# Multi-Scale Wake Characteristics of the Flow over a Cylinder with Different V-Groove Numbers

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Methods

#### 2.1. Experimental Models

#### 2.2. High-Speed PIV Measurement

^{3}/h is the flow quantity and S = 0.12 m

^{2}is the area of the cross-section). The free-stream velocity was fixed (${U}_{0}=0.37\text{}\mathrm{m}/\mathrm{s}$) in the PIV experiment corresponding to a Reynolds number of Re = 7400 based on the experimental model.

#### 2.3. Orthogonal Wavelet Multi-Resolution Procedure

## 3. Experimental Results and Discussion

#### 3.1. Profiles of Mean Velocity

#### 3.2. Time-Averaged Streamlines and Recirculation Region

#### 3.3. Reynolds Stress and Turbulent Kinetic Energy

#### 3.4. Instantaneous Flow Structures

#### 3.5. Select Wavelet Components Based on Relative Energy Percentage

#### 3.6. Spectral Analysis Calculation of Vortex Frequency

#### 3.7. Instantaneous Vorticity Contours of Different Scales

#### 3.8. Reynold Shear Stress Contour of Different Scales

## 4. Conclusions

- (1)
- The results of the time-averaged velocity profile showed that the v-groove can reduce the width of the reflux section and the velocity gradient. The v-groove can reduce the recirculation region, increase the Reynolds shear stress and the turbulent kinetic energy, and prevent the formation of Karman-like vortices. The number of grooves does not affect the recirculation region.
- (2)
- When comparing the smooth and 32-groove cylinders at different scales of vorticity, it was observed that at the large scale, the Karman-like vortices of the grooves were closer to the trailing edge of the cylinder. At the intermediate scale, there were more vortices behind the 32-groove cylinder, whereas at the small scale, strong vortex oscillations were observed behind the grooved cylinder.
- (3)
- When comparing the smooth and 32-groove cylinders at different shear stress scales, it was confirmed that at the large scale, the v-groove surface can reduce the recirculation region. At the intermediate and small scales, the shear layer instability creates vortices, increasing the turbulent kinetic energy and narrowing the wake region.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

d | the diameter of the smooth cylinder | ${S}_{1}$ | the low-frequency coefficients |

${f}_{0}$ | the center frequency | s | the area of the cross-section |

h | the height of the v-groove | st | the Strouhal number |

L | the length of the smooth cylinder | TKE | the turbulent kinetic energy |

${L}_{x}$ | the length of two vortices | $TK{E}_{max}$ | the maximum turbulent kinetic energy |

${L}_{y}$ | the width between the two foci points | ${U}_{0}$ | the flow velocity |

${M}^{N}$ | the wavelet basis matrix | $u$ | the horizontal velocities |

p | the distance between two measured points | ${u}_{l}^{\prime}$ | the flow-fluctuating velocities at different levels |

${P}^{N}$ | the permutation matrices | ${u}^{\prime N}$ | the fluctuation velocity |

Q | the quantity of flow | $\overline{u}/{U}_{0}$ | the normalized time-averaged stream-wise velocity |

Re | the Reynolds number | $\overline{{u}^{\prime}{w}^{\prime}}/{U}_{0}^{2}$ | the normalized Reynolds stress |

${R}_{l}$ | the spatial correlation coefficient | $v$ | the longitudinal velocities |

${{\displaystyle \int}}_{0}^{{p}_{max}}{R}_{l}\left(p\right)dp$ | the integral length scale | $W$ | the wavelet analysis matrix |

${r}_{1,2}^{s,g}$ | the relative energy | $w$ | the wall-normal velocities |

${R}_{l}\left(p\right)=0$ | the length from the reference point to the space point | ${\omega}_{z}d/{U}_{0}$ | the normalized vorticity |

S | the wavelet coefficient matrix | ${y}_{0}$ | the referenced point |

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**Figure 1.**Experimental models: (

**a**) smooth cylinder, (

**b**) 12-groove cylinder, (

**c**) 24-groove cylinder, (

**d**) 32-groove cylinder.

**Figure 13.**Relative energy ratio of eight wavelet levels in (x, y)- and (x, z)-plane. (a) Smooth cylinder in the (x, y)-plane (${r}_{1}^{s}$), (b) 32-groove cylinder in the (x, y)-plane (${r}_{1}^{g}$), (c) smooth cylinder in the (x, z)-plane (${r}_{2}^{S}$), (d) 32-groove cylinder in the (x, z)-plane (${r}_{2}^{g}$).

**Figure 14.**Power spectra of the measured and the three wavelet components at the points x/d = 0.6 and y/d = 0.5.

**Figure 15.**Multi-scale instantaneous streamlines and vorticity contours of the smooth and 32-groove cylinders in the (x, y)-plane.

**Figure 16.**Multi-scale instantaneous streamlines and vorticity contours of the smooth and 32-groove cylinders in the (x, z)-plane.

**Figure 17.**Multi-scale shear stress distributions of the smooth and 32-groove cylinders in the (x, y)-plane.

**Figure 18.**Multi-scale shear stress distributions of the smooth and 32-groove cylinder in the (x, z)-plane.

Authors | h/d | N | Re |
---|---|---|---|

Ying [5] | 0.07 | 24 | 1500 |

Yamagishi [6] | 0.01 | 20, 26, 32 | ${10}^{4}-{10}^{5}$ |

Quintavalla [7] | 0.0017, 0.0033 | 45, 31, 21 | $2\times {10}^{4}-1.2\times {10}^{4}$ |

Qi [10] | 0.02, 0.03, 0.04, 0.05 | 16 | $3\times {10}^{4}$ |

Talley [12] | 0.035, 0.07, 0.105 | 24 | $9\times {10}^{4}-2\times {10}^{5}$ |

Fujisawa [13] | 0.01, 0.017 | 35 | $2\times {10}^{4}-6\times {10}^{4}$ |

Present study | 0.05 | 12, 24, 32 | $3\times {10}^{4}$ |

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**MDPI and ACS Style**

Jiang, S.; Yan, F.; Zhang, J.; Song, B.
Multi-Scale Wake Characteristics of the Flow over a Cylinder with Different V-Groove Numbers. *Water* **2023**, *15*, 805.
https://doi.org/10.3390/w15040805

**AMA Style**

Jiang S, Yan F, Zhang J, Song B.
Multi-Scale Wake Characteristics of the Flow over a Cylinder with Different V-Groove Numbers. *Water*. 2023; 15(4):805.
https://doi.org/10.3390/w15040805

**Chicago/Turabian Style**

Jiang, Suyu, Fei Yan, Jian Zhang, and Bo Song.
2023. "Multi-Scale Wake Characteristics of the Flow over a Cylinder with Different V-Groove Numbers" *Water* 15, no. 4: 805.
https://doi.org/10.3390/w15040805