## 2 Description of the Numerical Tools  
### The Eulerian Solver--MaPFlow  
MaPFlow is a compressible, cell-centered finite volume CFD solver, which employs both structured and unstructured grids. The convective fluxes are discretized using [PERSON]'s approximate Riemann solver [18] together with the Venkatakrishnan limiter [19], while the viscous fluxes are discretized using a central 2 nd-order scheme. Turbulence closure includes several options; the one equation turbulence model of [PERSON] (SA) [20], as well as the two equation turbulence model of [PERSON] (\(k-\omega\) SST) [21]. MaPFlow can handle both steady and unsteady flows, the latter being an important requirement for aerodynamic tools employed in helicopter analysis. Time integration is achieved in an implicit manner permitting large CFL numbers. The unsteady calculations use a 2 nd order time accurate scheme combined with the dual time-stepping technique to facilitate convergence. Additionally, flows in the in-compressible regime are realized using Low Mach Preconditioning [22]. Let \(D\) denote a volume of fluid and \(\partial D\) its boundary. By integrating the governing equations over \(D\), the following integral form is obtained:  
\[\frac{\partial}{\partial t}\int_{D}U_{E}dD+\oint_{\partial D}(F_{c}-F_{v})dS= \int_{D}QdD \tag{1}\]where, \(U_{E}=\{\rho,\rho\vec{u},\rho E\}^{T}\), \(F_{c,i}=\left\{\,\rho\,u_{n},\,\rho u_{i}\,u_{n}+u_{i}\,p,\,\rho H\,u_{n}\right\}^ {T}\) and \(F_{v,i}=\{0,\,n_{j}\,\tau_{ij},\,n_{j}\,u_{i}\,\tau_{ij}\}^{T}\) are the convective and viscous fluxes, respectively; \(\rho\) is the fluid density; \(\vec{u}\) is the flow velocity; \(p\) is the pressure; \(E\) is the total internal energy; \(H\) is the total enthalpy; \(\overleftrightarrow{\tau}\) denotes the viscous stress tensor; \(u_{n}=\vec{u}\cdot\vec{n}\) the normal velocity; and \(Q\) the source terms (if any). Actuator Line Model  
In the present work, the rotor blades are modeled as rotating actuator lines that contribute the corresponding body forces to the source term \(Q\) of the previous equation. The loading is calculated on the basis of blade element analysis along the blade span and in conjunction with tabulated \(2D\) polars [23]. The blade is divided in a number of elements with a \(dr\) span-wise extent. The centers of these elements constitute the control points of the numerical process. A lift and drag resultant body force is assigned to each actuator line control point depending on the airfoil type, twist angle, chord length, and incoming flow velocity. Lift and drag are calculated (as shown in Figure 1), by the following equations:  
\[\phi=arctan\frac{W_{eff,z}}{W_{eff,x}} \tag{2}\]  
\[a_{eff}=\phi-\theta_{t}-\phi_{p} \tag{3}\]  
\[L=\frac{1}{2}\rho W_{eff}^{2}C_{L}(a_{eff})c\Delta r,D=\frac{1}{2}\rho W_{eff} ^{2}C_{D}(a_{eff})c\Delta r \tag{4}\]  
\[\vec{f}_{cp}=L\vec{e}_{L}+D\vec{e}_{D} \tag{5}\]  
where \(\phi\) is the inflow angle; \(\theta_{t}\) is the local twist angle; \(\phi_{p}\) is the pitch angle of the blade; \(a_{eff}\) is the effective angle of attack; \(C_{L}(a_{eff})\) and \(C_{D}(a_{eff})\) are the lift and drag coefficient for the specific angle of attack, respectively; \(\rho\) is the density of the fluid; \(W_{eff}\) is the local two dimensional inflow velocity; \(c\) is the blade element characteristic chord; \(\Delta r\) is the width of the blade element; and \(\vec{e}_{L}\), \(\vec{e}_{D}\) are the unit vectors in the direction of lift and drag, respectively. The reaction of this body force and the corresponding power are then imposed as source terms in the momentum and energy equations of the cells swept by the blades  
Figure 1: Velocity triangle and aerodynamic loads on a blade section. Actuator line analysis definitions. during their rotation. In order to avoid singularities, the body forces are distributed over a number of neighboring cells using a \(3D\) isotropic Gaussian distribution [24].