# Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Simulation Configurations

#### 2.1. Numerical Method

**U**and P are the averaged velocity and pressure, respectively; $\rho $ shows air density and $\nu $ refers to eddy viscosity; $\tau $ is the Reynolds stress which can be expressed as ${\tau}_{ij}={\nu}_{t}(\partial {u}_{i}/\partial {x}_{j}+\partial {u}_{j}/\partial {x}_{i})-2/3\rho k{\delta}_{ij}$ and ${\delta}_{ij}$ is the Kronecker delta.

_{1}:

^{−6}s. In terms of time step selection, to assess how the size of the time steps may affect the results, three time steps of 2 × 10

^{−5}, 1.0 × 10

^{−6}, and 1.0 × 10

^{−7}s are tested. At last, 1.0 × 10

^{−6}s is selected to balance the simulation time and the result accuracy. For all simulations, more than 1000 nondimensional time steps, corresponding to approximately 50 vortex-shedding cycles, which is much larger than the number of vortex-shedding cycles adopted in the previous studies [20,21], are taken for assuring reliable results.

#### 2.2. Computational Configuration and Mesh Arrangement

^{4}, which is consistent with that in [22]. At the outlet boundary, a zero static pressure is adopted ($p={p}_{0}=1.013\times {10}^{5}$, $\partial u/\partial n=0$). Two lateral boundaries are defined as a symmetry boundary condition ($v=0$, $\partial u/\partial n=0$), which is used to model zero-shear slip walls in viscous flows. An empty condition is enforced on the top and bottom surfaces of the computational domain, which is an ordinary method to solve the 2D problem in OpenFOAM. Two vertical-axis wind turbines are installed at two leading corners of the square cylinder and the nonslip wall boundary condition ($ui=0$, $\partial p/\partial n=0$) is applied to surfaces of the cylinder and the wind turbines, where $n$ refers to the normal direction, $u$ and $v$ represent the velocity components in x and y directions, respectively.

#### 2.3. Grid Size and Time Step Configuration

_{d}and F

_{l}are the drag and lift forces of the cylinder, respectively; f

_{s}is the vortex shedding frequency; W is the width of the cross section; and U is the oncoming flow velocity set at the inlet boundary condition. The specifications of the grids used are given in Table 1 and wind turbine rotation speed is 11.88 rev/s. The Strouhal number and RMS lift coefficients ${C}_{l}^{\mathit{RMS}}$ of G4 and G5 are identical. Considering accuracy and calculation speed, G4 is selected as the final grid scheme.

_{cd}is the relative error of $\overline{{C}_{d}}$. As shown in Table 2, the drag coefficient $\overline{{C}_{d}}$ for the plain cylinder is 1.99, which agrees well with that of 2.05 for a square cylinder reported in [24]. The relative error δ

_{cd}is nearly 5% when compared with all the experimental results. The RMS lift coefficient for the plain cylinder is 1.29, which matches well with the RMS lift coefficients of 1.37 for a square cylinder reported in [25].

#### 2.4. Simulation Cases

_{t}is the rotation speed of the wind turbines and f

_{s}is the vortex shedding frequency of the square cylinder calculated according to Strouhal number of 0.132 for the square cylinder provided by [22]. The rotation speed ranges from 0 to 4 times that of f

_{s}, and a rotation speed of 0 refers to a stationary case. In this study, the turbine rotates independently from the wind speed. According to [30], the rotation speed of the Savonius wind turbine is about 19.5 (r/s), which is closest to the rotation speed of R6. Thus, R6 is considered as an actual case. In order to conduct a comparative study on the influence of different rotational speeds on the aerodynamic characteristics of the square cylinder, we created 8 other working conditions.

## 3. Results and Discussions

#### 3.1. Mean Pressure Coefficients

#### 3.1.1. Stationary Wind Turbines at Cylinder Corners

#### 3.1.2. Rotating Wind Turbines at Cylinder Corners

_{t}/f

_{s}is less than 1 while Figure 6b shows the cases when V

_{t}/f

_{v}is equal to or larger than 1.

