# Comparative Analysis of the Hydrodynamic Performance of Dual Flapping Foils with In-Phase and Out-of-Phase Oscillations

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## Abstract

**:**

## 1. Introduction

## 2. Motion Model

#### 2.1. Motion Description

#### 2.2. Mechanical Parameters and Pumping Indicators

## 3. Numerical Method and Validation

#### 3.1. Governing Equation and Turbulence Model’

#### 3.2. Computational Domain and Mesh Generation

^{+}values. To enhance the accurate representation of the dynamic wall characteristics and mitigate numerical dispersion, it is imperative to build boundary layers on both sides of the flow channel and hydrofoil wall. Hence, the initial mesh size of the boundary layer adjacent to the hydrofoil surface is established at 0.03 mm, satisfying Y

^{+}< 1 and a growth rate of 1.2.

#### 3.3. Boundary Conditions and Parameter Settings

#### 3.4. Verification of the Irrelevance of the Time Steps and Grid Number

#### 3.5. Validation

## 4. Experimental Setup

#### 4.1. Bionic Pumping Device

#### 4.2. Uncertainty Analysis

## 5. Results and Analysis

#### 5.1. Influence of Two Oscillation Modes on Mechanical Properties

#### 5.2. Influence of Two Oscillation Modes on the Flow Field

#### 5.3. Influence of Two Oscillation Modes on the Pumping Performance

^{3}/s, namely the hump phenomenon. As shown in Figure 15, the graph shows that at a flow rate of 0.84 m

^{3}/s, the internal flow field under the corresponding operating conditions shows strong instability, which poses a significant threat to the safety, stability, and reliability of the bionic pumping device. Previous studies show that the humping phenomenon leads to the instability of the device [34,35,36,37]. Hence, the pumping device with the out-of-phase oscillation is more reliable compared with the in-phase oscillation. At the same time, the maximum head is smaller when the dual flapping foil is used to push water in the out-of-phase oscillation, which can meet the demand of ultra-low head conditions.

#### 5.4. Performance Test

## 6. Conclusions

- (1)
- The mechanical properties of the two oscillation modes exhibit notable distinctions. Out-of-phase oscillation consistently produces thrust throughout a motion cycle, whereas in-phase oscillation generates both thrust and drag forces. Furthermore, under the instantaneous lift coefficient curve, the out-of-phase oscillation is more symmetrical than the in-phase oscillation, indicating that the out-of-phase oscillation’s flow field is more uniform.
- (2)
- The form of the tail vortex structure is a crucial determinant affecting the hydraulic performance of the dual flapping foil. There is a significant difference in the tail vortex structure between the two oscillation modes, with in-phase oscillation forming a pair of vortex streets and out-of-phase oscillation forming two pairs of vortex streets. Furthermore, it influences the flow field, whereby in-phase oscillation results in the formation of a single straight jet, while out-of-phase oscillation led to the formation of two parallel straight jets.
- (3)
- Both oscillations have propulsive effects on the water body. The pumping efficiency of the out-of-phase oscillation is greater than that of the in-phase oscillation. Specifically, with the oscillation frequency f = 1 Hz, the pumping efficiency of the out-of-phase oscillation reaches 38.4%, which is 90.5% greater than that of the in-phase oscillation. Furthermore, it should be noted that the out-of-phase oscillation results in a greater outlet flow, a more uniform flow field structure, and a superior pumping effect. Experimental verification has demonstrated that the out-of-phase oscillation yields a greater outlet flow rate when compared to the in-phase oscillation.
- (4)
- The calculation results show that the dual flapping foil and conventional pumps have similar characteristic curves. However, in the flow rate range of 0.64~1 m
^{3}/s, the characteristic curve of the dual flapping foils with in-phase oscillation reveals an “S” type unstable oscillation phenomenon, namely the hump phenomenon, which will lead to the instability of the device. In contrast, out-of-phase oscillation does not exhibit this phenomenon, effectively extending its application range. In addition, the out-of-phase oscillating hydrofoil has a reduced applicable head, allowing it to better meet the requirements of ultra-low head conditions.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Position relationship of dual-oscillating hydrofoils over one cycle: (

**a**) out-of-phase oscillation; (

**b**) in-phase oscillation.

**Figure 3.**Computational domain mesh and boundary conditions: (

**a**) grid overview; (

**b**) leading edge grid; (

**c**) trailing-edge grid.

**Figure 5.**Comparison between the numerical simulation outcomes and the experimental data in the previous study [33].

**Figure 6.**Three-dimensional schematic diagram of flapping hydrofoil device: 1. synchronous belt linear module; 2. motor; 3. coupling; 4. support frame; 5. connecting rod; 6. foil.

**Figure 7.**(

**a**) Dual flapping foil device test bench; (

**b**) internal schematic diagram of the experimental flow passage.

**Figure 8.**Variation in the instantaneous thrust and lift coefficients for two oscillation modes: (

**a**) instantaneous thrust coefficients; (

**b**) instantaneous lift coefficients.

**Figure 9.**Velocity nephograms of two oscillation modes: (

**a**) in-phase oscillation; (

**b**) out-of-phase oscillation.

**Figure 10.**Characteristic diagram of the instantaneous flow field in the out-of-phase oscillation mode (t/T = 3/4).

**Figure 11.**Vorticity nephogram of two oscillation modes during one cycle: (

**a**) in-phase oscillation; (

**b**) out-of-phase oscillation.

**Figure 14.**Pump efficiency and head characteristic curves of two oscillation modes: (

**a**) out-of-phase oscillation; (

**b**) in-phase oscillation.

**Figure 15.**Vorticity nephogram of in-phase oscillation mode at a flow rate of 0.84 m

^{3}/s (t/T = 0).

Terms | Equipment | Type | Systematic Error |
---|---|---|---|

Flow | Doppler flowmeter | WIM-@ADV | ±1% |

Head | Differential pressure sensor | 3051 | ±0.2% |

Current | Clamp power meter | VC6412D | ±2.5% |

f/Hz | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | |
---|---|---|---|---|---|---|---|---|---|

$\mathrm{v}/\mathrm{m}\cdot {\mathrm{s}}^{-1}$ | Expeiment 1 | 0.020 | 0.047 | 0.070 | 0.097 | 0.120 | 0.152 | 0.172 | 0.201 |

Expeiment 2 | 0.021 | 0.045 | 0.075 | 0.095 | 0.125 | 0.151 | 0.170 | 0.207 | |

Expeiment 3 | 0.025 | 0.050 | 0.068 | 0.096 | 0.123 | 0.147 | 0.175 | 0.205 |

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**MDPI and ACS Style**

Hua, E.; Qiu, L.; Xie, R.; Su, Z.; Zhu, W.
Comparative Analysis of the Hydrodynamic Performance of Dual Flapping Foils with In-Phase and Out-of-Phase Oscillations. *Water* **2023**, *15*, 3275.
https://doi.org/10.3390/w15183275

**AMA Style**

Hua E, Qiu L, Xie R, Su Z, Zhu W.
Comparative Analysis of the Hydrodynamic Performance of Dual Flapping Foils with In-Phase and Out-of-Phase Oscillations. *Water*. 2023; 15(18):3275.
https://doi.org/10.3390/w15183275

**Chicago/Turabian Style**

Hua, Ertian, Linfeng Qiu, Rongsheng Xie, Zhongxin Su, and Wenchao Zhu.
2023. "Comparative Analysis of the Hydrodynamic Performance of Dual Flapping Foils with In-Phase and Out-of-Phase Oscillations" *Water* 15, no. 18: 3275.
https://doi.org/10.3390/w15183275