The rotor is placed at the center of the wind tunnel. The direction of inflow is X-axis, and the rotation axis of the rotor is Z-axis. Taking the diameter (D) of the rotor as the standard, the distance of the rotor with inlet and outlet is set as 10D and 15D. The area of the wind tunnel was 20D*20D (Dessoky et al., 2019a). The influence of rotation axis, beams bolts and other structures were ignored during the calculation. The boundary condition of the inlet and outlet were set at velocity inlet and pressure outlet. The velocity of the inlet was set as a constant value of 4 m/s, while the pressure value of the outlet was atmospheric. The surrounding wind tunnel was set as a pressure outlet to avoid affecting the simulation results. The blades and the WGD are set to a no-slip wall. In order to ensure the stability of the grid of the rotation process, the computational domain was divided into static domain and rotation domain. The sliding grid method was adopted of the interface of the computational domain, which could ensure that the grid does not cause deformation during the rotation process. The same grid setup was used for the interface boundary of the static domain and rotation domain to ensure gradient continuity. The grid growth rate is 1.1, and the maximum grid size is set to C with a grid quality of 0.28. As shown in Figure 3, on the blade and WGD surfaces, 30 layers of boundary layer grids were formed, with the first layer cell having a thickness of 3.93 × 10⁻⁵ m to guarantee that Y+ is nearly equal to 1 (Bangga et al., 2020).

The grid independence was studied because the grids have a greater impact on the results in numerical simulation. The optimal grid parameters were selected considering the accuracy and time spent. The torque coefficients of SB-VAWTs with WGD at 30° azimuth in this study are used to compare the results in steady simulation with different grid numbers. The size of the grid near the blade plays a decisive role in the calculation process, and the external grid has less influence on the calculation results, so the focus is on the grid near the blade. In the grid study, the number of blade section nodes of 100, 170, 240, 310, and 380 and the total number of grids of 5,253,067, 16345876, 27563886, 38645036, and 49034287, are used for comparison, respectively, under all other conditions being the same (Dessoky et al., 2019b). The values simulation result increased with the increase of the grid number when it was less than 38.6 million. When the number of grids exceeded 38.6 million, the values simulation result kept almost the same. The grid number was decided as 38.6 million, considering the accuracy and time spent. A small adjustment was made according to the changes in the shape of WGD, i.e., to increase the number of nodes at the upper edge and the polyline.

In order to verify the validity of the numerical simulation, a wind tunnel test was carried out on the static torque performance of the SB-VAWT without WGD at the wind speed of 4 m/s. The results are shown in Table 1. Both the numerical simulation and the tests are basically the same. Therefore, the simulation results can be used to evaluate the aerodynamic performance of the SB-VAWT to a certain degree. However, the data of simulation is higher than the tests. The two main reasons can be considered. First, the simulation model was simplified so that the shaft, beams, flanges, and bolts were ignored. However, during the wind tunnel tests, these components inevitably affected the overall aerodynamic performance. Second, compared with the stable conditions in the simulation process, the environment was more complicated during the tests, which led to differences in results.

After obtaining the optimal structural parameters of the WGD by numerical simulation, a real rotor with the optimal WGD was designed and made. Wind tunnel tests were carried out on the starting performance, power performance, and rotational revolutions performance of the SB-VAWT with and without WGD. The wind tunnel tests were conducted by using a low-speed open type one with a test section of 1 m × 1 m, and the turbulence intensity is 5% in Northeast Agricultural University in this study. The test model and parameters of the SB-VAWT and WGD are shown in Supplementary Figure S1 and Supplementary Table 2.

The range of wind speed supplied by the wind tunnel is from 1 m/s to 20 m/s with the ±3% precision. Four kinds of wind speeds, 4 m/s, 5 m/s, 8 m/s, and 10 m/s, were used for the test. An ultrasonic wind rotation sensor (GILL Wind Sonic, England) with ±0.1% precision was used to measure the wind speed of the wind tunnel. The rotor was set up at the center of the wind tunnel outlet and 0.6 m distance from the outlet. A three-phase asynchronous induction motor with brakes (Mitsubishi, Japan) was used to control rotational revolutions by a variable frequency drive, and the rotational revolutions were measured by intelligent digital. A torque meter (ONO SOKKI, Japan) with the ±0.2% precision was used to measure the torque and rotational revolutions of the rotor, which was placed between the rotor and three-phase asynchronous motor.

For the static torque measurements, the rotor axis was fixed by the brake of the induction motor. The static torque was measured at the interval of about 10° of azimuth angle and was averaged for the 60s to obtain the average starting torque M _as. Because there are four blades of the rotor, the rotation period is 90°. Therefore, only the static torque during the azimuth angles from 0° to 90° was measured.

For the measurement of dynamic characteristics, the rotor was operated with a constant rotational revolution under steady wind speeds. The dynamic torque was measured for the 60 s to obtain the average dynamic torque M. The whole dynamic torque characteristics were obtained by changing the conditions of rotational revolutions. The output power coefficient was calculated by using torque coefficients and tip speed ratios. The revolutions were tested under the condition of no loads on the rotor. The variation in rotational revolutions of the rotor from standstill to steady is measured (time not exceeding 600s).

