# Study of Airfoil Deflections for Unsteady Aerodynamics Optimization in Pitching Airfoils +
## 2 Materials and Methods
### Pitching Parameters  
Angle of attack changes during transients and pitching can yield unsteady separation and light or deep dynamic stall [29; 30]. With dynamic stall, the suction surface flow can develop a dynamic stall vortex (DSV) that induces undesirable variations in aerodynamic forces and moments and potentially unstable net-cycle pitch damping. These variations can be severe enough to negatively impact the wing control, vibrations, structural integrity, and noise production and therefore are important in a number of engineering applications involving pitching wings, such as helicopter rotors and wind turbines, and also others involving aggressive maneuvers, wing-gust encounters, etc. [29; 30]. It is noted that most of these studies have Reynolds number ranges similar to the current study (Re = 250 K), representing, as stated earlier, a medium flow regime configuration. The pitching frequency is set to the rotorcraft conditions (k = 0.04). The pitching motion is introduced to induce angle of attack variations, and here, it represents a rotorcraft blade rapidly changing the angle of attack as it rotates around the azimuth, constantly adjusting for the new flow field that it is going to encounter. To do so, the numerical scheme used a pitching motion described by Equation (1) [17; 28]  
\[\alpha(t)=\alpha_{P_{0}}+A_{P}cos(2\pi f_{P}t+\varphi_{P}) \tag{1}\]Here, \(A_{p}\) is the amplitude of the pitching motion around the center of pressure, and the term \(\varphi_{p}\) is used to adjust where the airfoil starts its pitching motion, for instance, to start at negative 2\({}^{\circ}\) and pitch up, a value of \(\varphi_{p}=\pi\) is used to offset the sinusoidal motion so that the airfoil can begin its pitching at negative 2\({}^{\circ}\) and advance into an upward pitching motion. The center of pivoting is the center of pressure \(x_{\rm p}\) with \(x_{\rm p}/c=1/3\), where \(c\) is chord length. The geometry and parameters are shown in Figure 3, where \(\theta\) represents LE and TE prescribed deflections and V\({}_{\infty}\) is the free stream velocity. In order to pitch the airfoil at a proper frequency (\(f\)) with a reduced frequency \(k\) of 0.04 based on nominal operation conditions for a rotorcraft, \(k\) is calculated from Equation (2), where \(c\) is chord and \(V_{\infty}\) free stream velocity:  
\[k=\frac{\pi fc}{V_{\infty}} \tag{2}\]  
ANSYS CFX 18.2 [26] is employed for further computational simulations and analysis of the modified VR-12 airfoil shapes. ANSYS CFX is a CFD solver utilizing the Finite Volume Method based on Elements with a Cell Vertex Formulation to solve the Reynolds Averaged Navier-Stokes (RANS) equations. It includes a wide range of well-validated turbulence modeling techniques, can handle moving/rotating meshes, and it is highly scalable for high-performance computing. Time-dependent, two-dimensional simulations are computed using the Shear Stress Transport (SST) k-\(\omega\) two-equation turbulence model [31] coupled with \(\gamma-Re_{\theta}\) transition model [32; 33]. The SST model is selected because of its strength and performance in separated flows as well as in free shear and wall-bounded flows. The coupling of the SST turbulence model with the \(\gamma-Re_{\theta}\) transition model allows for the capturing of the transitional nature of the flow. The \(\gamma-Re_{\theta}\) transition model is based on the concept of intermittency transport modeling. The onset of transition and extent of transitional region before flow becomes fully turbulent are computed through correlations based on experimental data representing a wide range of transitional flows [32; 33]. Experimental measurements in the wind tunnel obtained a tunnel turbulence intensity of 0.5%. Due to this low turbulence intensity, the flow near the leading edge of the airfoil may stay laminar, then transition to turbulent. In order to capture the transitional nature of the flow, the transition model was incorporated into the simulations.