# Aerodynamic Shape Optimization of an Arc-Plate-Shaped Bluff Body via Surrogate Modeling for Wind Energy Harvesting

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Characteristics and Optimization Objectives of a Galloping Bluff Body

#### 2.1. Characteristics of a Galloping Bluff Body

_{1}and a

_{2}, ranging from 0 to 1. The arc angle β = 2πa

_{1}, and the tail plate length L

_{t}= 2Da

_{2}. Changing the two variables can create a variety of different shapes. For instance, when the arc angle β is 2π and the plate length L

_{t}is 0, the bluff body is a circular cylinder; when the arc angle β is 2π and the plate length L

_{t}is not 0, the bluff body is a circular cylinder with a plate, following the aerodynamic shape studied by Song, Hu, Tse, Li, and Kwok [35]. When the angle β is π/2 and the plate length L

_{t}is 0, the bluff body follows the aerodynamic shape studied by Tucker Harvey, Khovanov, and Denissenko [40]. Therefore, the proposed arc-plate bluff body has a good ability to create a variety of aerodynamic shapes using two variables.

#### 2.2. Optimization Objectives of a Galloping Bluff Body

**x**, the general optimization problem can be expressed as follows:

**x**) is the p purpose objective function.

**x**

_{l}and

**x**

_{u}are design variables’ upper and lower limits, respectively. h

_{i}(

**x**) and g

_{j}(

**x**) are equality and inequality constraints, respectively. n

_{h}and n

_{g}are the number of constraints, respectively.

**x**)is the output energy of the galloping-based WEH with the shape of

**x**.

_{e}weighted by the probability density distribution function of wind speed as

_{p}(U) is the probability density distribution function of wind velocity, and U

_{U}and U

_{L}are the upper and lower limits of the wind speed, respectively. In this paper, the probability density distribution function of wind velocity f

_{p}is assumed as an uniform distribution in the tested wind velocity range, i.e.,

## 3. Surrogate Modeling

#### 3.1. The Kriging Surrogate Modeling Method

**x**is given by a linear combination of the function values at the original samples and their gradients at observed points, as follows:

**x**, ${w}^{\left(i\right)}$ is the weight coefficient for the function value ${y}^{\left(i\right)}$. To determine the weight coefficient, the output of a deterministic experiment is treated as a realization of a stochastic process, i.e.,

**Ys**is the response of the sample points. Solving this constrained minimization problem, ${w}^{\left(i\right)}$ can be found by solving the following linear equations in matrix form:

**R**is the correlation matrix representing the correlation between the observed points, and

**r**is the correlation vector representing the correlation between the untried point and the observed points. The kriging predictor can be written in the form

#### 3.2. Design of Experiments

_{s}sample points, n

_{v}design variables are evenly divided into n

_{sl}intervals. The total sampling interval is divided into ${\left({n}_{v}\right)}^{{n}_{s}}$ subintervals. When each sample point is projected into any dimension, there is only one sample point in each interval.

_{1}and a

_{2}in the range from 0 to 1. As shown in Figure 3a, 50 samples for the modeling were generated in the two-dimensional input space. The shapes of bluff bodies are shown in Figure 3b, corresponding to all the sample points. Taking shape 2 as an example, a

_{1}= 0.02 and a

_{2}= 0.69, the angle of arc β is 0.04π, and the plate length L

_{t}is 0.69D. The corresponding bluff body section is approximately T-shaped. For shape 47, where a

_{1}= 0.94 and a

_{2}= 0.10, arc angle β is 1.88π and the plate length L

_{t}is 0.20D. The related bluff body section is roughly circular. For shape 49, where a

_{1}= 0.98 and a

_{2}= 0.82, arc angle β is 1.96π and the plate length L

_{t}is 1.64D. The resultant bluff body section follows the aerodynamic shape studied by Song, Hu, Tse, Li, and Kwok [35]. It can be concluded that the arc-plate bluff body controlled by two variables can describe a variety of aerodynamic shapes using LHS.

## 4. Wind Tunnel Test

#### 4.1. Piezoelectric Wind Energy Harvesting Tests

^{3}to 2.4 × 10

^{4}.

#### 4.2. Force Measurement Tests

#### 4.3. Uncertainty Analyses

^{−6}V. The accuracy of the voltage DAQ module was ±0.03% of the measured value. The accuracy and resolution of the measuring instruments and equipment are shown in Table 1.