In this study, some performance parameters were used, including starting torque coefficient C _ms, power coefficient C _p, and the tip speed ratio λ. The computational formulas are shown as follows: C m s = M s 1 2 ρ A U 2 R ; (2) C P = P 1 2 ρ A U 3 = λ M 1 2 ρ A U 2 R ; (3) λ = 2 π n R 60 U , (4) where M _s is the static torque, (N• m); ρ is the air density, (kg/m³); A is the rotor swept area, (m²); U is the wind speed, (m/s); R is the radius of rotation of the rotor, (m); P is the power, (W); M is the dynamic torque, (N• m); n is the rotational revolutions, (r/min).

Glauert’s method (Glauert, 1933) has been used to correct the data for wind tunnel blockage effects. As shown in Eqs 5-7, u _T is the stream-wise velocity through the rotor, and C _m is the torque coefficient. Once U′ is known, the starting torque coefficient (C _m’) and the power coefficient (C _P’) can be calculated. U ′ = U ( ( u T / U ) 2 + C m / 4 ) u T / U ; (5) C m ′ = M 1 2 ρ A U ' 2 R ; (6) C P ′ = M ω 1 2 ρ A U ' 3 . (7)

In this study, the rotational revolutions, torque, and wind speed were measured, so the error transmission formula of C _m and C _p could be obtained as shown in Eqs 8, 9: W m C m = [ ( W n n ) 2 + ( W M M ) 2 + 4 ( W U U ) 2 ] 1 / 2 ; (8) W P C P = [ ( W n n ) 2 + ( W M M ) 2 + 9 ( W U U ) 2 ] 1 / 2 . (9)

Among them, W _m, W _P, W _n, W _M, and W _U represent the errors of torque coefficient, power coefficient, rotational revolutions, torque, and wind speed, respectively. According to the above equation, uncertainty values can be obtained. The errors of the ultrasonic wind speed sensor, rotation sensor, torque sensor, power coefficient, and torque coefficient are 3, 0.1, 0.2, 0.632, and 0.422%, respectively, as shown in Supplementary Table S3.

A total of two factors and five levels were coded for design. Based on the previous research results (Yan et al., 2018), the range of the upper edge radius R (m) was 0.36–0.48m, and the range of the distance ΔL (m) was 0.01–0.04 m. The coding table is shown in Supplementary Table S4.

The average starting torques (M _as) of the rotor with different structural parameters WGD were obtained by simulation, as shown in Table 2. The improvement effect of the parameters combination of the R (m) = 0.42 m and the △L (m) = 0.025 m was optimal, and the M _as is 0.058124 (N•m). The R (m) = 0.3776 m and the △L (m) = 0.0144 m was the slightest effect, and the M _as is 0.0470106 (N• m).

According to the tests design results, the response surface was obtained, as shown in Figure 4. Along with the increase of R (m) and △L (m), the torque shows a trend of first rising and then falling, which the optimal R (m) was in the range of 0.415–0.424m, the optimal range of △L (m) was from 0.025 to 0.0303 m. This is because when the upper edge radius was too small, the slope of the WGD section was too large, the stability of the airflow into the rotor and the WGD would be affected. When the radius of the upper edge was too large, i.e., the length of the upper edge was longer, a lot of process loss would be cured of the airflow through into the rotor. When the distance between WGD and rotor was small, the rotor was close to the low-speed area at the bottom edge of the WGD. However, the distance was too large, so the effect of WGD was weakened.

The equation was obtained as shown in Eq. 11. The optimal structural parameters of the hexagonal pyramid-shaped WGD were obtained. The R is 0.418m, the R ₁ is 0.3m, the H is 0.12m, and the ΔL is 0.03 m. Y = 10.32259 + 1.65901 * 10 − 3 x 1 + 2.35388 * 10 − 3 x 2 − 2.10963 * 10 − 6 x 1 x 2 − 1.9116 * 10 − 6 x 1 2 − 2.49016 * 10 − 5 x 12 2 , (11) where x ₁ is the radius of the upper edge-inscribed circle, and x ₂ is the distance between WGD and rotor.

Figure 5 shows the velocity contours of the WGD. It can be seen from the figure that the flow separation occurs close to the upper edge of the WGD so that the upper edge does not play a role in gathering wind. The distance between the flow separation position and the upper edge is 0.02m, and the cross-sectional-inscribed circle radius is 0.398 m. So, the line shape was optimized to a polyline shape at the separation position, which will improve the phenomenon of flow separation. The separation position had moved to the upper edge of WGD to increase the wind gathering.

Under the condition of the other WGD parameters keeping unchanged, the upper edge radius of WGD at the distance of 0.02 m was changed. The value range of the change of the upper edge was set from 0.418 to 0.458 m. The interval of the upper edge on numerical simulation was set as 0.01m, which equaled that the angle change was 2°. The M _as of SB-VAWT with WGD with hexagonal pyramid shape was 0.0586. The M _as of SB-VAWT with WGD with polyline shape was 0.059429 when the upper edge radius was 0.438 m. Finally, the optimal upper edge radius was decided as 0.438 m.

