# Wind-Tunnel Experiments on the Interactions among a Pair/Trio of Closely Spaced Vertical-Axis Wind Turbines

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Rotational directions;
- Gap ratios;
- Wind direction over 360°;

- Wind-tunnel experiments equivalent to the operation of small variable-speed VAWTs;
- Well-supported flow patterns obtained by flow visualization.

## 2. Methods

#### 2.1. Configuration of the Flow Field

_{eD}~4.7 × 10

^{4}and λ~0.75, respectively, for the experiments on a pair of turbines with a wind speed of V = 14 m/s. On the other hand, in experiments on a trio of turbines with a wind speed of V = 12 m/s (N~3400 rpm), the corresponding values were R

_{eD}~4.0 × 10

^{4}and λ~0.74. The Reynolds number using the length of the chord c was R

_{ec}~1.9/1.6 × 10

^{4}for the experiments on a pair or trio of turbines. More details about the effect of a Reynolds number on the performance in a VAWT are given in [20] with relevant references. The error in the model rotational speed measurements was ±10 rpm, which corresponds to ±0.25% or ±0.29% of the rotational speed N (~4000 rpm or ~3400 rpm) of a single rotor with an isolated setting at a uniform wind velocity of V = 14/12 m/s. Table 1 provides a comparison of the parameters adopted in related studies with those used in ours, including the cases of multiple rotors with more than three turbines. In Table 1, g

_{min}/D is the minimum rotor gap ratio investigated and ω indicates the angular velocity of the rotor.

#### 2.2. Tandem Layouts of a Pair of VAWTs

#### 2.3. Wind Directions of a Pair of VAWTs (16 Wind Directions)

#### 2.4. Wind Directions of a Trio of VAWTs (12 Wind Directions)

#### 2.5. Characteristics of a Single Rotor Configuration

_{p}with the tip speed ratio λ of a wind turbine (see Table 1). The relationship between the rotational speed N and the motor power (power P of the model turbine at equilibrium) is shown in Figure 5b, where the red circles are points interpolated from the experiment. The approximate black curve (P vs. N) in Figure 5b is expressed by Equation (1).

#### 2.6. Experimental Setup of a Pair of VAWTs and a Trio of VAWTs

_{SI}/N

_{SI}

_{0}placed y = 0 mm (see Figure 2, Figure 3 and Figure 4) at a uniform velocity V = 10 or 12 m/s. Here, N

_{SI}is the rotational speed of the single rotor and N

_{SI}

_{0}is that located at x/D = 0. Hereafter, the rotational speed in the experimental result means the value measured in the case without a power supply to the startup motor, i.e., the free rotational speed (see Figure 7). The origins of the x- and y-axes correspond to the centers of the two/three rotors. Note that the upstream rotor (R1) is fixed at x/D = –1 and only the streamwise position of the downstream rotor (R2) was adjusted according to the rotor gap g in the tandem experiment (Figure 2). We confirmed the constant rotational speed within the error of ±2% in the range of 0 ≤ x/D ≤ 12 (Figure 6). This 600 mm streamwise range covers the full length of the tandem experiment, including a pair of two rotors at the maximum gap of 10D = 500 mm.

## 3. Results and Discussion

#### 3.1. Rotational Speeds and Power of Closely Spaced VAWTs in Tandem Layouts

_{norm}, defined in Equation (2).

_{norm}obtained from the wind-tunnel experiment (Figure 10) with those obtained by Hara et al. [13] via 2D CFD analysis using the dynamic fluid body interaction (DFBI) method. The gap between the two rotors (g on the abscissa) is also nondimensionalized using the diameter of each rotor D. Regardless of the layout type (TCO and TIR), the normalized rotational speed decreased as the gap decreased in the experimental and CFD results.

_{norm}with the gap at g/D > 5, substantiates their appropriate gap ratio.

#### 3.2. Wind-Direction Dependence of a Pair of VAWTs (16 Wind Directions)

#### 3.3. Wind-Direction Dependence of a Trio of VAWTs (12 Wind Directions)

_{norm}in the 3CO and 3IR configurations. In each figure, the blue circles indicate the rotational speed N of Rotor 1 (R1), the green crosses plot the N of Rotor 2 (R2), and the black triangles show the N of Rotor 3 (R3). The red squares represent the average rotor speed of R1, R2, and R3, defined as N

_{norm}

_{,}

_{ave}.

