# Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Calculation Model

#### 2.1. Governing Equations of the Fluid Flow

#### 2.2. Numerical Model and Solution Setting

#### 2.3. Validity Investigation

## 3. Results and Discussion

#### 3.1. Analysis of Influence of $D/L$ and $h/D$ Parameters

#### 3.2. Detailed Results and Analysis of $D/L=0.3$ and $h/D=1/4$

#### 3.2.1. Aerodynamic Statistics and Frequency Characteristics

#### 3.2.2. The Mean Pressure Distribution Characteristics

#### 3.2.3. Aerodynamic Time History Analysis

#### 3.2.4. Wake Vortex Structure

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Computational mesh arrangement and domain partition, (

**a**) Computational mesh, (

**b**) Computational domain partition and local refined mesh.

**Figure 3.**Wind angle, rotation direction and pressure monitoring point layout ($h/D=1/4$, $D/L=0.3$), (

**a**) Inward rotation, (

**b**) Outward rotation, (

**c**) Clockwise co-rotation.

**Figure 4.**Comparisons of results of flow around a standard square cylinder and a single square cylinder, (

**a**) Time histories of lift and drag coefficients, (

**b**) Spectral analysis of ${C}_{l}$.

**Figure 5.**Vortex shedding in the wake of a standard square cylinder and a single square cylinder, (

**a**) Standard square cylinder, (

**b**) Single square cylinder.

**Figure 6.**Statistical values of lift and drag coefficients, Strouhal number, and control effects under different $D/L$ values, (

**a**) Statistical values of ${C}_{l},{C}_{d}$, and ${S}_{t}$, (

**b**) Control effects of ${C}_{l},{C}_{d}$.

**Figure 7.**Statistical values of lift and drag coefficients, Strouhal number, and control effects under different $h/D$ values, (

**a**) Statistical values of ${C}_{l},{C}_{d}$, and ${S}_{t}$, (

**b**) Control effects of ${C}_{l},{C}_{d}$.

**Figure 8.**Aerodynamic coefficient statistics of the square cylinder for the wind angles ($\theta $) and the velocity ratios ($k$) under inward rotation, (

**a**) ${\overline{C}}_{l}$, (

**b**) ${C}_{l}^{\prime}$, (

**c**) ${\overline{C}}_{d}$, (

**d**) ${C}_{d}^{\prime}$.

**Figure 9.**Aerodynamic coefficient statistics of the square cylinder for the wind angles ($\theta $) and the velocity ratios ($k$) under outward rotation, (

**a**) ${\overline{C}}_{l}$, (

**b**) ${C}_{l}^{\prime}$, (

**c**) ${\overline{C}}_{d}$, (

**d**) ${C}_{d}^{\prime}$.

**Figure 10.**Aerodynamic coefficient statistics of the square cylinder for the wind angles ($\theta $) and the velocity ratios ($k$) under co-rotation, (

**a**) ${\overline{C}}_{l}$, (

**b**) ${C}_{l}^{\prime}$, (

**c**) ${\overline{C}}_{d}$, (

**d**) ${C}_{d}^{\prime}$.

**Figure 11.**Amplitude spectra of Strouhal number under different wind angles with the change of velocity ratios (inward rotation), (

**a**) $k=0$, (

**b**) $k=2$, (

**c**) $k=4$.

**Figure 12.**Amplitude spectra of Strouhal number under different wind angles with the change of velocity ratios (outward rotation), (

**a**) $k=0$, (

**b**) $k=2$, (

**c**) $k=4$.

**Figure 13.**Amplitude spectra of Strouhal number under different wind angles with the change of velocity ratios (co-rotation), (

**a**) $k=0$, (

**b**) $k=2$, (

**c**) $k=4$.

**Figure 14.**Variation of Strouhal number for the velocity ratios and the wind angles under different rotation modes, (

**a**) Inward rotation, (

**b**) Outward rotation, (

**c**) Clockwise co-rotation.

**Figure 15.**Mean pressure distribution varies with wind angles and velocity ratios (inward rotation), (

**a**) $\theta ={0}^{\xb0}$, (

**b**) $\theta ={15}^{\xb0}$, (

**c**) $\theta ={30}^{\xb0}$, (

**d**) $\theta ={60}^{\xb0}$, (

**e**) $\theta ={90}^{\xb0}$, (

**f**) $\theta ={120}^{\xb0}$, (

**g**) $\theta ={150}^{\xb0}$, (

**h**) $\theta ={180}^{\xb0}$.

**Figure 16.**Mean pressure distribution varies with wind angles and velocity ratios (outward rotation), (

**a**) $\theta ={0}^{\xb0}$, (

**b**) $\theta ={30}^{\xb0}$, (

**c**) $\theta ={60}^{\xb0}$, (

**d**) $\theta ={90}^{\xb0}$, (

**e**) $\theta ={120}^{\xb0}$, (

**f**) $\theta ={150}^{\xb0}$, (

**g**) $\theta ={180}^{\xb0}$.

**Figure 17.**Mean pressure distribution varies with wind angles and velocity ratios (co-rotation), (

**a**) $\theta ={0}^{\xb0}$, (

**b**) $\theta ={30}^{\xb0}$, (

**c**) $\theta ={60}^{\xb0}$, (

**d**) $\theta ={90}^{\xb0}$, (

**e**) $\theta ={120}^{\xb0}$, (

**f**) $\theta ={150}^{\xb0}$, (

**g**) $\theta ={180}^{\xb0}$.

**Figure 18.**Lift and drag coefficient time histories of the square cylinder with different velocity ratios at $\theta ={0}^{\xb0}$ (inward rotation), (

