# Research on Dynamic Modeling and Transition Flight Strategy of VTOL UAV
## 2 Aerodynamic Expression
### Aerodynamic Model of Lift Duct  
The advantages of ducted propellers over isolated propellers have been analyzed in Section 2.1. The following is the aerodynamic modeling of the ducted propellers. It is assumed that the density of the gas through the duct remains unchanged and the influence of viscosity is ignored. According to the continuity equation, the three sections in the Figure 5 satisfy \(v_{1}A_{1}=v_{p}A_{p}=v_{2}A_{2}\), where V is the axial flow velocity of the flow tube, and A is the section area of the flow tube. The tension, power and torque generated by the propeller can be calculated by BEMT theory above. Tension coefficient expression  
\[C_{T_{p}}=\int\limits_{r=0}^{r=1}d\mathrm{C}_{T_{p}}=\frac{1}{2}\int\limits_{0 }^{1}\sigma_{p}C_{l}r^{2}dr, \tag{30}\]  
Power coefficient expression  
\[C_{P_{d}}=\int\limits_{r=0}^{r=1}\lambda_{d}dC_{T_{d}}+\int\limits_{0}^{1} \frac{1}{2}\sigma_{d}C_{d}r^{3}dr, \tag{31}\]  
Figure 4: Lift fan force diagram. Figure 5: Schematic diagram of speed and pressure before and after the disc. Torque coefficient expression  
\[C_{M_{d}}=C_{P_{d}}/2\pi, \tag{32}\]  
The final propeller tension, power and torque are expressed as follows  
\[ \begin{array}{l}T_{p}=\rho A(\Omega R)^{2}C_{T_{p}}\\ P_{d}=\rho A(\Omega R)^{3}C_{P_{d}}\\ M_{d}=\rho AR(\Omega R)^{2}C_{M_{d}}\end{array} , \tag{33}\]  
The factor \(q\) of duct tension ratio is also introduced to establish the total tension model of the lift duct as follows \(T_{d}=T_{s}+T_{p}=qT_{d}+T_{p}\). The velocity at the propeller disc is \(v_{p}\). According to the momentum theory, the propeller tension and the power driving the propeller (equal the kinetic energy increased by the flow) can be expressed as  
\[T_{p}=\rho A_{2}v_{2}(v_{2}-v_{1}), \tag{34}\]  
\[P_{d}=v_{p}T_{p}=\rho A_{2}v_{2}v_{p}(v_{2}-v_{1})=\frac{1}{2}\rho A_{2}v_{2} \Big{(}v_{2}^{2}-v_{1}^{2}\Big{)}=\Delta E_{kin}, \tag{35}\]  
The useful work done by the propeller can be expressed as  
\[W_{p}=T_{p}v_{1}=\rho A_{2}v_{1}v_{2}(v_{2}-v_{1}), \tag{36}\]  
Then, the propeller efficiency can be expressed as  
\[\eta_{p}=\frac{W_{p}}{P_{p}}=\frac{\rho A_{2}v_{1}v_{2}(v_{2}-v_{1})}{\frac{1 }{2}\rho A_{2}v_{2}\Big{(}v_{2}^{2}-v_{1}^{2}\Big{)}}=\frac{2v_{1}}{v_{1}+v_{2 }}=\frac{2}{1+\frac{v_{2}}{v_{1}}}, \tag{37}\]  
In an ideal condition, the contracted wake area is \(A_{2}=\pi a^{2}R^{2}\), where \(a=0.707\), \(A_{2}/A_{p}=0.5\). In other words, under the ideal gas flow condition, the airflow velocity of the rear vena contracta of the duct can be obtained as \(v_{2}=v_{1}+\Delta v/a^{2}\). At this time, \(v_{1}\) is the axial free flow velocity, and the dimensionless velocity \(\lambda\) at the vena contracta of the flow tube can be calculated according to the BEMT theory mentioned above in relation (13), and then converted into \(v_{2}\). \[v_{2}=\lambda\Omega R. \tag{38}\]  
The aerodynamic force and torque of the lift duct were converted to the coordinate axis of the body, and the force and geometric relationship are shown in the Figure 6.  
The coordinate transformation matrix of lift duct force and torque to the plane body coordinate system is \(\mathbf{R}_{bd}\), of which the dimension is \(3\times 3\). \[\mathbf{R}_{bd}=\left[ \begin{array}{ccc}\cos\theta_{d}&0&\sin\theta_{d}\\ 0&0&0\\ -\sin\theta_{d}&0&\cos\theta_{d}\end{array} \right], \tag{39}\]  
Figure 6: Diagram of force and geometry of lift duct. Under the coordinate system of the body, the force of the lift duct is expressed as: \(\mathbf{F}_{b,d,i}=\mathbf{R}_{bd}\cdot\mathbf{F}_{d,i}\), where the dimension of \(\mathbf{F}\) is \(3\times 1\).The lift duct torque is expressed in the body coordinate system as \(\mathbf{M}_{b,d\Lambda,i}=\mathbf{R}_{bd}\cdot\mathbf{M}_{d,i}\), where the dimension of \(\mathbf{M}\) is \(3\times 1\).