# Analysis of Flow Characteristics around a Square Cylinder with Boundary Constraint

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation Method for Flow around a Square Cylinder

#### 2.1. Mathematical Model

#### 2.2. Computational Domain and Boundary Conditions

#### 2.3. Computational Grids

#### 2.4. Model Analysis

## 3. Characteristics of the Flow around a Square Cylinder under Boundary Constraint

#### 3.1. Feature of Wake Flow Patterns of the Flow around a Square Cylinder

#### 3.2. Boundary Constraint Effect in Vortex Street Flow

#### 3.3. Fluid Characteristics of the Flow around a Square Cylinder

#### 3.4. Flow Pattern Division of Flow around a Square Cylinder

## 4. Conclusions

- (1)
- The flow pattern around a square cylinder can be divided into four types under different Reynolds numbers: L1 (stable attached vortex phase), L2 (unstable attached vortex phase), L3 (laminar vortex street phase), and T (turbulence phase). The boundary constraint does not change the flow pattern around a square cylinder.
- (2)
- In the flow pattern of a laminar vortex street, the boundary constraint squeezes the vortex shapes, pushes the vortex centres to the channel boundaries, and reduces the distance between the vortices along the flow direction, but the boundary constraint cannot change the attenuation law and magnitude of the vorticity along the flow direction.
- (3)
- When the constraint degree of boundaries exceeds a specific level (G/D < 3.5), the time-averaged drag coefficient of the cylinder and the shedding frequency of the attached vortices can increase significantly.
- (4)
- The variation law of the drag coefficient of the flow around a square cylinder and the shedding frequency of attached vortices with the Reynolds number will not be affected by the boundary constraint degree.
- (5)
- The boundary constraint rectifies the wake flow pattern around a square cylinder, and the appearance of turbulence requires a larger Reynolds number. With the increase of boundary constraint degree, the expansion of the of Reynolds number range corresponding to the flow pattern of the laminar vortex street is the most significant.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Computational grids distribution of the computational domain; (

**a**) total computational grids distribution; (

**b**) detailed computational grids distribution close to the square cylinder.

**Figure 4.**Different classifications of flow distribution around a square cylinder; (

**a**) Classification L1 (${R}_{e}$ = 40); (

**b**) Classification L2 (${R}_{e}$ = 95); (

**c**) Classification L3 (${R}_{e}$ = 400); and (

**d**) Classification T (${R}_{e}$ = 1000).

**Figure 5.**Flow pattern around a square cylinder under different boundary constraint degrees when ${R}_{e}=400$. (

**a**) G/D = 5.5; (

**b**) G/D = 3.5; (

**c**) G/D = 2.5; (

**d**) G/D = 1.5; (

**e**) G/D = 1.0; and (

**f**) G/D = 0.5.

**Figure 6.**Trajectory of the vortex centres of the flow around a square cylinder under different boundary constraint degrees when ${R}_{e}=400$. (

**a**) G/D = 5.5; (

**b**) G/D = 3.5; (

**c**) G/D = 2.5; (

**d**) G/D = 1.5; (

**e**) G/D = 1.0; and (

**f**) G/D = 0.5.

**Figure 7.**Average distance of the vortex centres of the flow around a square cylinder $\overline{{d}_{x}}$ along the flow direction under different boundary constraint degrees when ${R}_{e}=400$.

**Figure 8.**Variation of the vorticity of the flow around a square cylinder along the flow direction under different boundary constraint degrees.

**Figure 9.**Relationship between ${\mathsf{\Omega}}_{1}/{\mathsf{\Omega}}_{0}$ and G/D in the vortex street flow pattern.

**Figure 10.**Relationship of $\overline{{C}_{d}}$ nd G/D, ${R}_{e}$. (

**a**) $ln\overline{{C}_{d}}$ versus $lnRe$ and $G/D$; (

**b**) $\overline{{C}_{d}}$ versus $G/D$; and (

**c**) $ln\overline{{C}_{d}}$ versus $lnRe$.

**Figure 11.**Relationship of $\overline{{C}_{d}}$ and G/D, ${R}_{e}$. (

**a**) ${S}_{t}$ versus $Re$ and $G/D$; (

**b**) ${S}_{t}$ versus $G/D$; and (

**c**) ${S}_{t}$ versus $Re$.

**Figure 12.**Flow pattern classification map of the flow around a square cylinder under boundary constraint.

Flow Pattern Classification | Flow Characteristics | |
---|---|---|

L1 | Stable attached vortex phase | Stable and symmetrical attached vortices formed by the separated flow behind a cylinder (Taneda, 1956 [4]; Coutanceau and Bouard, 1977 [5]). |