_{norm}

_{,}

_{ave}) reaching a maximum value at g/D = 0.5 and 1 (Figure 17a,c). At θ = 60°, called 3COF, one of the downwind rotors, R3, stops rotating at g/D = 0.5 and 1 (Figure 17a,c). Similarly, at θ = 90°, called 3TCOD, the downwind rotor R2 stops at g/D = 0.5 and 1 (Figure 17a,c). In this layout, a significant decrease in the rotational speed of R2 occurs, even in the largest gap of g/D = 2 (Figure 17e). As a result, the average rotational speed N

_{norm}

_{,}

_{ave}is remarkably low in the 3TCOD. As explained in Section 2.4, the same layout occurs in every 120° in the 3CO configuration. In fact, the experimental results (Figure 17a,c,e) show the excellent rotational symmetry for the average normalized rotational speed N

_{norm}

_{,}

_{ave}.

_{norm}

_{,}

_{ave}, in the 3IR configuration reaches its maximum value at θ = 0° (i.e., CDB) at g/D = 0.5, 1, and 2 (Figure 17b,d,f). Hence, N

_{norm}

_{,}

_{ave}in other layouts tends to have a much smaller value than it does in CDB, especially in the cases with a smaller gap distance. Although we do not explain all the layouts in detail here, the experimental results shown in Figure 17b,d,f prove that there is no rotationally symmetric distribution in N

_{norm}

_{,}

_{ave}, as expected by the 12 independent layouts in the 3IR configuration explained in Section 2.4.

_{norm}

_{,}

_{ave}value (Figure 17).

_{foot}and the footprint area A

_{foot}of a trio of rotors, as in Equations (3) and (4). R

_{foot}is the radius of the circular region that connotes the three rotors with a gap distance of g. The second term of the right hand of Equation (3) represents a distance from the center of the rotor trio to the corner of an equilateral triangle (lavender dash-dotted line in Figure 4). Since the diameter of each rotor D is 50 mm, the values of R

_{foot}at g = 25, 50, and 100 mm are 68.30, 82.74, and 111.60 mm, respectively. The A

_{foot}values at g = 25, 50, and 100 mm are then 0.01466, 0.02150, and 0.03913 m

^{2}, respectively. We define the simple average of the N

_{norm}

_{,}

_{ave}of the 12 wind directions as the N

_{norm}

_{, 12-wind}. Table 3 lists the values of N

_{norm}

_{, 12-}

_{wind}based on the wind-tunnel experiments and those multiplied by 3 (i.e., the number of rotors) divided by the corresponding A

_{foot}.

_{norm}

_{, 12-wind}increases with an increase in g/D. This gap dependence is emphasized by the plots in Figure 18, in which N

_{norm}

_{,}

_{ave}of the different rotor spacings are presented in one radar chart for 3CO and 3IR, respectively. In each figure, the blue circles indicate the average normalized rotational speed N

_{norm}

_{,}

_{ave}at g/D = 0.5, the green triangles plot the N

_{norm}

_{,}

_{ave}at g/D = 1, and the red squares show the N

_{norm}

_{,}

_{ave}at g/D = 2.

_{norm}

_{, 12-wind}/A

_{foot}in a unit footprint area shows the opposite tendency, i.e., the advantage of a smaller rotor spacing (see Table 3). However, it is crucial to contemplate not only the performance (wind-direction dependence in N

_{norm}

_{, 12-wind}or 3N

_{norm}

_{, 12-wind}/A

_{foot}) of a trio of turbines, but also the velocity deficit of the wake flow.

_{norm}

_{,}

_{bi-wind}, in Equation (5). Here, N

_{norm}

_{,}

_{ave}

_{, 0°}and N

_{norm}

_{,}

_{ave}

_{, 180°}are average normalized rotational speeds of parallel-like layouts at θ = 0° and θ = 180°, respectively. Therefore, N

_{norm}

_{,}

_{bi-wind}indicates the performance of a trio of turbines in an isotropic bidirectional wind speed. An example of the bi-wind in nature is a daytime sea breeze or a nighttime land breeze.

