## 3 3D BEM-DMST Model: Basic Formulation and High-Order Submodels  
The in-house BEM tool presented in this work was conceived to assess the performance of VAWT's featuring any general shape of the rotor, employing the most recent improvements available in the literature, and with the aim of reproducing the effects of 3D flow around VAWT's. The code is based on the steady-state approach of the Blade Element Momentum Theory, reformulated according to the so-called Double Multiple Stream Tube (DMST) model inherited by [PERSON] [4]; the variable induction factor formulation improves the level of local load prediction accuracy by considering upwind and downwind induced velocities as a function of the rotation angle of each blade. The chosen reference frame is depicted in Figure 1. The basic formulation is enriched by sub-models and corrections making the DMST able to account for several 2D and 3D phenomena affecting  
Figure 1: DMST model reference frame. \begin{table}
\begin{tabular}{l c} \hline \hline Blades & 3 \\ Blade height & 1.902 m \\ Rotor diameter & 2.028 m \\ Solidity & 0.310 \\ Chord & 0.101 m \\ Blade airfoil & DU06W200 \\ Rotor tilt & 0\({}^{\circ}\)/15\({}^{\circ}\) \\ Swept area & 2.63 m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: DeepWind rotor geometric parameters. VAWTs aerodynamics and performance. The following sub-sections present in detail all the corrections, which are now briefly listed. First considering 2D flow features, the unsteady operation of the VAWT blades induces large variations in angle of attack on the airfoils with high peak values (both positive and negative); the operation at large angle of attack might make the blade to operate in the post stall region, for part of the revolution, requiring the extension of profile polars beyond the static stall limits. Beside this, the change of relative velocity along the chord produced by the blade cycloidal motion, requires the introduction of a so-called \"flow curvature correction\" [34]. The inherent unsteadiness of VAWTs makes the aforementioned incidence variation to occur in relatively short time scales and partly within the post stall region, resulting in a hysteresis loop of lift and drag forces, phenomenon known as \"dynamic stall\". Semi-empirical models have been treated in the literature and show a relative ease of implementation in DMST codes such as the one proposed by [PERSON] [35] for helicopter applications and its modifications by [PERSON] [36], [PERSON] [2] and [PERSON] [37]. Further studies brought to indicial approaches, e.g., the most renowned proposed by [PERSON] [38]. These physical-based models consider distinct contributions for the unsteady aerodynamics, leading edge and trailing edge flow separation. The implementation of these methods still relies on the availability of empirical constants which are Reynolds-dependent: literature offers a wide range of data at high Reynolds regimes that infrequently find space in small scale VAWT applications featured by typical low Reynolds number levels, for that a lack of experimental tuning of the empirical constants can be shown. Focusing on 3D flow features, we first note that the finite length of the blade leads to the spanwise change in the aerodynamic forces (and hence in the vorticity) and eventually results in the formation of the tip vortex, so a model has to be adopted to consider the effect, either in an integral or in a distributed way. Moreover, VAWTs comprise supporting structures such as the spokes and pole which contribute to power losses and are modelled as a substantial drag augmentation affecting the torque harvested by the rotor. The effects of struts are evaluated as function of the azimuthal angle. Losses caused by wake-blade interaction still remain at an higher order of resolution, unaffordable for BEM codes. However, the wake can be considered nearly vanished when the blade passes through the downstream region as inferred by [PERSON] et al. [39] simulating the wake structure interaction using CFD.