L2 | Unstable attached vortex phase | The attached vortices behind a cylinder are asymmetric, fluctuating sinusoidally. |

L3 | Laminar vortex street phase | Attached vortex shedding appearing behind a cylinder and forming periodic laminar vortex streets (Thoman and Szewczyk, 1969 [6]; Collins and Dennis, 1973 [7]). |

T | Turbulence phase | Vortex streets are unstable, attached vortex shedding appearing non-sinusoidally, and turbulence is formed behind a cylinder. |

Zone | Grid Length | Grid Width |
---|---|---|

1 | $3\delta $ | $3\delta $ |

2 | $\delta $ | $3\delta $ |

3 | $3\delta $ | $3\delta $ |

4 | $3\delta $ | $\delta $ |

5 | $\delta $ | $\delta $ |

6 | $3\delta $ | $\delta $ |

7 | $3\delta $ | $3\delta $ |

8 | $\delta $ | $3\delta $ |

9 | $3\delta $ | $3\delta $ |

**Table 3.**Comparison of $\overline{{C}_{d}}$ and ${S}_{t}$ of the flow around a square cylinder under different grid resolutions.

Grids | $\mathit{\delta}/\mathit{D}$ | Total Elements | Re = 40 | Re = 1000 | |
---|---|---|---|---|---|

$\overline{{\mathit{C}}_{\mathit{d}}}$ | $\overline{{\mathit{C}}_{\mathit{d}}}$ | ${\mathit{S}}_{\mathit{t}}$ | |||

G1 | 0.125 | 5444 | 1.49 | 1.88 | 0.117 |

G2 | 0.075 | 15,194 | 1.54 | 2.02 | 0.120 |

G3 | 0.050 | 33,636 | 1.59 | 2.17 | 0.121 |

G4 | 0.025 | 134,544 | 1.64 | 2.29 | 0.121 |

G5 | 0.0125 | 535,076 | 1.67 | 2.39 | 0.122 |

G6 | 0.0075 | 1,487,034 | 1.69 | 2.42 | 0.122 |

**Table 4.**Comparison of $\overline{{C}_{d}}$ and ${S}_{t}$ of the flow around a square cylinder under different ${R}_{e}$ values between the present and earlier results.

${\mathit{R}}_{\mathit{e}}$ | Source | $\overline{{\mathit{C}}_{\mathit{d}}}$ | ${\mathit{S}}_{\mathit{t}}$ |
---|---|---|---|

40 | Sen et al. [15] | 1.67 | - |

Lan et al. [49] | 1.72 | - | |

Present | 1.67 | - | |

200 | Lan et al. [49] | 1.49 | 0.143 |

Gera et al. [50] | - | 0.145 | |

Okajima [51] | 1.50 | 0.141 | |

Present | 1.49 | 0.141 | |

1000 | Lan et al. [49] | 2.23 | 0.122 |

Okajima [51] | 2.10 | 0.120~0.130 | |

Wang [52] | 2.40 | 0.124 | |

Present | 2.39 | 0.122 | |

22,000 | Trias et al. [33] | 2.19 | 0.132 |

Lyn [53] | 2.18 | 0.134 | |

Bouris et al. [54] | 2.10 | 0.135 | |

Fureby et al. [55] | 2.11–2.30 | 0.126~0.138 | |

Cao et al. [56] | - | 0.130 | |

Bearman et al. [57] | 2.05 | 0.122 | |

Lee [58] | 2.18 | 0.132 | |

Present | 2.06 | 0.124 |

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**MDPI and ACS Style**

Xu, Z.; Wu, S.; Wu, X.; Xue, W.; Wang, F.; Gao, A.; Zhang, W.
Analysis of Flow Characteristics around a Square Cylinder with Boundary Constraint. *Water* **2023**, *15*, 1507.
https://doi.org/10.3390/w15081507

**AMA Style**

Xu Z, Wu S, Wu X, Xue W, Wang F, Gao A, Zhang W.
Analysis of Flow Characteristics around a Square Cylinder with Boundary Constraint. *Water*. 2023; 15(8):1507.
https://doi.org/10.3390/w15081507

**Chicago/Turabian Style**

Xu, Zhun, Shiqiang Wu, Xiufeng Wu, Wanyun Xue, Fangfang Wang, Ang Gao, and Weile Zhang.
2023. "Analysis of Flow Characteristics around a Square Cylinder with Boundary Constraint" *Water* 15, no. 8: 1507.
https://doi.org/10.3390/w15081507