_{rms}is the root mean square of the voltage. The uncertainty was assessed using the Taylor series method (TSM) [58]. The correlated random error terms were assumed to be zero. The standard uncertainty of average output power P is expressed as

_{V}and S

_{R}are standard uncertainties of voltage and resistance, respectively. The random uncertainty values for output power P is ±1.0%.

_{Fy}is expressed as

_{Fy}. The uncertainty was calculated using the Taylor series method (TSM) [58]. It was assumed that only U, B, L, and F

_{y}contributed to the random uncertainty. These correlated random error terms were assumed to be zero. The standard uncertainty of force coefficient is expressed as

_{u}, S

_{Fy}, S

_{B}, and S

_{L}are standard uncertainties of the wind velocity, force, width, and length of the bluff body. The random uncertainty values for C

_{Fy}using the TSM are shown in Table 2. The standard uncertainty of force coefficient was caused by the error of the anemometer. The accuracy of the anemometer was ±(0.03 m/s + 5% of measured value), and the force was proportional to the square of the wind velocity.

## 5. Results and Discussions

#### 5.1. Surrogate Model

_{1}= 0.20 and a

_{2}= 0.63. The arc angle β and tail length L

_{t}were 0.40π and 1.26D, respectively. The WEH with shape 18 had an excellent performance when a

_{1}= 0.35 and a

_{2}= 0.57. The arc angle β and tail length L

_{t}were 0.70π and 1.14D, respectively. The surrogate model had a peak region for the average output power P

_{e}, as in the red colored area with an inclination of 45° shown in Figure 6. The WEHs corresponding to the peak region output have abundant power because the bluff body of the WEHs is prone to gallop. For the other areas in Figure 6, the output power declines rapidly and decays to zero. Based on the surrogate model, the impacts of the shape modification on the efficiency of the WEH are further evaluated in the subsequent sections.

#### 5.2. Output Power

#### 5.3. Force Coefficient

_{d}and lift force F

_{l}at each attack angle α in the process of oscillation are the same as the values measured at the corresponding steady attack angle α. The relative attack angle α = arctan(ẏ/U) and forces were defined, as shown in Figure 9. The transverse force is the resultant of both drag force F

_{d}and lift force F

_{l}

_{y}was directly measured in the wind tunnel using the force balance. The transverse force coefficient C

_{Fy}is expressed as

_{eff}is effective wind velocity.

_{Fy}are in the same direction, as shown in Figure 10, the aerodynamic force is negative. Negative aerodynamic damping is an essential prerequisite for the galloping instability. With the increase of the wind velocity, d decreases gradually and may be negative, and then the system is prone to instability. In terms of the well-known Den Hartog criterion [62,63], the necessary condition for the occurrence of the galloping instability is ${\frac{\partial {C}_{Fy}}{\partial \alpha}|}_{\alpha ={0}^{\xb0}}>0$. In other words, a positive slope of the C

_{Fy}versus α curve at α = 0° is an essential prerequisite for galloping instability.

_{Fy}with wind attack angle α for bluff bodies with different arc angles is shown in Figure 11. The slopes of the transverse force coefficient of all the three bluff bodies are positive when wind attack angle α = 0°. As a result, the corresponding WEHs work at high wind velocity, as shown in Figure 7. With the increase of the wind attack angle, the transverse force coefficient of shape 18 is much more significant than those of shapes 30 and 42. This is because the transverse force provides the negative aerodynamic force, and then the output power of the WEH with the shape 18 is much larger than those with the shapes 30 and 42.

_{Fy}with wind attack angle α for bluff bodies with different tail lengths is shown in Figure 12. The slope of the transverse force coefficient of shape 17 is positive at α = 0°. In other words, shape 17 satisfies the essential prerequisite of galloping. The transverse force coefficient remains positive, but the value of the transverse force coefficient is relatively small, less than 0.1. The WEH with the shape 17 hardly works, because the bluff body cannot produce the aerodynamic damping force. The slope of the transverse force coefficient of shape 19 is positive at α = 0°. The bluff body also satisfies the essential prerequisite of galloping. The transverse force coefficient of shape 19 increased at the initial wind attack angle and decreased with the wind attack angle α > 6°. Although the WEH with the shape 19 can work, the performance was barely satisfactory.

_{Fy}with wind attack angle α for bluff bodies with excellent vibration performance is shown in Figure 13, for different wind attack angles α. The selected bluff bodies are within the region of the surrogate model showing peak output energy. For the three bodies shown in Figure 13, the slopes of the transverse force coefficient C

_{Fy}are all positive at α = 0. The transverse force coefficients of the bluff bodies increase rapidly with α. The transverse force coefficient of shape 25 is less than those of the other two when α ≥ 14°, and hence the output power of the WEH with the shape 25 bluff body outputs was less than those of the other two.