_{norm}

_{,}

_{bi-wind}for different gap ratios. Table 4 also contains the values of 3N

_{norm}

_{,}

_{bi-wind}/A

_{foot}, which indicates the advantage of a smaller rotor spacing. As shown in Table 4, the 3IR configuration yielded a higher average rotational speed than the 3CO arrangement at any rotor spacing in the ideal bidirectional wind conditions. It is interesting that the change in wind from 12-wind-direction (Table 3) to 2-wind-direction (Table 4) acts negatively in the 3CO configuration but positively in the 3IR configuration, on VAWT performance.

_{norm}

_{,}

_{ave}value is an advantageous wind direction against θ = 60° in the 3CO (3COF).

_{norm}

_{,}

_{ave}value is a disadvantageous wind direction against θ = 270° in 3CO (TCOF). In total, the value of N

_{norm}

_{, 12-wind}(the simple average of the N

_{norm}

_{,}

_{ave}of the 12 wind directions) of 0.801 of the 3IR is 3% larger than 0.778 of the 3CO (see Table 3).

_{norm}

_{,}

_{ave}at θ = 60° in the 3CO configuration (i.e., 3COF), R2 rotates at the fastest speed for the gap/D = 0.5 and 1. This is the same result in three Savonius turbines in a 3COF-like layout independent of rotor gap ratio conducted by Shaheen et al. [27] (see their Figure 21) and on three Darrieus turbines in a 3COF layout by Silva and Danao [28] for gap/D = 1.

_{norm}

_{,}

_{ave}in this layout takes the maximum value of 0.9618 among the whole 16 independent layouts (see Table 2). This value is 106% of 0.9087 obtained in the case of 3COB (θ = 0° in the 3CO layout). The corresponding 2D CFD shows 103% in this comparison (CDB to 3COB).

_{norm}

_{,}

_{ave}at θ = 0° in the 3IR configuration (i.e., CDB) is R1 > Ave. > R2/R3, as seen in Figure 17d or Figure 17f for gap/D = 1 or 2. This agrees with the order of the power coefficient for CDB obtained by Zheng et al. [23] (see their Figure 19) based on 2D CFD.

_{norm}

_{,}

_{ave}at the CDF layout (θ = 60° in the 3IR configuration) over N

_{norm}

_{,}

_{ave}at 3COF, as seen in Figure 21 (indicated by CDF > 3COF).

_{norm}

_{,}

_{ave}value in the CFD (see Figure 20a). Remember that the TCOU layout in Figure 22b for θ = 30° is equivalent to that in θ = 150° and 270° only for the 3CO configuration, as explained in Table 2. There is a remarkable difference in N

_{norm}

_{,}

_{ave}between the TIRU-CU layout and the TCOU layout in the same wind direction (see Figure 21).

_{norm}

_{,}

_{ave}value in the 3IR configuration always exceeds that in the 3CO configuration in the experiments (see Figure 21a). In contrast, the N

_{norm}

_{,}

_{ave}value in the 3IR configuration falls below that in the 3CO configuration regarding the six tandem-like layouts (θ = 30°, 90°, 150°, 210°, 270°, and 330°).

## 4. Conclusions

- The decrease in the rotational speed and rotor power of a pair of turbines arranged in tandem was demonstrated;
- The amount of decrease depended on the g/D ratio, with the value of the rotational speed of the downwind rotor 75–80% of that of an isolated rotor even at g/D = 10, although the value of the upstream rotor was 100%;
- The corresponding power value of the downwind rotor was approximately 80%.

- The “origin-symmetrical” distribution of the average rotational speed of two rotors in the CO pair configuration and the “line-symmetrical” distribution in the IR pair configuration were demonstrated;
- The existence of a deflected wake flow accounted for the decreasing tendency of the rotational speed at θ = 90°, 112.5°, 270°, and 292.5° (showing the origin symmetry) in the CO pair configuration;
- A wake interaction caused a slowdown tendency at θ = 90°, 112.5°, 270°, and 247.5° (showing the line symmetry) in the IR pair configuration.

