## 3 3D BEM-DMST Model: Basic Formulation and High-Order Submodels
### Tip Losses  
Tip losses are commonly referred to as the difference between two-dimensional and three-dimensional configuration of a lift-driven device due to pressure equalization across the suction and pressure side at the tip. The effect was first considered through tip loss factor \(F\) defined by [PERSON] as the ratio between actual circulation (and hence lift) on the blade at the local spanwise section and the optimal circulation of Betz, assuming a helical rigid wake accounting for the rotational component without wake expansion. [PERSON]'s tip loss factor found in the literature is in the form of Equation (6) where \(F\) is a factor which tends to zero as the propeller radius \(r\) approaches the tip of the blade. Several expressions of \(F\) derived from [PERSON]'s theory for screw propellers (and excellent review is reported in [44]). By virtue of the different character of wake dynamics in VAWTs respect to HAWTs and propellers, [PERSON]'s modified version proposed in [45] is adopted in this work, where \(g\) is defined in Equation (7) as a function of the vertical coordinate \(z\) and the pitch of the wake \(s\).  
\[F=\frac{2}{\pi}\arccos\left(e^{-g(z)}\right) \tag{6}\]  
\[g(z)=\frac{-\pi(H-|z|)}{s} \tag{7}\]  
\[s=\frac{\pi V_{e}}{N\,\Omega} \tag{8}\]  
The linkage between Blade Element Theory and Momentum Theory inherited by traditional DMST formulation is employed to reach convergence on induction factors \(a\) to both upstream and downstream actuator discs.  
Once two-dimensional solution is converged, the tip loss factor \(F\) is applied to correct force coefficients including three-dimensional tip losses, following the approach proposed in [46].  
\[C_{N}^{r}=FC_{N} \tag{9}\]  
\[C_{T}^{r}=FC_{T} \tag{10}\]  
Flow parameters are calculated according to the original DMST approach so that relative velocity and angle of attack are obtained for each blade position \(\theta\).  
\[W_{rel,j}=\sqrt{\left[(V_{D,j}\,cos\theta_{j}+\Omega r)\,cos\gamma_{c} \right]^{2}+(V_{D,j}\,sin\theta_{j}\,cos\delta_{i})^{2}} \tag{11}\]\[\alpha_{j}=\beta_{c}+\arcsin\left(\frac{V_{D,j}\sin\theta_{j}\cos\delta_{i}}{W_{rel, j}}\right) \tag{12}\]  
where subscript \(j\in[0,\,2\pi]\), \(H\) is the blade semi-height, \(z\) is the vertical coordinate (\(z=0\) at equatorial plane) and \(V_{e}\) corresponds to the inter-disc velocity, whereas \(V_{D}\) assumes the values of \(V\) or \(V^{\prime}\) when respectively the upstream or downstream disc is of concern. The stagger and sweep angles, for unstaggered and straight blades, likewise for those examined in this research, \(\beta_{c}\) and \(\gamma_{c}\) are both null and \(\delta\) is the angle between the blade normal and the equatorial plane.  
The proposed distributed formulation of the tip loss correction is particularly relevant in this work, since it perfectly fits for the 3D character of present BEM code. Since the discretization along the turbine height is anyway necessary to consider general rotor shapes, it can be exploited to calculate \(F\) at each spanwise location. Conversely, BEM analyses found in the literature for VAWT frequently apply the Prandtl-Lanchester [47] as single overall correction to introduce the effects of the finite blade length and the induced drag losses due to tip vortices. The correction is applied to the midsection where only two-dimensional effects are assumed and predicts a loss of performance enforcing a depletion of angle of attack over the blade span. The hypothesis underlying Prandtl-Lanchester theory refers to elliptical loading under lifting line and linear aerodynamic behaviour of the airfoils, assumptions which do not hold in common VAWT operation for that post-stall conditions and dynamic stall hysteresis occurs. For these reasons, the proposed 3D formulation of the tip loss correction appears more physically consistent and of more general validity with respect to the lumped-parameter approach of the Prandtl-Lanchester one.