#### 5.4. Comparision and Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The aerodynamic shapes of optimized galloping bluff bodies, such as (

**a**) square prism with leading-edge fins; (

**b**) circular cylinder with different shape attachments; (

**c**) circular cylinder with roughness strips; (

**d**) square prism with Y-shaped fins near the trailing edge; (

**e**) circular cylinder with a splitter plate; (

**f**) blade bluff bodies.

**Figure 6.**Kriging model for the average output power in (

**a**) three-dimensional and (

**b**) two-dimensional cartesian.

**Figure 7.**Variation of output power with wind velocity for WEHs with different arc angle bluff bodies.

**Figure 8.**Variation of output power with wind velocity for WEHs with different tail lengths bluff bodies.

**Figure 9.**Variation of output power with wind velocity for WEHs with prism and arc-plate-shaped bluff bodies.

**Figure 11.**Variation of transverse force coefficient C

_{Fy}with wind attack angle α for bluff bodies with different arc angles.

**Figure 12.**Variation of transverse force coefficient C

_{Fy}with wind attack angle α for bluff bodies with different tail lengths.

**Figure 13.**Variation of transverse force coefficient C

_{Fy}with wind attack angle α for bluff bodies with excellent vibration performance.

**Figure 14.**The schematic of flow separation for (

**a**) square cross-section and (

**b**) arc-plate-shaped cross-section.

Variable | Resolution | Accuracy |
---|---|---|

Velocity | 0.01 m/s | ±(0.03 m/s + 5% of the measured value) |

Width of bluff body | 0.025 mm | ±2.4% |

Length of bluff body | 0.025 mm | ±0.4% |

Force | 0.00625 N | ±1% |

Wind direction | 0.017° | ±0.5% |

Resistance | 1000 Ω | ±1% |

Voltage | 3.6 × 10^{−6} V | ±0.03% |

U (m/s) | S_{CFy} (%) |
---|---|

1 | 16.2 |

2 | 13.3 |

3 | 12.3 |

4 | 11.8 |

5 | 11.5 |

6 | 11.3 |

7 | 11.2 |

Bluff Body | Power (mW) | Wind Velocity (m/s) | Efficiency (%) | ||||
---|---|---|---|---|---|---|---|

No. | Author | Shape | Width (cm) | Height (cm) | |||

1 | Zhao [66] | Square | 2 | 10 | 1.25 | 5 | 0.77 |

2 | Sirohi [67] | D-shape | 3 | 23.5 | 1.14 | 4.7 | 0.24 |

3 | Hu [33] | Cylinder with rods | 4.8 | 24 | 0.053 | 5.5 | 0.004 |

4 | Hu [7] | Square with fins | 2.4 | 24 | 0.034 | 5 | 0.007 |

5 | Song [35] | Cylinder with plate | 4.8 | 24 | 0.014 | 5 | 0.002 |

6 | Our work | Square | 5 | 10 | 0.50 | 5 | 0.12 |

7 | Our work | Arc-plate | 5 | 10 | 1.01 | 5 | 0.25 |

8 | Our work | Square | 5 | 10 | 1.12 | 7 | 0.10 |

9 | Our work | Arc-plate | 5 | 10 | 2.31 | 7 | 0.21 |

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**MDPI and ACS Style**

Shi, T.; Hu, G.; Zou, L.
Aerodynamic Shape Optimization of an Arc-Plate-Shaped Bluff Body via Surrogate Modeling for Wind Energy Harvesting. *Appl. Sci.* **2022**, *12*, 3965.
https://doi.org/10.3390/app12083965

**AMA Style**

Shi T, Hu G, Zou L.
Aerodynamic Shape Optimization of an Arc-Plate-Shaped Bluff Body via Surrogate Modeling for Wind Energy Harvesting. *Applied Sciences*. 2022; 12(8):3965.
https://doi.org/10.3390/app12083965

**Chicago/Turabian Style**

Shi, Tianyi, Gang Hu, and Lianghao Zou.
2022. "Aerodynamic Shape Optimization of an Arc-Plate-Shaped Bluff Body via Surrogate Modeling for Wind Energy Harvesting" *Applied Sciences* 12, no. 8: 3965.
https://doi.org/10.3390/app12083965