# Study on the Mechanism of the Variable-Speed Rotor Affecting Rotor Aerodynamic Performance
## 2 Modeling  
The rotor vortex theory assumes that rotor wake could be treated as a potential flow field, so the induced velocity is computed by the Biot-Savart Law. As Figure 1a shows, the current work segments start as straight line elements. The induced velocity on interest point P by a certain element is  
\[d\mathbf{v}_{i}=\frac{\Gamma}{4\pi}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right) \frac{\mathbf{r}_{1}\times\mathbf{r}_{2}}{r_{1}r_{2}+\mathbf{r}_{1}\cdot \mathbf{r}_{2}} \tag{1}\]  
where \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) are vectors from the straight line end-points to P.  
The vortex element position, always within the rotor disc, is expressed as  
\[z_{v}=-\mu_{z}\Delta\psi-\lambda_{i0}\left(1-2\mu_{x}y+E\left(\frac{8}{15\pi}+ \cos\psi_{t}+\frac{\mu_{x}\left(\psi_{b}-\psi_{t}\right)}{2r}-\left|y\right|^{3} \right)\right)\Delta\psi. \tag{2}\]  
The vortex element position, always beyond the rotor disc, is expressed as  
\[z_{v}=-\mu_{z}\Delta\psi-2\lambda_{i0}\left(1-2\mu_{x}y+E\left(\frac{8}{15\pi }-\left|y\right|^{3}\right)\right)\Delta\psi. \tag{3}\]  
The position of a vortex element, convecting from the inside to the outside of the disc, is expressed as  
\[z_{v}=-\mu_{z}\Delta\psi-\frac{2\lambda_{i0}x_{e}}{\mu_{x}}\left(1-2\mu_{x}y+E \left(\frac{8}{15\pi}-\left|y\right|^{3}\right)\right) \tag{4}\]  
where \(x_{e}\) is the \(x\)-axis position when the vortex element convects outside the disc, and \(y\) is the vortex element \(y\)-axis position. Variable \(E\) equals \(\frac{1}{2}\tan^{-1}\left(\frac{\mu_{x}}{-\mu_{z}+\lambda_{i}}\right)\). The vortex core model is introduced to eliminate the singularity of Equation (6):  
\[r_{c}=\sqrt{r_{0}^{2}+4\pi v\delta t} \tag{5}\]  
where \(r_{0}\) is the initial core radius, and is chosen to be 5% of the chord in the current work. The Lamb-Oseen constant, \(\alpha\), equals 1.25643. The induced velocity on P is computed by the core-modification Bivot-Savart Law:  
\[d\mathbf{v}_{i}=\frac{\Gamma\hbar^{2}}{4\pi\left(r_{c}^{4}+\hbar^{4}\right)^{ 0.5}}\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\frac{\mathbf{r}_{1}\times \mathbf{r}_{2}}{r_{1}r_{2}+\mathbf{r}_{1}\cdot\mathbf{r}_{2}}. \tag{6}\]  
In forward flight, upwash can appear from the advancing blade to the retreating blade. Some interactions may be caused, as shown in Figure 0(b), between the wake trailing from the front blade and the following blade because of small convection, the so-called blade-vortex interaction (BVI). The lift-surface correction [23] is applied in the current work to consider this phenomenon. The lift coefficient by lift-surface modeling is  
\[ \begin{split}\frac{L_{ls}}{\rho v\Gamma}=&\frac{r \sin\Lambda}{\left(r\sin\Lambda\right)^{2}+\left(h+c_{0}\right)^{2}}-a_{o}^{ \prime}\frac{b_{0}x_{0}}{x_{0}^{2}+4\left(r\sin\Lambda\right)^{2}\left(h+c_{0 }^{\prime}\right)^{2}}\\ &-\sum_{k=1}^{2}a_{k}\left(\frac{d}{dr\sin\Lambda}\right)^{2k}- \frac{\left(r\sin\Lambda+b_{1}\right)\cos b_{2}+\left(h+c_{k}\right)\cos b_{2 }}{\left(r\sin\Lambda+b_{1}\right)^{2}+\left(h+c_{k}\right)^{2}}.\end{split}  \tag{7}\]  
Figure 1: Rotor wake modeling. The lift coefficient by lift-line modeling is  
\[\frac{L_{ll}}{\rho v\Gamma}=\frac{r\sin\Lambda}{\left(r\sin\Lambda\right)^{2}+ \left(h+c_{0}\right)^{2}}-a_{1}\left(\frac{d}{dr\sin\Lambda}\right)^{2}\frac{-r \sin\Lambda}{\left(r\sin\Lambda\right)^{2}+\left(h+c_{1}\right)^{2}}. \tag{8}\]  
The parameters in Equations (7) and (8) are expressed as functions of the collision angle, \(\Lambda\)[23]. When the distance, \(d\), between a certain vortex element and the computing point, P, is smaller than a predefined value, 10 times the chord in the current work, the vortex element-induced velocity is computed as  
\[d\tilde{\mathbf{v}}_{i}=d\mathbf{v}_{i}\min\left(\left|\frac{L_{ls}}{L_{ll}} \right|,1\right). \tag{9}\]  
Both the blade motion and the airfoil freestream vary periodically along with the rotor revolution. With an increase in AoA, flow separation could appear on the airfoil surface.