_{s}. As shown in Figure 6b, the suction on the two sides has an obvious reduction when the wind turbine rotation speed increases from f

_{s}(R5) to 2f

_{s}(R6). Lastly, the suction will not be lower than R6 as the rotation speed further increases.

#### 3.2. Force Coefficients and Vortex Shedding Characteristics

#### 3.2.1. Stationary Wind Turbines at Cylinder Corners

#### 3.2.2. Rotating Wind Turbines at Cylinder Corners

_{s}(R1), $\overline{{C}_{d}}$ of the cylinder with wind turbines dramatically decreases by about 34.2%, and then the drop rate gradually decreases as the rotating speed increases before reaching f

_{s}. When the rotation speed increases to 2f

_{s}(R6), the drop rate of $\overline{{C}_{d}}$ again increases to 34.2%. Similarly, all cases of the square cylinder with rotating corner wind turbines have lower ${C}_{l}^{RMS}$ than the plain cylinder. As rotation speed increases, the variation in the drop rate of ${C}_{l}^{RMS}$ is similar to that of $\overline{{C}_{d}}$, but the reduction in ${C}_{l}^{RMS}$ is more significant than $\overline{{C}_{d}}$. Compared with the plain cylinder, ${C}_{l}^{RMS}$ of the cylinder with corner wind turbines dramatically decreases by about 86.0% as the wind turbine rotation speed is 0.2f

_{s}(R1) and the drop rate becomes 71.3% when the rotating speed increases to 2f

_{s}(R6). When the rotation speed is 4f

_{s}(R8), both $\overline{{C}_{d}}$ and ${C}_{l}^{RMS}$ have the least reduction. It is worth noting that even though $\overline{{C}_{d}}$ and ${C}_{l}^{RMS}$ have a significant reduction when the wind turbine has a rotating speed of 0.2f

_{s}and 2f

_{s}, it is still smaller than the reduction in the case with the stationary wind turbines (R0). Figure 8c–e present the comparison of time history of Cl of R1, R6, and R8. It can be found that the fluctuation of Cl of R1 is smallest, which corresponds to the largest drop rate of ${C}_{l}^{RMS}$ of R1. Figure 8d,e shows that the fluctuation of Cl of R6 is also effectively suppressed, but the effect of fluctuation inhibition of R8 is not obvious.

_{t}/f

_{s}less than 1 and Figure 11b shows the cases with V

_{t}/f

_{s}equal to or larger than 1. For Strouhal number (${S}_{t}$) shown in Figure 11, all of the rotating cases have larger Strouhal numbers than the plain cylinder, except for R6. The vortex shedding energy represented by the peak value of the spectrum of the nine rotating cases is evidently smaller than that of the plain cylinder. Moreover, the vortex shedding energy of the cylinder with corner wind turbines has the lowest value when the wind turbine rotation speed is 2f

_{s}(R6).

#### 3.3. Flow Pattern around the Cylinders

#### 3.3.1. Stationary Wind Turbines at Cylinder Corners

#### 3.3.2. Rotating Wind Turbines at Cylinder Corner

_{s}, the normalized instantaneous vorticity distribution during one period, shown in Figure 15c, is similar to S2, which means that R1 has a similar mechanism of drag reduction as S2.

## 4. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**Views of grids for a square cylinder with wind turbines at two leading corners: (

**a**) Grid distribution in x–y plane; (

**b**) Grid surrounding square cylinder; (

**c**) Grid surrounding upper wind turbine; (

**d**) Grid surrounding lower wind turbine.

**Figure 3.**Comparisons of mean pressure distribution between previous studies and present numerical simulations.

**Figure 5.**Mean pressure coefficients around the cylinder for the four stationary cases and plain cylinder.

**Figure 6.**Mean pressure coefficients around the cylinder for the nine rotating cases and the plain cylinder: (

**a**) V

_{t}/f

_{s}< 1; (

**b**) V

_{t}/f

_{s}≥ 1.

