# Multidisciplinary Aerodynamic Design of a Rotor
## 2 Methodology
### Modeling  
#### 2.2.1 Forward Flight Trim  
Affected by blade flap motion and rotor speed, the air velocity around a 2-D airfoil can be expressed by  
\[\left( \begin{array}{c}u_{r}\\ u_{t}\\ u_{p}\end{array} \right)=\left( \begin{array}{ccc}\cos\beta&0&\sin\beta\\ 0&1&0\\ -\sin\beta&0&\cos\beta\end{array} \right)\left( \begin{array}{ccc}\cos\psi& \sin\psi&0\\ -\sin\psi&\cos\psi&0\\ 0&0&1\end{array} \right)\left( \begin{array}{c}-\mu_{nx}\\ -\mu_{hy}\\ \lambda_{h}\end{array} \right)\omega R+\left( \begin{array}{c}0\\ \omega\widetilde{r}\\ \hat{\beta}(r-\varepsilon R)\end{array} \right) \tag{1}\]  
where \(\widetilde{r}=\varepsilon R+(r-\varepsilon R)\cos\beta\). The blade section aerodynamic forces can be formulated by airfoil aerodynamic coefficients, air dynamic pressure and the section chord. \[\left\{ \begin{array}{c}dL=\frac{1}{2}\rho u_{a}^{2}C_{L}cdr\\ dD=\frac{1}{2}\rho u_{a}^{2}C_{D}cdr\end{array} \right. \tag{2}\]  
where section lift \(dL\) is normal to inflow direction locally, and section drag \(dD\) is parallel to the inflow direction. The airfoil lift coefficient and drag coefficient are both interpolated by the C81 table. Flap motion is modeled by moment balance at the flap hinge. \[M_{a}=-(M_{c}+M_{i}+M_{k}) \tag{3}\]  
where \(M_{c}=-\omega^{2}\big{(}I_{\beta}\cos\beta+\varepsilon RM_{\beta}\big{)}\sin\beta\), representing centrifugal moment; \(M_{i}=-I_{\beta}\bar{\beta}\), representing flap inertia moment; \(M_{k}=-K_{\beta}\beta\), expressing restrain moment at the hinge. Aerodynamic moment \(M_{a}\) is expressed by the integral of \(dF_{z}\) along the blade span. \[M_{a}=\int_{\varepsilon}^{1}RdF_{z}=R\int_{\varepsilon}^{1}\cos\phi dL-\sin \phi dD \tag{4}\]  
In the hypothesis of quasi-steady motion, the flap can be solved as  
\[\beta=\beta_{0}+\beta_{c}\cos\psi+\beta_{s}\sin\psi \tag{5}\]  
This study adopts [PERSON] prescribed wake method to model tip vortex, and calculates induced velocity. The position of the certain vortex is expressed following Kutta condition [15]. \[\left(\frac{\partial\mathbf{r}}{\partial\psi_{b}}+\frac{\partial\mathbf{r}}{ \partial\psi_{t}}\right)=\mathbf{\mu}+\Lambda(\mathbf{r},t) \tag{6}\]  
where the vortex position \(\mathbf{r}(\psi_{b},\psi_{t})\) can be parameterized by present blade azimuth \(\psi_{b}\) and shed azimuth \(\psi_{t}\); the freestream velocity vector is expressed by \(\mathbf{\mu}\); and the induced velocity vector is expressed by \(\lambda\). The Z-axis component of the vortex position can be expressed as  
\[z_{v}=-\mu_{z}(\psi_{b}-\psi_{t})+\int_{\psi_{t}}^{\psi_{b}}\lambda_{i}d\psi \tag{7}\]  
Linear inflow distribution is assumed by Beddoes. Induced velocity within the rotor disc can be expressed as  
\[\lambda_{i}(x,y)=-\lambda_{io}\bigg{[}1+E\bigg{(}\frac{8}{15\pi}+x-|y|^{3} \bigg{)}-2\mu_{x}y\bigg{]} \tag{8}\]Induced velocity in the far field  
\[\lambda_{i}(x,y)=-2\lambda_{i0}\bigg{[}1+E\bigg{(}\frac{8}{15\pi}-|y|^{3}\bigg{)} -2\mu_{x}y\bigg{]} \tag{9}\]  
where \(|y|^{3}\) reflects vortex roll-up near the blade tip, and \(-2\mu_{y}\), from [PERSON] [16], represents lateral variation. The constant term \(8/15\pi\) is introduced by [PERSON] [17], in agreement with momentum theory. Beddoes applied \(E=\chi\) to represent longitudinal non-uniformity, while \(E=\chi/2\) is adopted in this paper, instead, which was found to be better in agreement with experiments and the free vortex wake method [15], where \(\chi=\arctan\left(\frac{\mu_{x}}{\mu_{y}+\lambda}\right)\). The \(x\) and \(y\) components of vortex position can be expressed as  
\[\left\{ \begin{array}{c}x_{v}=r_{v}\cos\psi_{t}+\mu_{x}(\psi_{b}-\psi_{t})\\ y_{v}=r_{v}\sin\psi_{t}\end{array} \right. \tag{10}\]  
where \(r_{v}\) is the radius station vortex shed.