- The inverse-rotating trio (3IR) configuration takes a higher average rotational speed than the co-rotating trio (3CO) configuration at any rotor gap under the ideal bidirectional wind conditions;
- The maximum average rotational speed can be obtained at a wind direction of θ = 0° in the 3IR configuration, which is 6% faster than that in the 3CO configuration;
- The average rotor speed of the three rotors N
_{norm}_{,}_{ave}is remarkably low at θ = 90°, 210°, and 330° in the 3CO configuration (3TCOD). The 3CO configuration demonstrates the “rotational symmetry” for N_{norm}_{,}_{ave}in every 120°; - The 3IR configuration does not show the rotational symmetry for N
_{norm}_{,}_{ave}, as expected.

- The relationship 3COB > 3COF for N
_{norm}_{,}_{ave}is explained by accelerated gap flows at θ = 0° and the decelerating effect on the flow at θ = 60°, in the 3CO configuration; - The relationship CDF > 3COF for N
_{norm}_{,}_{ave}is explained by straight wakes at θ = 60° in the 3IR configuration and asymmetric wakes at θ = 60° in the 3CO configuration.

- We are confident that our research will serve as a base for future studies on designing a wind farm consisting of sets of these turbine pairs and trios. Further investigation on the wake interference between them is essential for future research on the design. We are currently making preparations for the wind-tunnel experiments with 12 BWTs to determine the optimal arrangement, supported by JSPS KAKENHI below.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Two tandem layouts against the wind direction in a closely spaced VAWT pair: (

**a**) tandem co-rotating (TCO); (

**b**) tandem inverse-rotating (TIR).

**Figure 3.**Definition of 16-wind-direction configurations in a closely spaced VAWT pair: (

**a**) co-rotation (CO); (

**b**) inverse-rotation (IR).

**Figure 4.**Definition of 12-wind-direction configurations in a closely spaced VAWT trio: (

**a**) co-rotation trio (3CO); (

**b**) inverse-rotation trio (3IR).

**Figure 5.**Experimental data of a 3D-printed miniature rotor: (

**a**) power coefficient as a function of tip speed ratio; (

**b**) power as a function of rotational speed (reproduced with permission from Jodai and Hara [20]).

**Figure 9.**Experimental setup of a trio of 3D-printed VAWT models with a gap of g/D = 1 in the 3IR configuration at θ = 120° (2COB).

**Figure 10.**Variations in rotational speed with the gap between the rotors in tandem layouts at V = 10 m/s: (

**a**) TCO; (

**b**) TIR.

**Figure 12.**Comparison between the wind-tunnel experiment and CFD (reproduced with permission from Hara et al. [13]) on VAWTs in tandem layouts at V = 10 m/s.

**Figure 13.**CO configuration of 16-wind-direction dependence on two VAWTs at V = 14 m/s: (

**a**) g/D = 0.5; (

**b**) g/D = 1.

**Figure 14.**IR configuration of 16-wind-direction dependence on two VAWTs at V = 14 m/s: (

**a**) g/D = 0.5; (

**b**) g/D = 1.

**Figure 15.**Comparison between the experimental and CFD results on 16-wind-direction dependence of average normalized rotational speed in a pair of VAWT models in CO configuration at V = 14 m/s: (

**a**) g/D = 0.5; (

**b**) g/D = 1.

**Figure 16.**Comparison between the experimental and CFD results on 16-wind-direction dependence of average normalized rotational speed in a pair of VAWT models in IR configuration at V = 14 m/s: (

**a**) g/D = 0.5; (

**b**) g/D = 1.

**Figure 17.**Configurations of 12-wind-direction dependence of normalized rotational speed in a trio of VAWT models at V = 12 m/s: (

**a**) 3CO configuration, g/D = 0.5; (

**b**) 3IR configuration, g/D = 0.5; (

**c**) 3CO configuration, g/D = 1; (

**d**) 3IR configuration, g/D = 1; (

**e**) 3CO configuration, g/D = 2; (

**f**) 3IR configuration, g/D = 2.

**Figure 18.**Experimental results of the gap dependence of the 12-wind-direction for the average normalized rotational speed in a trio of VAWT models at V = 12 m/s in: (

**a**) 3CO configuration; (

**b**) 3IR configuration.

**Figure 19.**Comparison between the experimental and CFD results on 12-wind-direction dependence of normalized rotational speed in a trio of VAWT models at g/D = 1 in the 3CO configuration: (

