*2.1. Literature Background*

The CFD 2D numerical model presented in this paper was based on a wide literature background. The literature dealt mainly with larger rotors that in general operated in more stable conditions and higher Reynolds numbers, therefore their 3modelling3 usually led to accurate results. On the other hand, the present study focused on the development of a numerical 3model3 specifically dedicated to micro H-Darrieus rotor in which highly unstable conditions like boundary layer instability, dynamic stall, and blade-wake interaction affected the rotor operation more than in large rotors. However, numerous advices were found in the literature as reported hereinafter. Balduzzi et al. [13–15] provided essential guidelines for the development of accurate CFD 2D models of VAWTs, particularly concerning the spatial and temporal independency study. Rezaeiha et al. [16] provided the guidelines for the development of accurate CFD simulations for H-Darrieus rotors, focusing on the domain dimension and the azimuthal increment, which imposed the temporal discretization size. In this case, the authors found a good compromise using a smaller domain than Balduzzi et al. Actually, there is no accordance concerning the computational domain size, which, however, must be sufficiently large to guarantee that the flow around the rotor is not affected by the domain dimensions. Through wind tunnel experiments, Raciti Castelli et al. [17] validated a CFD 3modelling3 strategy, which regarded the near blade spatial discretization depending on the turbulence models. Recently, Rogowski et al. [18] developed a 2D CFD model of a two bladed H-Darrieus rotor. The model was validated using experimental data. Bangaa et al. [19] performed a numerical study on a single bladed VAWT under dynamic stall conditions using CFD. Additionally, the authors of the present paper previously developed a 2D CFD 3model for moderately large H-Darrieus [20], demonstrating the good accuracy of the transition turbulence model by Menter in cases in which the laminar to turbulent boundary layer transition played an essential role.

The above showed that, beyond the domain size, there was no strong agreement about the spatial and temporal discretization as well. All the reviewed papers agreed with the fact that the SST k-ω based turbulence models demonstrated best accuracy among the RANS turbulence models when used with sufficiently refined spatial and temporal discretization. For example, a y<sup>+</sup> less than one near the blades was universally considered an essential constraint. A time step size, which limited the azimuthal increment to less than 0.5 degree, was equally important. When boundary layer transition phenomena are detected, a transition turbulence model strongly increased the accuracy of the CFD models [20,21]. However, other authors evidenced the superior accuracy of advanced turbulence 4models4 like hybrid RANS-LES formulations, specifically when the rotors are subjected to unstable conditions. Strong dynamic stall, high blade-wake interaction, and vortices related to the flow separation were certainly more accurately predicted through the use of hybrid RANS-LES models like DES and Delayed DES. For example, Lei et al. [22] demonstrated that improved DDES simulation was able to capture real flow characteristics, like those generated by vortices related to dynamic stall phenomena, that were not predicted by the SST k- ω model. Li et al. [23] optimized the blade pitch in a two-blade H-Darrieus turbine using a 2D CFD model based on DDES simulation, therefore evidencing its superior predictive capability with respect to the RANS models. Thè et al. [24], in their thorough review, showed that DDES simulations were the best compromise between high accuracy and computation requirements for the simulation of VAWTs under unstable conditions. Simão Ferreira et al. [25,26] performed a complete numerical-experimental comparison between RANS, DES, and LES turbulence models using accurate Particle Image Velocimetry (PIV) data. They proved that "the DES model is not only able to predict the generation and shedding of vorticity and its convection, it also shows an acceptable sensitivity to grid refinement (both space and time), thus making it suitable for simulations where validation data are limited or non-existent. URANS models proved insufficient because of their inability to correctly model the large eddies, and the influence of this in the development of forces in the downwind passage of the rotor. The LES performed worse than the DES model, probably because of a less accurate 4modelling4 of the wall region." Again, DDES techniques demonstrated an ability to accurately predict massively separated turbulent flow structures in an Orthopter-type VAWT [27] in which the operating conditions are inherently unstable. Abdellah et al. [28] 4analyzed the effect of the spatial discretization when DDES simulations were implemented for VAWT rotors. Wang et al. [29] also demonstrated the high accuracy of DDES turbulence models for deep dynamic stall simulations at low Reynolds numbers of the NACA 0012 airfoil. The paper fixed grid and time step requirements such that the LES region was able to capture at least 80% of the turbulent kinetic energy of the flow. Furthermore, they evidenced the superior accuracy of the SST transition model for the RANS region among the RANS models at low Reynolds numbers. The above suggested that a DDES, based on a transition formulation for the RANS region, would be the optimal choice for the simulation of airfoil subjected to deep dynamic stall at low Reynolds number. This was certainly the typical operating condition of the H-Darrieus rotor in the present work. This option was studied by Sa et al. [30] on the flow past an Eppler 387 wing. The results indicated that the hybrid DES/Transition model predicted both strong laminar/turbulent transition phenomena, including the laminar separation bubble, and flow separation at high angles of attack. Therefore, the DDES turbulence model with the transition turbulence model by Menter [31–33], recently implemented in ANSYS Fluent solver, appeared to be very suitable for the scope of the present work. In order to support this assumption a comparison between the RANS fully turbulent SST k-ω model, the SST transition model and the DDES, coupled to the SST transition model, was carried out in the present paper.

