Keywords

1 Introduction

Small-scale wind power systems are increasingly focused on Vertical Axis Wind Turbines (VAWT) due to their improved performance in turbulent conditions compared to Horizontal Axis Wind Turbines (HAWT). Particularly, the H-Darrieus VAWT solution has received significant attention due to benefits such as low wind speed and omnidirectional operation. Equations 1 and 2 express the most common operational and performance parameters of this type of turbomachinery, Tip Speed Ratio (TSR) and Power Coefficient (Cp).

$$ Cp = \frac{T\omega }{{\frac{1}{2}\rho AU_\infty^3 }} $$
(1)
$$ TSR = \frac{\omega R}{{U_\infty }} $$
(2)

where T is the torque generated by the turbine, ω is the angular velocity, ρ the specific mass of the air, A the frontal area swept by the rotor, U∞ the undisturbed wind speed and R the radius of the rotor.

Computational fluid dynamics (CFD) simulations are valuable tools for predicting wind turbine performance. It reduces the need for physical testing, but sometimes the computational cost can be high or even prohibitive [1, 2]. Regarding H-Darrieus VAWT, 2D simulations are reliable but require solving an unsteady flow, which increases computational cost. Proper simulation setup is crucial for balancing accuracy and computational time. This study proposes a simplifying methodology for a CFD simulation of a three-bladed H-Darrieus VAWT by using dimensionless numbers to assess discretization and simulation stability.

2 Methodology

Simulations were run on a workstation with Intel Xeon E-2236 6-core processors. The model set up followed article [3], where authors carried H-Darrieus VAWT simulation using ANSYS Fluent ®. Convergence criteria was based on Cp variability during turbine rotation and the total cpu-clock time was recorded. Aerodynamic performance results were associated to dimensionless numbers Grid-Reduced Vorticity and Reference Courante Number Co* [4] which indicating mesh and time step suitability, respectively.

Equation 3 represents the dimensionless Courant number (Co):

$$ Co = V\frac{\Delta t}{{\Delta x}} $$
(3)

where V is the necessary velocity for a particle to cross an element of size Δx, and Δt is the time-step adopted in the simulation. For Co*, Δx must represent the perimeter of the rotor blade divided by the number of elements that the discrete and V the peripheral velocity of the blade. According to [3, 4], the recommended value of Co* for turbomachinery should be less than 10.

Equation 4 represents the dimensionless number GRV:

$$ GRV = \frac{\omega_v }{{{\raise0.7ex\hbox{${V_0 }$} \!\mathord{\left/ {\vphantom {{V_0 } {L_0 }}}\right.\kern-0pt}\!\lower0.7ex\hbox{${L_0 }$}}}} $$
(4)

where ωv represents the vorticity, V0 the local velocity and L0 the representative length or local element size. Authors of [4] recommends GRV values of up to 0.01 in the critical regions of the flow, such as near the blades and in areas of vortex shedding.

2.1 Computational Domain

Figure 1 shows the computational domain, including auxiliary lines, boundary conditions, and their locations. The rotor has an 850 mm radius (Rturb) and 246 mm chord length (c) of cambered NACA0018. The domain comprises a rectangular stationary zone (SZ) which surrounding a circular rotating zone (RZ). Blade position along its circular path is represented by azimuthal angle (θ).

Fig. 1.
figure 1

Computational Domain and boundary conditions locations.

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2.2 Spatial Discretization

Non-structured triangular elements mesh was generated, except for the use of quadrilateral elements in the near-wall regions of blade surfaces. Sizing controls tools were utilized on auxiliary lines to regulate element size, ensuring Skewness and Orthogonal Quality mesh metrics were maximum 0.6 and minimum 0.5 respectively. Figure 2 displays mesh details.

Fig. 2.
figure 2

Grid details. (a) SZ, (b) RZ, (c) trailing edge, (d) leading edge.

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2.3 Modelling setup

The ANSYS Fluent® solver software was configured with a 2nd order upwind scheme for spatial discretization and a bounded 2nd order implicit scheme for temporal discretization. The k-ω SST turbulence model, which requires a y+ value close to 1, was found to be the best performer for the case. This is supported by references [5,6,7], which demonstrate satisfactory results obtained by using this model for turbomachinery. Moreover, pressure-based solver type, coupled solution algorithm, and 30 iterations per time step were used. The boundary conditions included a velocity inlet of 8 m/s in the x-axis direction, a pressure outlet of 0 Pa (gauge), symmetry on the side edges, a sliding interface between the SZ and RZ, and a wall on the blades with a no-slip condition and stationary in relation to the RZ.

3 Results and Discussion

The authors of [3] suggest using a convergence criterion of 0.1% deviation in Cp value over two subsequent rotor revolutions. In this study, the criterion was met prematurely and produced unrealistic Cp values, leading to a revision requiring at least three consecutive revolutions with four decimal place precision. Table 1 displays Cp results for five operating points along with relative percentage error compared to [3], number of revolutions required for convergence, and simulation time.

Table 1. Cp error in relation to reference [3] results.
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Comparing with CFD data from reference [3], errors were below 10%, considered sufficient for methodology validation. Total simulation time using one reference workstation was estimated at 1600 h, and with a mesh sensitivity study, it would be at least three times longer.

Concerning spatial discretization, it's worth mentioning that y+ results, not included here for conciseness, indicated reliability of the k-ω SST turbulence model. The GRV number showed similar magnitudes and behaviors throughout a complete rotor revolution. Figure 3 displays a variation between 0.022 and 0.032 in maximum values.

Fig. 3.
figure 3

GRV along a complete rotor’s revolution at different operating points.

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Comparing Fig. 3 with Table 1, it can be observed that at TSR of 1.1 (operating point with highest prediction errors), the GRV behavior is more chaotic and has a larger amplitude between minimum and maximum values. Conversely, at other operating points, the amplitude is smaller, and behavior is more uniform.

Table 2 shows that Co* is below the recommended upper limit of 10 in temporal discretization, indicating potential to increase the time-step, resulting in decreased simulation convergence time. Like GRV, Co* does not significantly vary after a few rotor revolutions, allowing evaluation of the time-step without waiting for convergence.

Table 2. Maximum value of Co* during one complete revolution at different operating points.
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4 Conclusions

The CFD methodology applied to a small Darrieus H TEEV in reference [3] was successfully replicated in this study. Results in Table 1 showed small errors compared to reference methodology, demonstrating its validity. However, modifications and adaptations were necessary, such as the adaptation of the simulation convergence criterion. It is recommended to consider the simulation converged after Cp does not vary for three consecutive rotor revolutions at four decimal place precision.

The high computational demands of the simulation (1600 h for five operating points) emphasize the significant resources required for predicting aerodynamic performance of H-Darrieus VAWT. Monitoring GRV from the start of the simulation allows for early assessment of spatial discretization adequacy, avoiding the need to wait for Cp convergence and simplifying methodologies requiring multiple simulations, such as optimization, by eliminating mesh sensitivity studies. Results in Fig. 3 suggest that GRV values up to 0.022 indicate adequate mesh and reliable Cp results for the three-bladed H-Darrieus VAWT.

Finally, after establishing an appropriate GRV number, the time-step should be adjusted to result in Co* < 10. Like GRV, Co* stabilizes in the early revolutions, making it useful for determining the time-step. Setting limits on GRV and Co* can be beneficial in cases of limited computational resources or tasks requiring many simulations.