**a**) $k=1$, (

**b**) $k=2$, (

**c**) $k=3$, (

**d**) $k=4$.

**Figure 19.**Lift and drag coefficient time histories of the square cylinder with different velocity ratios at $\theta ={15}^{\xb0}$ (inward rotation), (

**a**) $k=1$, (

**b**) $k=2$, (

**c**) $k=3$, (

**d**) $k=4$.

**Figure 20.**Comparisons of lift and drag coefficient time histories of the square cylinder while starting MSBC at different time under $\theta ={0}^{\xb0}$ and inward rotation, (

**a**) ${C}_{l}$, (

**b**) ${C}_{d}$.

**Figure 21.**Lift and drag coefficient time histories of the square cylinder with different velocity ratios at $\theta ={30}^{\xb0}$ (outward rotation), (

**a**) $k=2$, (

**b**) $k=4$.

**Figure 22.**Lift and drag coefficient time histories of the square cylinder with different velocity ratios at $\theta ={120}^{\xb0}$ (outward rotation), (

**a**) $k=2$, (

**b**) $k=4$.

**Figure 23.**Lift and drag coefficient time histories of the square cylinder with different velocity ratios at $\theta ={60}^{\xb0}$ (co-rotation), (

**a**) $k=2$, (

**b**) $k=4$.

**Figure 24.**Lift and drag coefficient time histories of the square cylinder with different velocity ratios at $\theta ={90}^{\xb0}$ (co-rotation), (

**a**) $k=2$, (

**b**) $k=4$.

**Figure 25.**Vorticity contours in the wake of the square cylinder under inward rotation mode, (

**a**) $\theta ={0}^{\xb0}$, (

**b**) $\theta ={15}^{\xb0}$.

**Figure 26.**Velocity contours and streamlines in the near wake of the square cylinder under inward rotation mode and $\theta ={0}^{\xb0}$, (

**a**) $k=0$, (

**b**) $k=1$, (

**c**) $k=2$, (

**d**) $k=3$, (

**e**) $k=4$.

**Figure 27.**Vorticity contours in the wake of the square cylinder under outward rotation mode, (

**a**) $\theta ={30}^{\xb0}$, (

**b**) $\theta ={120}^{\xb0}$.

**Figure 28.**Vorticity contours in the wake of the square cylinder under co-rotation mode, (

**a**) $\theta ={60}^{\xb0}$, (

**b**) $\theta ={90}^{\xb0}$.

**Table 1.**Effect of mesh refinement on the calculation result of the flow around a single square cylinder at $\mathrm{Re}=200$.

Mesh Density | ${\mathit{N}}_{\mathit{c}}$ | ${\mathit{N}}_{\mathit{m}\mathit{e}\mathit{s}\mathit{h}}$ | ${\mathit{N}}_{\mathit{n}\mathit{o}\mathit{d}\mathit{e}\mathit{s}}$ | ${\overline{\mathit{C}}}_{\mathit{d}}$ | ${\mathit{C}}_{\mathit{d}}^{\prime}$ | ${\mathit{C}}_{\mathit{l}}^{\prime}$ | ${\mathit{S}}_{\mathit{t}}$ |
---|---|---|---|---|---|---|---|

Scheme 1, coarsest | 40 | 30,344 | 15,387 | 1.523 | 0.024 | 0.446 | 0.146 |

Scheme 2, coarse | 55 | 36,208 | 18,349 | 1.489 | 0.022 | 0.404 | 0.148 |

Scheme 3, normal | 70 | 42,092 | 21,323 | 1.465 | 0.020 | 0.373 | 0.148 |

Scheme 4, dense | 85 | 48,590 | 24,602 | 1.456 | 0.019 | 0.358 | 0.149 |

**Table 2.**Comparison of the force coefficients statistics and Strouhal number of flow around a single square cylinder at $\mathrm{Re}=200$.

