# Study on the Mechanism of the Variable-Speed Rotor Affecting Rotor Aerodynamic Performance
## 2 Modeling
Under the effects of periodic motion and freestream, the airfoil aerodynamic forces and moment behaviors are unsteady. With a further increase in AoA, the airfoil can experience dynamic stall which also affects the aerodynamic forces and moments. The unsteady forces and moments are related to the airfoil motion, freestream, AoA and their rates of change. When lowering the rotor speed, the advance ratio increases the blade flap and feathering equivalently, and the AoA range increases; therefore, simulation of the unsteady behavior is of great significance when discussing the variable-speed rotor aerodynamics. The current work applies the Leishman-Beddoes algorithm to simulate the process of airfoil dynamic stall. The model separates the unsteady dynamic stall process into three sub-processes--attached flow, separation flow, and deep dynamic stall, and introduces some one-step delays based on airfoil steady and dynamic experiments to simulate the aerodynamics via the indicial response method and Duhamel's integral. The attached flow is the base for simulating the unsteady aerodynamic process. The Leishman-Beddoes algorithm applies two parts--initial implusive loading, \(C_{N}^{I}\), and circulatory loading, \(C_{N}^{C}\), tending towards a steady state--to simulate this process:  
\[C_{N}^{P}\left(k\right)=C_{N}^{C}\left(k\right)+C_{N}^{I}\left(k\right). \tag{10}\]  
The circulatory loading caused by variation in AoA is expressed as  
\[C_{N}^{C}\left(k\right)=C_{N}\left(Ma\right)\left(\alpha\left(k\right)-X\left( k\right)-Y\left(k\right)\right)=C_{N}\left(Ma\right)\alpha_{E}\left(k\right) \tag{11}\]  
where \(X\) and \(Y\) are called deficiency functions and  
\[\left\{ \begin{array}{c}X\left(k\right)=X\left(k-1\right)e^{-b_{1}\beta^{2} \Delta s}+A_{1}\Delta\alpha_{k}e^{-b_{1}\beta^{2}\Delta s/2}\\ Y\left(k\right)=Y\left(k-1\right)e^{-b_{2}\beta^{2}\Delta s}+A_{2}\Delta \alpha_{k}e^{-b_{2}\beta^{2}\Delta s/2}\end{array} \right. \tag{12}\]  
where \(A_{1}=0.3\), \(b_{1}=0.14\), \(A_{2}=0.7\), \(b_{2}=0.53\). The term, \(C_{N}\left(Ma\right)\), is the slope of the airfoil normal force to AoA, obtained through an airfoil steady experiment. The terms \(\alpha\) and \(\alpha_{E}\) mean AoA and equivalent AoA, respectively, at three-quarters of the chord. The variation \(\beta\) expresses the Prandtl-Glauert compression factor and the non-dimensional time constant, \(\Delta s=\frac{2V\Delta t}{c}\). The impulsive loading caused by the variation in AoA, \(\Delta\alpha\), and the pitch rate, \(\Delta q=\frac{\Delta\delta c}{V}\), can be expressed as  
\[C_{N}^{I}\left(k\right)=C_{N\alpha}^{I}\left(k\right)+C_{N\alpha}^{I}\left(k\right) \tag{13}\]where the part by \(\Delta\alpha\) is  
\[\left\{ \begin{array}{l}C_{Na}^{I}\left(k\right)=\frac{4T_{\alpha}}{Ma}\left( \frac{\Delta\alpha_{k}}{\Delta t}-D_{1}\left(k\right)\right)\\ D_{1}\left(k\right)=D_{1}\left(k-1\right)e^{\frac{-\Delta t}{Ta}}+\left(\frac {\Delta\alpha_{k}-\Delta\alpha_{k-1}}{\Delta t}\right)e^{\frac{-\Delta t}{Ta}} \end{array} \right. \tag{14}\]  
and the part by \(\Delta q\) is  
\[\left\{ \begin{array}{l}C_{Nq}^{I}\left(k\right)=\frac{T_{q}}{Ma}\left(\frac {\Delta q_{k}}{\Delta t}-D_{2}\left(k\right)\right)\\ D_{2}\left(k\right)=D_{2}\left(k-1\right)e^{\frac{-\Delta t}{Ta}}+\left(\frac {\Delta q_{k}-\Delta q_{k-1}}{\Delta t}\right)e^{\frac{-\Delta t}{Ta}}.\end{array} \right. \tag{15}\]  
The time constants can be expressed as  
\[\left\{ \begin{array}{l}T_{\alpha}=\frac{0.75}{1-Ma+\pi\beta^{2}Ma^{2}\left( A_{1}b_{1}+A_{2}b_{2}\right)}\frac{c}{a}\\ T_{q}=\frac{0.75}{1-Ma+2\pi\beta^{2}Ma^{2}\left(A_{1}b_{1}+A_{2}b_{2}\right)} \frac{c}{a}.\end{array} \right. \tag{16}\]  
As the AoA becomes relatively large, flow separation may appear in the leading edge. Under this unsteady condition, there is so-called dynamic stall delay. This means that the leading edge pressure distribution occurs later than normal force variation, and the normal force variation is delayed after the AoA variation.