**a**) Rotor 1; (

**b**) Rotor 2; (

**c**) Rotor 3; (

**d**) Average.

**Figure 20.**Comparison between the experimental and CFD results on 12-wind-direction dependence of normalized rotational speed in a trio of VAWT models at g/D = 1 in the 3IR configuration: (

**a**) Rotor 1; (

**b**) Rotor 2; (

**c**) Rotor 3; (

**d**) Average.

**Figure 21.**Comparison between 3CO and 3IR configurations on the average normalized rotational speed in a trio of VAWT models at g/D = 1: (

**a**) experimental results; (

**b**) CFD results.

**Figure 22.**Photographs of smoke flow through three turbines at g/D = 1 under V = 2 m/s in the 3CO configuration: (

**a**) θ = 0°; (

**b**) θ = 30°; (

**c**) θ = 60°; (

**d**) θ = 90°.

**Figure 23.**Photographs of smoke flow through three turbines at g/D = 1 under V = 2 m/s in the 3IR configuration: (

**a**) θ = 0°; (

**b**) θ = 270°; (

**c**) θ = 60°; (

**d**) θ = 90°.

**Figure 24.**Color contours of flow velocity at g/D = 1 under V = 10 m/s in the 3CO configuration: (

**a**) θ = 0°; (

**b**) θ = 30°; (

**c**) θ = 60°; (

**d**) θ = 90°.

**Figure 25.**Color contours of flow velocity at g/D = 1 under V = 10 m/s in the 3IR configuration: (

**a**) θ = 0°; (

**b**) θ = 270°; (

**c**) θ = 60°; (

**d**) θ = 90°.

**Table 1.**Comparison of the parameters of the vertical-axis turbine. CO and IR represent co-rotating and inverse-rotating configurations, respectively. U is the uniform speed; D is the rotor diameter; R

_{ec}is the chord-based Reynolds number; λ = Rω/V is the tip speed ratio; and σ = Bc/(πD) is the solidity. Values marked with † are examples of nonequilateral-triangular cluster design consisting of three VAWTs.

Study | Layout | U (m/s) | D (m) | R_{ec}/10^{4} (-) | λ (-) | σ (-) | g_{min}/D (-) |
---|---|---|---|---|---|---|---|

Ahmadi-Baloutaki et al. [15] | 0° trio (IR) | 6–14 | 0.30 | 1.8–4.2 | ~0.05–0.3 | 0.239 | 0.5 ^{†} |

Bangga et al. [19] | six-parallel (CO&IR) | 8.0 | 2.0 | 14 | 1.5–3.0 | 0.0844 | 1.0 |

Dabiri [3] |
$$\{\begin{array}{c}\mathrm{over}360\xb0\mathrm{over}\mathrm{pair}\left(\mathrm{IR}\right)\\ \mathrm{elbow}-\mathrm{like}\mathrm{trio}\left(\mathrm{IR}\right)\\ \mathrm{six}-\mathrm{staggered}\left(\mathrm{IR}\right)\end{array}$$
| 5.7 (7.8) | 1.2 | 4.2 (6) | 1.5–3.0 | 0.102 | 0.65 0.65 3.0 |

Dessoky et al. [7] | tandem pair (CO) | 8.0 | 2 | 14 | 0.75 | 0.0844 | 1.5 |

De Tavernier et al. [12] | over 360° pair (CO&IR) | 1.0 | 20 | 6.7 | 2.5, 3.5 | 0.032 | 0.01 |

Hezaveh et al. [4] | 0°, 20°, 40°, 60° trio (CO) | 12 | 1.2 | 8.8 | 2.18 | 0.0875 | 2.0 |

Kuang et al. [8] | tandem pair (CO) | 8.0 | 0.8 | 10.7 | 0.4–1.5 | 0.239 | 1.0 |

Sahebzadeh et al. [11,21] | over ±90° pair (CO) | 9.3 | 1 | 15.7 | 4 | 0.0191 | 0.25 |

Zanforlin and Nishino [6] | over 360° pair (IR) | 8.0 | 1.2 | 6.8 | 2.3–3.2 | 0.102 | 0.5 |