**Figure 7.**Comparisons of RMS lift coefficients, mean drag coefficients, and Strouhal number for the four stationary cases and plain cylinder.

**Figure 10.**Comparisons of RMS lift coefficients, mean drag coefficients, and Strouhal number for the nine rotating cases and the plain cylinder.

**Figure 11.**Power spectra of lift forces for the nine rotating cases and plain cylinder: (

**a**) V

_{t}/f

_{s}< 1; (

**b**) V

_{t}/f

_{s}≥ 1.

**Figure 12.**Time-averaged pressure coefficient distribution: (

**a**) plain cylinder; (

**b**) S2; (

**c**) S4; (

**d**) R1; (

**e**) R6; (

**f**) R8.

**Figure 13.**Time-averaged streamlines around the cylinder: (

**a**) plain cylinder; (

**b**) S2; (

**c**) S4; (

**d**) R1; (

**e**) R6; (

**f**) R8.

**Figure 14.**Normalized time-averaged vorticity distributions: (

**a**) plain cylinder; (

**b**) S2; (

**c**) S4; (

**d**) R1; (

**e**) R6; (

**f**) R8.

**Figure 15.**Normalized instantaneous vorticity distributions during one period: (

**a**) plain cylinder; (

**b**) S2; (

**c**) R1; (

**d**) R6.

Grid Schemes | Cell Numbers | ${\mathit{y}}^{+}\left(\mathbf{Max}\right)$ | $\mathsf{\delta}/\mathbf{W}$ | $\overline{{\mathit{C}}_{\mathit{d}}}\text{}$ | ${\mathit{C}}_{\mathit{l}}^{\mathit{R}\mathit{M}\mathit{S}}\text{}$ | ${\mathit{S}}_{\mathit{t}}\text{}$ |
---|---|---|---|---|---|---|

G1 | 1.1 × 10^{5} | 0.66 | 4 × 10^{−5} | 1.43 | 0.69 | 0.136 |

G2 | 1.3 × 10^{5} | 0.52 | 4 × 10^{−5} | 1.41 | 0.63 | 0.136 |

G3 | 1.7 × 10^{5} | 0.55 | 4 × 10^{−5} | 1.42 | 0.61 | 0.136 |

G4 | 2.4 × 10^{5} | 0.57 | 4 × 10^{−5} | 1.47 | 0.78 | 0.134 |

G5 | 3.0 × 10^{5} | 0.61 | 4 × 10^{−5} | 1.48 | 0.78 | 0.134 |

**Table 2.**Comparisons of force coefficients between previous studies and present numerical simulations.

Data from Previous Studies | Re | $\overline{{\mathit{C}}_{\mathit{d}}}\text{}$ | ${C}_{l}^{RMS}$ | ${S}_{t}$ | δ_{cd} |
---|---|---|---|---|---|

Experimental results: | |||||

[24] | 1.76 × 10^{5} | 2.05 | - | 0.122 | 2.93% |

[26,27] | 2.14 × 10^{4} | 2.1 | - | 0.132 | 5.24% |

[28] | 2.20 × 10^{4} | 2.09 | 0.95 | 0.123 | 4.78% |

Simulation results: | |||||

[22] | 6.0 × 10^{4} | 2.26 | 1.52 | 0.132 | 11.9% |

[25] | 1.9 × 10^{4} | 2.37 | 1.37 | 0.122 | 16.0% |

[29] | 2.2 × 10^{4} | 2.26 | 1.60 | 0.136 | 11.9% |

Present simulation results: | 6.0 × 10^{4} | 1.99 | 1.29 | 0.123 | - |

Cases | Plain | R0 | R1 | R2 | R3 | R4 | R5 | R6 | R7 | R8 |
---|---|---|---|---|---|---|---|---|---|---|

V_{t}/f_{s} | - | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 | 2 | 3 | 4 |

V_{t} (r/s) | - | 0 | 2.38 | 4.75 | 7.13 | 9.50 | 11.88 | 23.76 | 35.64 | 47.52 |

