# 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 |

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**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 |

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**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