Based on the simulation, a WGD with the optimal structural parameters under the condition of this study was obtained. Wind tunnel tests carried out the static torque of the SB-VAWT with and without WGD. Figure 6 shows the results at wind speeds of 4 m/s, 5 m/s, 8 m/s, and 10 m/s.

As mentioned in Figure 2 of Section 3.1, there are four kinds of the mounting angle of WGD against wind direction, including 0°, 15°, 30°, and 45°. Wind tunnel tests were carried out under all four conditions. In order to facilitate the result comparison of the rotor without WGD, the test data of the rotor with the WGD at different mounting angles are shown in the shaded region. As shown in Figure 6, the starting torque coefficients of the rotor with WGD have been greatly improved at almost the azimuth angles for all wind speeds. This indicates the effectiveness of the WGD on the starting performance improvement for the SB-VAWT. The best improvement of starting torque coefficient at the azimuth angles of 30°, which the increasing percentage was 34.18, 25.39, 21.95, and 21.67% at the four wind speeds. This indicates that the WGD has a better function on the starting performance of the SB-VAWT at lower wind speeds.

Figure 7 shows the streamline around the rotor with and without WGD at the azimuth angle θ = 30°. According to Figure 7, the disorderly flow field improved when the WGD was installed. More wind flowed into the rotor through WGD. The vortex on the tip of the blade had been weakened, which means that the loss of energy on the tip of the blade had been reduced, and the performance of the upstream blade also will be improved. The vortex near the blades had moved away from the downstream blades, which concentrated on the WGD slope and reduced the downstream drag blades to improve the aerodynamic characteristics of the SB-VAWT.

The schematic diagram of the different cross-sections is shown in Supplementary Figure S2. The center and the top position of the rotor were selected as z = 0 and z₁ = 0.25. The mounting angle φ = 15° of the WGD was taken as an example to compare the pressure contours of the rotor with and without WGD at the plane of z and z_1, as shown in Figure 8.

It can be seen from Figure 8 that the pressure distribution in the z and z₁ planes was significantly different for SB-VAWT. The pressure difference on the blades at the z plane was higher than the z₁ plane. The aerodynamic force was provided from the middle part of the blade. The loss of energy occurred on the tip of the blades. After WGD was installed, the negative pressure near the blade at the z plane was expanded. At the same time, both of the range of positive and negative pressure in the z₁ plane were enlarged. It means that the pressure difference was increased on the blade, and the loss of energy on the tip of the blade was reduced. More energy will be applied by the rotor, which will improve the starting performance of the SB-VAWT significantly.

The velocity distribution diagrams of the rotor with and without WGD in z and z₁ planes at azimuth angles of 30° are shown in Figure 9. It can be seen from the figure that the airflow speed through WGD into the SB-VAWT was increased and obvious at the z₁ plane. In the wake area, the velocity of the rotor with WGD was lower, which showed that more energy was absorbed by the rotor, causing larger velocity attenuation in the wake. In summary, the installation of WGD has improved the internal flow field. The velocity increase of the airflow into the SB-VAWT and the velocity attenuation of the wake part increased significantly.

The power coefficient of the SB-VAWT with and without WGD was obtained by wind tunnel test, as shown in Figure 10. It is the same as the previous data processing method of the static torque coefficient in Section 5.2.1, the test data of the rotor with the WGD at different mounting angles are shown in the shaded region in Figure 10. It can be seen from the figure that the power coefficients of the rotor with the WGD have been improved at almost all wind speeds. For the lower wind speeds condition such as 4 m/s and 5 m/s, the improvement of the power coefficient is significant. Compared with the rotor without WGD, the maximum power coefficient increased by 151.03 and 94.56%, respectively. Furthermore, a different degree of improvement of power coefficient can be found for almost all tip speed ratios. Therefore, it can be said that the WGD can increase the power performance of the SB-VAWT at lower wind speeds. For the higher wind speeds such as 8 m/s and 10 m/s, the improvement of the power coefficients by the WGD can also be found. The maximum power coefficient just increased by 28.02 and 16.07%. This indicates that the WGD can positively affect the power performance improvement at lower wind speeds.

Figure 11 shows the pressure distribution of the rotor with and without WGD at different cross-sections at a tip speed ratio of 1.0 and azimuth of 0°. According to the figure, both in the z and z1 planes, the pressure difference near the blades was effectively increased when the WGD was installed. This results in increased torque and promotes SB-VAWT rotation, which improves aerodynamic performance.

Figure 12 shows the speed distribution at different cross-sections of the rotor with and without WGD at the tip speed ratio of 1.0 and azimuth of 20°. The difference in the high-velocity area of the rotor with and without WGD at the z plane seems inconspicuous. However, at the z₁ plane, the overall airflows velocity of the rotor with WGD was increased. The low-velocity area of the wake with WGD was expanded, which means that the rotor absorbed more energy in the airflow. Based on the flow characteristics, the power performance of the rotor with WGD was improved significantly.