# The Ground Effect in Anguilliform Swimming

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Fish Body Kinematics and Non-Dimensional Parameters

#### 2.2. The Numerical Method

#### 2.3. Computational Details

#### 2.4. Calculation of Hydrodynamic Forces and Swimming Efficiency

## 3. Results and Discussion

#### 3.1. Effects of Ground Swimming Performance

#### 3.2. Flow Field

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a | amplitude envelope of lateral motion |

${a}_{max}$ | maximum amplitude |

A | maximum lateral excursion of the tail over a cycle |

${C}_{f}$ | force coefficient |

${C}_{p}$ | pressure force coefficient |

${\overline{C}}_{f}$ | mean axial force coefficient |

${C}_{v}$ | viscous force coefficient |

D | drag |

f | tail-beat frequency |

F | force exerted on the fish body by the flow |

h | lateral excursion of the body |

$\dot{h}$ | velocity of the lateral undulations |

k | wave number of the body undulations |

L | fish length |

PIV | particle image velocimetry |

${\overline{P}}_{side}$ | mean power loss |

$Re$ | Reynolds number |

$St$ | Strouhal number |

t | time |

T | thrust |

$\overline{T}$ | mean thrust force |

U | swimming speed |

V | speed of the backward undulatory body wave |

z | axial flow direction measured along the fish axis from the tip of the fish’s snout |

$\beta $ | slip velocity |

$\eta $ | Froude propulsive efficiency |

$\lambda $ | wavelength |

$\nu $ | kinematic viscosity of water |

$\rho $ | fluid density |

$\tau $ | tail-beat period |

${\tau}_{ij}$ | viscous stress |

$\omega $ | angular frequency |

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**Figure 1.**Eel geometry within computational grid; (

**a**) 3D view, (

**b**) two dimensional view with ground. Every other grid point is shown for better visualization.

**Figure 2.**Time history of the non-dimensional velocity of the anguilliform swimmer for near ground and free swimming.

**Figure 3.**Time history of the pressure force (${C}_{p}$) and viscous force coefficient (${C}_{v}$) of the anguilliform swimmer swimming for distances: (a) close to the ground (b) at the center; during steady swimming. Positive values indicate thrust force, and negative values indicate drag force.

**Figure 4.**Three-dimensional (3D) vortical structures visualized by the isosurface of q-criterion (Q = 40) showing 3-D wake structures of the anguilliform swimmer swimming (

**a**) near ground, and (

**b**) freely.

**Figure 5.**The top view of the instantaneous velocity vectors with vorticity contours of the anguilliform swimmer swimming (

**a**) near ground, and (

**b**) freely. Only every third vector is plotted for clarity.

**Figure 6.**The side view of the instantaneous velocity vectors with vorticity contours of the anguilliform swimmer swimming (

**a**) near ground, and (

**b**) freely. The brown thick line represents the ground. Only every third vector is plotted for clarity.

**Table 1.**Calculated swimming performances for the anguilliform mode of fish swimming during steady swimming using the computational fluid dynamics (CFD) method. $\eta $ is the swimming efficiency, ${\overline{C}}_{T}$ is the thrust coefficient, and ${\overline{C}}_{{p}_{side}}$ is the power coefficient.

Swimming Conditions | $\mathit{\eta}$ | ${\overline{\mathit{C}}}_{\mathit{T}}$ | ${\overline{\mathit{C}}}_{{\mathit{p}}_{\mathbf{s}\mathbf{i}\mathbf{d}\mathbf{e}}}$ |
---|---|---|---|

Near ground swimming | 19.92% | $3.77\times {10}^{-4}$ | $1.50\times {10}^{-3}$ |

Free swimming | 20.73% | $3.78\times {10}^{-4}$ | $1.50\times {10}^{-3}$ |

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**MDPI and ACS Style**

Ogunka, U.E.; Daghooghi, M.; Akbarzadeh, A.M.; Borazjani, I.
The Ground Effect in Anguilliform Swimming. *Biomimetics* **2020**, *5*, 9.
https://doi.org/10.3390/biomimetics5010009

**AMA Style**

Ogunka UE, Daghooghi M, Akbarzadeh AM, Borazjani I.
The Ground Effect in Anguilliform Swimming. *Biomimetics*. 2020; 5(1):9.
https://doi.org/10.3390/biomimetics5010009

**Chicago/Turabian Style**

Ogunka, Uchenna E., Mohsen Daghooghi, Amir M. Akbarzadeh, and Iman Borazjani.
2020. "The Ground Effect in Anguilliform Swimming" *Biomimetics* 5, no. 1: 9.
https://doi.org/10.3390/biomimetics5010009