Zanforlin [22] (tidal turbines) | over 360° trio (CO) | 1.5 | 1.0 | 27 | 1.75 | 0.175 | 2.0 |

Zhang et al. [14] | 0°, 60° trio (CO) | 4.01 | 1.48 | 2 | 3.7 | 0.0323 | 2.0 ^{†} |

Zheng et al. [23] | 0°, 60° trio (IR) | 10.6 | 1.2 | 9 | 2.3 | 0.102 | 0.6 |

Present |
$$\{\begin{array}{c}\mathrm{tandem}\mathrm{pair}(\mathrm{CO}\mathrm{IR})\\ \mathrm{over}360\xb0\mathrm{pair}(\mathrm{CO}\mathrm{IR})\\ \mathrm{over}360\xb0\mathrm{trio}(\mathrm{CO}\mathrm{IR})\end{array}$$
| 10, 12 14 12 | 0.050 | 1.3, 1.6 1.9 1.6 | ~0.8 | 0.382 | 1.0 0.5 0.5 |

**Table 2.**Definition of the names of specific layouts in 12-wind-direction configurations (3CO and 3IR) in a closely spaced VAWT trio. Layouts with * are repeated every 120°.

Wind Direction (°) | 3CO (Co-Rotation Trio) | 3IR (Inverse-Rotation Trio) |
---|---|---|

0 | 3COB | CDB |

30 | TCOU | 2TCOD |

60 | 3COF | CDF |

90 | 3TCOD | TIRD-CO |

120 | 3COB * | 2COB |

150 | TCOU * | TIRU-CO |

180 | 3COF * | CUF |

210 | 3TCOD * | 2TCOU |

240 | 3COB * | CUB |

270 | TCOU * | TIRU-CU |

300 | 3COF * | 2COF |

330 | 3TCOD * | TIRD-CD |

**Table 3.**Comparison of average normalized rotational speed N

_{norm}

_{, 12-wind}and 3N

_{norm}

_{, 12-wind}/A

_{foot}in a trio of VAWT models in an isotropic 12-directional wind speed.

Average Speed | g/D = 0.5 | g/D = 1 | g/D = 2 | |||
---|---|---|---|---|---|---|

3CO | 3IR | 3CO | 3IR | 3CO | 3IR | |

N_{norm}_{, 12-wind} (-) | 0.686 | 0.677 | 0.778 | 0.801 | 0.918 | 0.869 |

3N_{norm}_{, 12-wind}/A_{foot} (-/m^{2}) | 140.4 | 138.6 | 108.5 | 111.7 | 70.4 | 66.6 |

**Table 4.**Comparison of normalized rotational speed N

_{norm}

_{,}

_{bi-wind}and 3N

_{norm}

_{,}

_{bi}

_{-wind}/A

_{foot}in a trio of VAWT models in an isotropic bidirectional wind speed.

Average Speed | g/D = 0.5 | g/D = 1 | g/D = 2 | |||
---|---|---|---|---|---|---|

3CO | 3IR | 3CO | 3IR | 3CO | 3IR | |

N_{norm}_{,}_{bi-wind} (-) | 0.629 | 0.763 | 0.762 | 0.903 | 0.960 | 0.961 |

3N_{norm}_{,}_{bi-wind}/A_{foot} (-/m^{2}) | 128.8 | 156.1 | 106.3 | 126.0 | 73.6 | 73.7 |

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## Share and Cite

**MDPI and ACS Style**

Jodai, Y.; Hara, Y.
Wind-Tunnel Experiments on the Interactions among a Pair/Trio of Closely Spaced Vertical-Axis Wind Turbines. *Energies* **2023**, *16*, 1088.
https://doi.org/10.3390/en16031088

**AMA Style**

Jodai Y, Hara Y.
Wind-Tunnel Experiments on the Interactions among a Pair/Trio of Closely Spaced Vertical-Axis Wind Turbines. *Energies*. 2023; 16(3):1088.
https://doi.org/10.3390/en16031088

**Chicago/Turabian Style**

Jodai, Yoshifumi, and Yutaka Hara.
2023. "Wind-Tunnel Experiments on the Interactions among a Pair/Trio of Closely Spaced Vertical-Axis Wind Turbines" *Energies* 16, no. 3: 1088.
https://doi.org/10.3390/en16031088