Cases | Plain | S1 | S2 | S3 | S4 |
---|---|---|---|---|---|

$\mathsf{\theta}$ | - | $0\xb0$ | $45\xb0$ | $90\xb0$ | $135\xb0$ |

Case | Cell Numbers | ${\mathbf{y}}^{+}\left(\mathbf{Max}\right)$ | $\mathsf{\delta}/\mathbf{W}$ | $\overline{{\mathit{C}}_{\mathit{d}}}\text{}$ | ${\mathit{C}}_{\mathit{l}}^{\mathit{R}\mathit{M}\mathit{S}}\text{}$ | ${\mathit{S}}_{\mathit{t}}\text{}$ |
---|---|---|---|---|---|---|

Plain cylinder | 2.0 × 10^{4} | 1.4 | 2 × 10^{−4} | 1.99 | 1.29 | 0.123 |

S1 | 2.4 × 10^{5} | 0.41 | 4 × 10^{−5} | 1.26 | 0.19 | 0.144 |

S2 | 2.4 × 10^{5} | 0.43 | 4 × 10^{−5} | 1.24 | 0.12 | 0.145 |

S3 | 2.4 × 10^{5} | 0.44 | 4 × 10^{−5} | 1.35 | 0.21 | 0.141 |

S4 | 2.4 × 10^{5} | 0.59 | 4 × 10^{−5} | 1.48 | 0.43 | 0.143 |

Case | Cell Numbers | ${\mathbf{y}}^{+}\left(\mathbf{Max}\right)$ | $\mathsf{\delta}/\mathbf{W}$ | $\overline{{\mathit{C}}_{\mathit{d}}}\text{}$ | ${\mathit{C}}_{\mathit{l}}^{\mathit{R}\mathit{M}\mathit{S}}\text{}$ | ${\mathit{S}}_{\mathit{t}}\text{}$ |
---|---|---|---|---|---|---|

Plain cylinder | 2.0 × 10^{4} | 1.4 | 2 × 10^{−}^{4} | 1.99 | 1.29 | 0.123 |

R0 (S2) | 2.4 × 10^{5} | 0.43 | 4 × 10^{−}^{5} | 1.24 | 0.12 | 0.145 |

R1 | 2.4 × 10^{5} | 0.46 | 4 × 10^{−}^{5} | 1.31 | 0.18 | 0.142 |

R2 | 2.4 × 10^{5} | 0.49 | 4 × 10^{−}^{5} | 1.34 | 0.25 | 0.142 |

R3 | 2.4 × 10^{5} | 0.53 | 4 × 10^{−}^{5} | 1.37 | 0.33 | 0.147 |

R4 | 2.4 × 10^{5} | 0.50 | 4 × 10^{−}^{5} | 1.39 | 0.52 | 0.127 |

R5 | 2.4 × 10^{5} | 0.57 | 4 × 10^{−}^{5} | 1.47 | 0.78 | 0.134 |

R6 | 2.4 × 10^{5} | 0.46 | 4 × 10^{−}^{5} | 1.31 | 0.37 | 0.114 |

R7 | 2.4 × 10^{5} | 0.51 | 4 × 10^{−}^{5} | 1.45 | 0.58 | 0.129 |

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**MDPI and ACS Style**

Wang, Z.; Hu, G.; Zhang, D.; Kim, B.; Xu, F.; Xiao, Y.
Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners. *Appl. Sci.* **2022**, *12*, 3515.
https://doi.org/10.3390/app12073515

**AMA Style**

Wang Z, Hu G, Zhang D, Kim B, Xu F, Xiao Y.
Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners. *Applied Sciences*. 2022; 12(7):3515.
https://doi.org/10.3390/app12073515

**Chicago/Turabian Style**

Wang, Zhuoran, Gang Hu, Dongqin Zhang, Bubryur Kim, Feng Xu, and Yiqing Xiao.
2022. "Aerodynamic Characteristics of a Square Cylinder with Vertical-Axis Wind Turbines at Corners" *Applied Sciences* 12, no. 7: 3515.
https://doi.org/10.3390/app12073515