude and the dashed line represents the trailing edge velocity. The background colors represent time periods of acceleration (red) and deceleration (blue) the yellow circles represent the approximate timing of primary vortex shedding. (

**A**) Case 1. (

**B**) Case 2. (

**C**) Case 3. (

**D**) Case 4.

**Figure 14.**Spanwise vorticity (${\omega}_{z}=\pm [1,4,9,16]$ s${}^{-1}$) contours are shown here for cases 1 through 4 when $t/T=1.00$. Positive spanwise vorticity is shown in red and negative in blue: (

**A**) Case 1. (

**B**) Case 2. (

**C**) Case 3. (

**D**) Case 4.

**Figure 15.**(Case 1: SG1, KG1) Spanwise vorticity (${\omega}_{z}=\pm [1,4,9,16]$ s${}^{-1}$) contours are shown here for case 1 which has ${\theta}_{T,o}\approx 0.{00}^{\circ}$, ${\theta}_{C,o}=12.{90}^{\circ}$, and $St=0.31$. Positive spanwise vorticity is shown in red and negative in blue. The dashed arrow highlights the linear trajectory of the secondary vortex. The trailing edge motion profile can be found in Figure 13A: (

**A**) $t/T=0.40$. (

**B**) $t/T=0.52$. (

**C**) $t/T=0.64$. (

**D**) $t/T=0.76$. (

**E**) $t/T=0.84$. (

**F**) $t/T=1.00$.

**Figure 16.**(Case 4: SG1, KG4) Spanwise vorticity (${\omega}_{z}=\pm [1,4,9,16]$ s${}^{-1}$) contours are shown here for case 4 which has ${\theta}_{T,o}\approx 3.{44}^{\circ}$, ${\theta}_{C,o}\approx {0}^{\circ}$, and $St=0.27$. Positive spanwise vorticity is shown in red and negative in blue. The trailing edge motion profile can be found in Figure 13A: (

**A**) $t/T=0.40$. (

**B**) $t/T=0.52$. (

**C**) $t/T=0.64$. (

**D**) $t/T=0.76$. (

**E**) $t/T=0.84$. (

**F**) $t/T=1.00$.

**Figure 17.**Time averaged x-direction velocity ($U/{U}_{\infty}-1=\pm [0.05,0.10,0.15,0.20,0.30]$) contours are shown here for cases 1 through 4 where velocity surplus is red and velocity deficit is blue: (

**A**) Case 1. (

**B**) Case 2. (

**C**) Case 3. (

**D**) Case 4.

**Table 1.**Kinematic parameters for each of the cases investigated.

**SG**is the Strouhal number group and

**KG**is the kinematic group. $\pm {\theta}_{T,o}$ is the measured tail angle amplitude with a misalignment of $\Delta {\theta}_{T}$. $\pm {\theta}_{C,o}$ is the measured caudal fin angle amplitude with a misalignment of $\Delta {\theta}_{C}$. $\varphi $ is the phase offset with positive $\varphi $ representing the tail leading the caudal fin. A is the maximum excursion of the trailing edge. $St$ is the Strouhal number where $St=fA/{U}_{\infty}$.

Case | SG | KG | ${\mathit{\theta}}_{\mathit{T},\mathit{o}}{(}^{\circ})$ | $\mathbf{\Delta}{\mathit{\theta}}_{\mathit{T}}{(}^{\circ})$ | ${\mathit{\theta}}_{\mathit{C},\mathit{o}}{(}^{\circ})$ | $\mathbf{\Delta}{\mathit{\theta}}_{\mathit{C}}{(}^{\circ})$ | $\mathit{\varphi}{(}^{\circ})$ | $\mathit{A}(\mathbf{mm})$ | ${\mathit{u}}_{\mathit{\infty}}(\frac{\mathbf{mm}}{\mathit{s}})$ | $\mathit{f}({\mathit{s}}^{-1})$ | $\mathit{St}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 0.24 | 0.09 | 12.90 | −3.4 | - | 25.1 | 81.5 | 1.0 | 0.308 |

2 | 1 | 2 | 2.01 | 0.20 | 9.03 | −2.1 | 70 | 22.7 | 81.5 | 1.0 | 0.279 |

3 | 1 | 3 | 3.08 | 0.18 | 5.48 | −1.3 | 70 | 23.3 | 81.5 | 1.0 | 0.286 |

4 | 1 | 4 | 3.44 | 0.09 | 0.18 | −0.9 | - | 22.1 | 81.5 | 1.0 | 0.271 |

5 | 2 | 1 | 0.22 | −0.07 | 11.18 | −1.5 | - | 22.1 | 59.5 | 1.0 | 0.371 |

6 | 2 | 2 | 2.06 | −0.04 | 9.12 | −0.7 | 70 | 24.0 | 59.5 | 1.0 | 0.403 |

7 | 2 | 3 | 3.04 | −0.09 | 4.91 | 0.0 | 70 | 21.9 | 59.5 | 1.0 | 0.368 |

8 | 2 | 4 | 3.63 | −0.12 | 0.68 | −1.0 | - | 22.5 | 59.5 | 1.0 | 0.378 |

© 2019 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**

Brooks, S.A.; Green, M.A.
Experimental Study of Body-Fin Interaction and Vortex Dynamics Generated by a Two Degree-Of-Freedom Fish Model. *Biomimetics* **2019**, *4*, 67.
https://doi.org/10.3390/biomimetics4040067

**AMA Style**

Brooks SA, Green MA.
Experimental Study of Body-Fin Interaction and Vortex Dynamics Generated by a Two Degree-Of-Freedom Fish Model. *Biomimetics*. 2019; 4(4):67.
https://doi.org/10.3390/biomimetics4040067

**Chicago/Turabian Style**

Brooks, Seth A., and Melissa A. Green.
2019. "Experimental Study of Body-Fin Interaction and Vortex Dynamics Generated by a Two Degree-Of-Freedom Fish Model" *Biomimetics* 4, no. 4: 67.
https://doi.org/10.3390/biomimetics4040067