**2. Materials and Methods**

The numerical simulation of the Savonius turbine with deformable blades was performed within the FSI method available in Ansys Workbench v19.2. The geometry and position of the rotor blades change continuously and, thus, transient simulations were performed both for a structural analysis with the ANSYS solver and a fluid flow analysis with the ANSYS fluent solver.

Full 3-dimensional (3D) simulations of the turbine were not possible due to enormous requirements of the coupled fluid flow and structural solvers. Therefore, it was decided to solve the problem as the 2D one, actually quasi-2D, as the structural solver demanded a three-dimensional model to be applied. Despite its limitations as 3D effects at the blade ends are disregarded, a 2D approach is a frequently applied simplification, which allows one to learn about performance of the turbine configuration. It is especially useful to observe changes in the performance if different configurations of the blades are compared. However, one can keep in mind that significant differences can be obtained comparing a 2D prediction with the full three-dimensional one, especially when the aspect ratio of Savonius turbines is low [34,35].

The geometry of the blades, the guidance system and the fluid domain were prepared in SolidWorks. In general, the guide ring can be of any arbitrary smooth shape, preferably elliptical. However, in this first study the guide ring of the constant diameter *D* = 1 m was selected. It was also the value of the diameter of the reference turbine rotor with fixed-shape blades, where the eccentricity was equal to zero. The blades of that reference rotor had a semi-circular shape. In the case of the rotor with eccentricity, the arc length of the blades was the same as in the reference rotor, but the distance between outer edges of the deformable blades was variable during the rotor revolution. In order to simplify the numerical model, it was decided to disregard the rotor shaft. Its impact on the flow around the rotor blades is rather limited and it can be assumed to be similar in all configurations. The blade overlap can have a positive effect on the turbine performance. However, in those investigations, it was decided not to overlap the blades.

As one can see in Figure 2, the fluid domain was divided into two regions. The internal domain of the diameter 1.5*D* including turbine blades was surrounded by the external one. The total length of the domain was 60*D* in the flow direction, with the turbine axis located 20*D* from its inlet and in the middle of the domain height, which was set to 40*D*. The domain blockage was similar to our previous studies [34] or in [18] and it did not affect comparisons between different turbine configurations.

Due to the complexity of blade deformations, combined with their rotation, a tool outside the fluid flow solver was needed to define the instantaneous rotor geometry. It was decided to use the structural ANSYS solver in order to take advantage of the FSI method implemented in the Ansys Workbench. A one-way system coupling was defined between simulation components, where the deformation of blades obtained in the structural analysis was transferred to the fluid flow solver. Because the pressure variation around the blade for the considered wind speed (*v* = 4 m/s) was less than 100 Pa, the two-way coupling, where the pressure load on blades would be transferred from the flow to the structural analysis, was disregarded. The one-way system coupling method was successfully used and presented in [27].

**Figure 2.** Computational domain scheme (external domain dimensions do not correspond to the scale).

In order to satisfy a good coupling of the structural and flow parts of the problem, the same timestep of transient simulations was applied to both of them. The value of the timestep was selected on the basis of solution stability tests performed for the rotors with the highest magnitude of blade deformation. The typically accepted timestep corresponding to the revolution of the rotor by 1◦ was selected initially in the Savonius rotor simulations. However, due to solution instabilities during the remeshing procedure, it was successively reduced. In the case of the timestep equal to 0.001 s, the numerical errors resulting from the mesh deformation and the remeshing algorithm were very limited and the computations were successful. The simulations were carried out for the tip speed ratio *TSR* = 0.8, for which Savonius rotors of typical aspect ratios (*AR* = *H*/*D* = 0.8–1.5) reach the maximal value of the power coefficient *Cp* [34,36]. For this *TSR*, the angular velocity of the turbine was 6.4 rad/s. Thus, the selected timestep of 0.001 s corresponded to a revolution of the rotor by approximately 0.367◦, resulting in 982 steps per one revolution, which is sufficiently low as far as the time discretization is concerned [16]. The total time of simulations was 10 s, which corresponds to more than 10 full revolutions, thus eliminating an influence of the initial conditions onto the simulation results.

The tip speed ratio *TSR* Equation (1) and the power coefficient *Cp* Equation (2), i.e., the energy extracted by the turbine to the available wind energy, were defined as follows:

$$TSR = \frac{aR}{v} \,\,\,\,\,\tag{1}$$

$$\text{Cp} = \frac{T\omega}{0.5\rho v^3 A'} \tag{2}$$

where: ω—angular velocity [rad/s], *R*—turbine radius, *v*—wind speed [m/s], *T*—output torque [Nm], ρ—air density [kg/m3], *A*—projected area of the rotor (*DH*) [m2], *D*—rotor diameter (2*R*) [m], *H*—rotor height [m].
