# Study on the Mechanism of the Variable-Speed Rotor Affecting Rotor Aerodynamic Performance
## 2 Modeling
The Leishman-Beddoes algorithm introduces a one-step delay to consider this phenomenon and uses a substitutive normal force, \(C_{N}^{\prime}\), to trigger leading edge flow separation. \[\left\{ \begin{array}{l}C_{N}^{\prime}\left(k\right)=C_{N}^{P}\left(k\right) -D_{P}\left(k\right)\\ D_{P}\left(k\right)=D_{P}\left(k-1\right)e^{\frac{-\Delta k}{Tp}}+\left(C_{N} ^{P}\left(k\right)-C_{N}^{P}\left(k-1\right)\right)e^{\frac{-\Delta k}{Tp}} \end{array} \right. \tag{17}\]  
where \(T_{p}\) is a function of the Mach number, whose value is obtained from the airfoil data. The leading edge flow separation affects the boundary layer and trailing edge separation, which causes circulatory loading loss and nonlinear variation in the aerodynamic force and moment. The algorithm considers this when computing the circulatory loading by the Kirchhoff plate flow solution. So, the unsteady normal force during trailing edge separation is  
\[C_{N}^{f}\left(k\right)=C_{N}^{fC}\left(k\right)+C_{N}^{I}\left(k\right) \tag{18}\]  
where \(C_{N}^{fC}\) represents the normal force after the Kirchhoff flow correction. During modeling, the point when the boundary layer starts to flow in reverse is treated as the trailing edge separation point, \(f^{\prime\prime}\); thus, the corrected normal force can be expressed as  
\[C_{N}^{fC}\left(k\right)=C_{N}\left(Ma\right)\left(\frac{1+\sqrt{f^{\prime \prime}\left(k\right)}}{2}\right)^{2}\left(\alpha_{E}\left(k\right)-\alpha_{0} \right). \tag{19}\]  
Due to the dynamic stall delay, the algorithm introduces a one-step delay when computing the separation point:  
\[\left\{ \begin{array}{l}f^{\prime\prime}\left(k\right)=f^{\prime}\left(k \right)-D_{f}\left(k\right)\\ D_{f}\left(k\right)=D_{f}\left(k-1\right)e^{\frac{-\Delta k}{Tp}}+\left(f^{ \prime}\left(k\right)-f^{\prime}\left(k-1\right)\right)e^{\frac{-\Delta k}{Tp}} \end{array} \right. \tag{20}\]where the separation point before delay correction is expressed by a piecewise function of AoA:  
\[\left\{ \begin{array}{ll}f^{\prime}=1-0.3e^{\left(\alpha_{eff}-\alpha_{1}\right)s _{1}^{-1}}&\alpha_{eff}\geq\alpha_{1}\\ f^{\prime}=0.04+0.66e^{\left(\alpha_{1}-\alpha_{eff}\right)s_{2}^{-1}}&\alpha_{ eff}<\alpha_{1}\end{array} \right. \tag{21}\]  
where the term \(\alpha_{1}\) represents steady stall AoA, and \(s_{1}\) and \(s_{2}\) are steady stall parameters, obtained through an airfoil steady experiment and data fitting via Equations (19) and (21). The effective AoA \(\alpha_{eff}\) is computed by the substituting normal force, \(C_{N}^{\prime}\), mentioned previously:  
\[\alpha_{eff}=\frac{C_{N}^{\prime}\left(k\right)}{C_{N}\left(Ma\right)}. \tag{22}\]  
The deep dynamic stall process characterizes the formation, convection and detachment of the leading edge vortex. The leading edge vortex contributes to the airfoil normal force with a vortex lift:  
\[C_{v}\left(k\right)=C_{N}^{C}\left(k\right)\left(1-\left(\frac{1+\sqrt{f^{ \prime\prime}\left(k\right)}}{2}\right)^{2}\right). \tag{23}\]  
In an unsteady flow, the decay and accumulation of the leading edge vortex exist simultaneously:  
\[C_{N}^{v}\left(k\right)=C_{N}^{v}\left(k-1\right)e^{\frac{-\Delta k}{\tau_{v} }}+\left(C_{v}\left(k\right)-C_{v}\left(k-1\right)\right)e^{\frac{-\Delta k}{ \tau_{v}}}. \tag{24}\]  
When the critical condition, \(C_{N}^{\prime}>C_{NI}\), is satisfied, the leading edge vortex forms and convects downstream. During this process, Equation (24) is applied. When it comes to the trailing edge, the accumulation effect stops. The critical normal force, \(C_{NI}\), is obtained through an airfoil experiment. During the deep dynamic stall process, the leading edge vortex may form and detach continuously. Its frequency is determined by \(T_{s}=\frac{2\left(1-f\right)}{5\tau_{v}}\), where the Strouhal number, \(S_{tr}\), equals 0.2. If the endurance of the current vortex, \(t_{v}\), is larger than \(T_{s}\), Equations (23) and (24) are implemented again.