## 3 3D BEM-DMST Model: Basic Formulation and High-Order Submodels
### Limitations of the Model  
The in-house BEM code contains most of the sub-models in the literature, formulated to be applied to general VAWT architectures (hence, to the H-shaped and Troposkein rotors considered in this study).  
As a first note, polar curves have a dramatic impact on the performance estimate and they have been found to be highly sensitive to procedure followed in their construction: this remains a critical step for all those algorithms which rely on aerodynamic look-up tables; experimental and numerical simulations, CFD or panel codes are very sensitive to set-ups (e.g., sensitivity to turbulence and surface roughness [52]), especially at low Reynolds number regimes.  
Moreover, the dynamic stall model here employed is based on semi-empirical constants (e.g., \(\gamma_{L}\), \(\gamma_{D}\) and \(K_{1}\)), which are obtained by experimental data fitting on a wide range of airfoil series, hence a margin of uncertainty has to be accepted since no dedicated experiments are available on airfoils such as NACA0021 and DU06W200. A study on the uncertainty propagation ascribed to dynamic stall constants can be found in [53]. Alternative to the present approach, several authors employ the Leishman-Beddoes model, since this formulation includes the effects of unsteady airfoil aerodynamics both within and beyond the stall limit, including the different effects of stall occurring at leading and trailing edge. Anyhow, this model is still based on empirical constants, which vary with airfoil shape and that are only available for high Reynolds number regimes: thus, so far the model does not find a proper application in the field of small-scale VAWTs.  
Finally, the code does not consider stream tube expansion from the upstream to the downstream discs, thus the mass conservation through the actuator discs is not rigorously fulfilled. Despite this, small VAWTs are not particularly sensitive to flow expansion [18], hence the present formulation is deemed suitable for the validation of test cases and application to distributed scale machines (though simplified in the application to the large scale DeepWind turbine).  
A great influence on micro and small VAWTs is represented by the role of turbulence. Turbulence intensity and scales dramatically affect VAWT performance, although it is very challenging to take into account turbulence effects in engineering codes. The way they can be addressed is to consider the effect on airfoil polars: (1) a parametric analysis of the airfoil performance based on several CFD computations varying the inlet turbulence level (2) a parametric analysis with panel codes such as XFOIL (used in this study) varying the number \(N_{crit}\) which is the parameter linked to the turbulent intensity used in the \(e^{N}\) method. Since BEM analyses, for the here reported validation, have been carried out with models tested in the PoliMi wind tunnel where turbulence level was very low (\(<\)1%), 3D BEM code primarily simulates study cases where turbulence impact is small as first instance. However, turbulence modelling in 3D BEM code is particularly suited for open-field applications such as the one treated herein. A future work is conceived to assess turbulence effects trough a sensitivity analyses on \(N_{crit}\) value with XFOIL tool.