Investigation | ${\overline{\mathit{C}}}_{\mathit{d}}$ | ${\mathit{C}}_{\mathit{d}}^{\prime}$ | ${\mathit{C}}_{\mathit{l}}^{\prime}$ | ${\mathit{S}}_{\mathit{t}}$ |
---|---|---|---|---|

Okajima [41], Experimental | 1.45 | - | - | 0.14~0.148 |

Sohankar et al. [39], Numerical, 2D | 1.462 | - | 0.377 | 0.15 |

Cheng et al. [40], Numerical, 2D | 1.45 | - | 0.372 | 0.15 |

Jan and Sheu [42], Numerical, 2D | - | - | - | 0.148 |

Abograis and Alshayji [43], Numerical, 2D | 1.488 | 0.027 | 0.332 | 0.153 |

Present, Numerical, 2D (${N}_{c}=70$) | 1.465 | 0.020 | 0.373 | 0.148 |

**Table 3.**Lift and drag coefficient statistics, Strouhal number, and control effects of flow around an uncontrolled ($k=0$) and controlled ($k=2$) square cylinder with $h/D=1/4$.

$\mathit{D}/\mathit{L}$ | $\mathit{k}=0$ | $\mathit{k}=2$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{l}}^{\prime}$ | ${\overline{\mathit{C}}}_{\mathit{d}}$ | ${\mathit{C}}_{\mathit{d}}^{\prime}$ | ${\mathit{S}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{l}\_\mathit{c}}^{\prime}$ | ${\overline{\mathit{C}}}_{\mathit{d}\_\mathit{c}}$ | ${\mathit{C}}_{\mathit{d}\_\mathit{c}}^{\prime}$ | ${\mathit{S}}_{\mathit{t}\_\mathit{c}}$ | ${\mathit{E}}_{{\mathit{C}}_{\mathit{l}}}(\%)$ | ${\mathit{E}}_{{\mathit{C}}_{\mathit{d}}}(\%)$ | |

0.05 | 0.3466 | 1.2873 | 0.0186 | 0.15 | 0.2584 | 1.2255 | 0.0141 | 0.1625 | 25.45 | 4.80 |

0.1 | 0.3433 | 1.2274 | 0.0180 | 0.15 | 0.1063 | 1.0828 | 0.0051 | 0.1786 | 69.04 | 11.78 |

0.2 | 0.3309 | 1.1159 | 0.0155 | 0.15 | 0.0306 | 0.9144 | 0.0012 | 0.1961 | 90.75 | 18.06 |

0.3 | 0.3238 | 1.0145 | 0.0128 | 0.15 | 0.0089 | 0.7927 | 0.0008 | 0.2000 | 97.25 | 21.86 |

0.4 | 0.3189 | 0.9091 | 0.0093 | 0.15 | 0.0018 | 0.7027 | 0.0011 | 0.1832 | 99.44 | 22.70 |

**Table 4.**Lift and drag coefficient statistics, Strouhal number, and control effects of flow around an uncontrolled ($k=0$) and controlled ($k=2$) square cylinder with $D/L=0.3$.

$\mathit{D}/\mathit{L}$ | $\mathit{k}=0$ | $\mathit{k}=2$ | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{l}}^{\prime}$ | ${\overline{\mathit{C}}}_{\mathit{d}}$ | ${\mathit{C}}_{\mathit{d}}^{\prime}$ | ${\mathit{S}}_{\mathit{t}}$ | ${\mathit{C}}_{\mathit{l}\_\mathit{c}}^{\prime}$ | ${\overline{\mathit{C}}}_{\mathit{d}\_\mathit{c}}$ | ${\mathit{C}}_{\mathit{d}\_\mathit{c}}^{\prime}$ | ${\mathit{S}}_{\mathit{t}\_\mathit{c}}$ | ${\mathit{E}}_{{\mathit{C}}_{\mathit{l}}}(\%)$ | ${\mathit{E}}_{{\mathit{C}}_{\mathit{d}}}(\%)$ | |

$1/8$ | 0.3388 | 1.0956 | 0.0150 | 0.15 | 0.0009 | 0.7770 | 0.0039 | 0.18 | 99.73 | 29.08 |

$3/16$ | 0.3318 | 1.0534 | 0.0140 | 0.15 | 0.0032 | 0.7870 | 0.0020 | 0.20 | 99.04 | 25.29 |

$1/4$ | 0.3238 | 1.0145 | 0.0128 | 0.15 | 0.0089 | 0.7927 | 0.0008 | 0.20 | 97.25 | 21.86 |

$5/16$ | 0.3226 | 0.9821 | 0.0121 | 0.15 | 0.0181 | 0.7996 | 0.0013 | 0.20 | 94.39 | 18.58 |

$3/8$ | 0.3254 | 0.9524 | 0.0115 | 0.15 | 0.0314 | 0.8026 | 0.0006 | 0.18 | 90.35 | 15.73 |

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**MDPI and ACS Style**

Song, T.; Liu, X.; Xu, F.
Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder. *Appl. Sci.* **2022**, *12*, 1632.
https://doi.org/10.3390/app12031632

**AMA Style**

Song T, Liu X, Xu F.
Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder. *Applied Sciences*. 2022; 12(3):1632.
https://doi.org/10.3390/app12031632

**Chicago/Turabian Style**

Song, Te, Xin Liu, and Feng Xu.
2022. "Moving Surface Boundary-Layer Control on the Wake of Flow around a Square Cylinder" *Applied Sciences* 12, no. 3: 1632.
https://doi.org/10.3390/app12031632