## 2 Methodology
### The Aerodynamic Model  
Concerning aerodynamics, MaPFlow [33; 34; 35] is used, which solves the compressible Unsteady Reynolds Averaged Navier-Stokes (URANS) equations under a cell-centered finite volume spatial discretization scheme. MaPFlow can handle both structured and unstructured grids, it is parallelized under the MPI protocol, and the grid partitioning is performed using the METIS library [36]. The convective fluxes are evaluated by solving the preconditioned local Riemann problem between the neighboring cells of each face, using the [PERSON]'s approximate Riemann solver [37] with the Venkatakrishnan limiter [38]. The viscous fluxes are discretized using a central 2 nd order scheme. For the reconstruction of variables at the interface a piecewise linear interpolation scheme is used. The evaluation of the spatial gradients of the primitive variables is done using the Green-Gauss formula, with a centered scheme approximation. Multiple options are available for turbulence modeling, such as the one-equation model of [PERSON] [39] or the two-equation model \(k\)-\(w\)\(SST\) of [PERSON] [40]. Regarding laminar to turbulent flow transition modeling, the \(\gamma\)-\(Re_{\theta}\) model of [PERSON] [41] is used. A Delayed Detached Eddy Simulation (DDES) approach is also implemented in MaPFlow, following the suggestions of [42]. Unsteady simulations are performed through an implicit second order backwards difference scheme [43], along with a dual time-stepping technique [44] in order to facilitate convergence. Finally, the implicit operator inversion is accomplished with the use of the Gauss-Seidel iterative method alongside with the Reverse Cuthill-Mckee reordering scheme. In the AL methodology implemented within MaPFlow, the blade loads are calculated on specific control points along the blade span through a blade element analysis. The control points aerodynamic forces are then applied to the flow as source terms of the flow equations. In order to avoid numerical instabilities, these source terms are smeared across the cells surrounding the rotating blades through a \(3D\) Gaussian convolution:  
\[\mathbf{f}_{CFD}=\frac{\mathbf{F}_{cp}}{\left(e\sqrt{\pi}\right)^{3}}\exp \left[-\left(\frac{r}{e}\right)^{2}\right]\]  
where:  
* \(\mathbf{f}_{CFD}\) is the projected force per unit volume [N/m\({}^{3}\)];
* \(\mathbf{F}_{cp}\) is the calculated force on the actuator line emission (control) point [N];
* \(r\) is the distance between the actuator line emission point and the point where the force is applied [m];
* \(e\) is the Gaussian kernel that defines the projection width [m]. There are numerous advantages when following this approach:1. CFD mesh is not forced to follow the blades rotation;
2. Overset grids methodologies can be avoided when interaction between rotor wake and other configurations (e.g., WT tower, helicopter fuselage, other rotors) is to be assessed;
3. Aeroelastic simulations can be performed without many implementation difficulties, as the the computational grid is not burdened to deform along with the blades. The above render the AL model befitting for complex aeroelastic analyses, provided that the loads are accurately predicted. The main weakness of AL is its dependence on tabulated \(2D\) airfoil data. The correct estimation of aerodynamic loads in stall conditions is critical in aeroelastic simulations when the aerodynamic options are based on 2D polars. For this reason, a dynamic stall model needs to be used. In simulations of this research, the extended ONERA model [45] has been used. The delay parameter of the original ONERA model, which accounts for dynamic stall delay in deep dynamic stall conditions is omitted. Moreover, the leading edge vortex and the vortex shedding effect considered by [PERSON] [46] in more recent versions of the ONERA model are not included in the present implementation.