#### *2.2. Computational Domain and Experimental Setup*

The experimental rotor was a four bladed H-Darrieus rotor with a ratio between height and diameter equal to one. Two endplates were used for the reduction of the tip losses in such a way to make the experimental results, as much as possible, consistent with the 2D simulations. Even though a 3D simulation would have been very interesting, the computation time would have been unaffordable, therefore a 2D simulation was the only way to gain a thorough insight into the blade aerodynamics. The blades were built in a resin 3D printer, which allowed for very high accuracy of the details and a very fine surface roughness. The endplates were 3D printed in PLA material. A steel threaded bar was used as shaft. Table 1 reports the geometrical and experimental rotor features while Figure 1 shows an image of the rotor assembled on the experimental setup.


**Table 1.** Experimental rotor features.

**Figure 1.** Experimental rotor.

The rotor was widely tested in the subsonic wind tunnel owned by the University of Catania. The wind tunnel was a closed circuit wind tunnel with a test section of 0.5 × 0.5 m, a maximum achievable flow speed of 31 m/s, and a maximum measured turbulent intensity of 0.4%. The test was carried out in open test section configuration. More details about the wind tunnel are reported in [11,34]. The flow speed was measured through the use of a Pitot probe, placed at the center of the test section inlet and at half diameter from it. The rotational speed was measured by means of a digital laser tachometer. The torque was evaluated by using a specifically designed braking system based on the principle of the belt brake. The instantaneous brake force was measured through a load cell and the torque was obtained. The rotor was anchored at the support structure through two needle roller bearings and the torque losses were experimentally evaluated as a function of the rotational speed. To measure the torque, the wind tunnel was set in such a way to obtain the maximum achievable flow speed in open test chamber. The rotor was free to rotate, without braking load, until an equilibrium

rotational speed was reached. A constant braking load was then applied, slowing the rotor down until a new equilibrium rotational speed was reached. The net instantaneous breaking force was measured through the load cell, and the average torque for the operating point was calculated. Then, keeping fixed the braking load, the flow speed was reduced by 2 m/s and the new torque was measured each time together with the new equilibrium rotational speed. This process was repeated in steps of 2 m/s until the rotor stopped. The experiment was with three different braking loads, five times for each. In order to obtain the fluid dynamic power, to be compared to the numerical simulation results for the validation, the power losses in the bearings were added at each measured operating point. The flow and rotational speed range is reported in Table 1, while in Figure 2 a sketch of the experimental setup is shown.

**Figure 2.** Experimental setup and detail of the mechanical brake system.

The CFD simulations were implemented for five different couples of flow and rotational speed to cover the entire range of measured tip speed ratio. Since there was no agreement about the adequate domain dimensions to have independent results, the first step in the development of the CFD 2D model was the definition of the suitable computational domain. In this case, the best compromise is shown in Figure 3. It was verified that larger domains did not affect the solution further in terms of torque prediction. The use of the symmetry condition for the lateral boundaries reduced the possible influences of the domain dimensions on the flow-field.

**Figure 3.** Computational domain dimensions and boundary conditions.

The boundary conditions used in the present work are shown in Figure 3. A velocity inlet condition was used for the inlet and a pressure outlet condition for the outlet. For the rotating zone, the domain was divided into three sub-domains in order to implement the unsteady sliding mesh model (SMM) in a rotating ring, which contained the four blades as in [20]. The internal circle and the outer domain remained stationary. The external and internal circumferences of the ring were thus set as sliding interfaces as highlighted in Figure 3.
