# Research on Dynamic Modeling and Transition Flight Strategy of VTOL UAV
## 2 Aerodynamic Expression
### Aerodynamic Model of Lift Fan
where \(\lambda=v/\Omega R\), \(\lambda_{u}=v_{u}/\Omega R\), \(\lambda_{\infty}=V_{\infty}/\Omega R\), \(v=V_{\infty}+v_{u}\); and \(r=y/R\) stands for dimensionless radial position. Therefore, the tension coefficient of the annular area can be written as  
\[dC_{Tu}=4\lambda\lambda_{u}rdr=4\lambda(\lambda-\lambda_{\infty})rdr, \tag{5}\]  
Similarly, the power coefficient of the annular area can be expressed as  
\[dC_{Tu}=\lambda dC_{Tu}=4\lambda^{2}\lambda_{u}rdr=4\lambda^{2}(\lambda- \lambda_{\infty})rdr. \tag{6}\]  
The tip loss in BEMT can be calculated by means of Prandtl tip-loss equation, and the expression of the correction factor F of tip loss is as follows [18]  
\[F=(2/\pi)\cos^{-1}(\exp(-f)), \tag{7}\]  
where \(f\) is given through the number of blades \(N_{b}\) and the radial position r of the blade element, and the expression is as follows  
\[f=\frac{N_{b}}{2}\bigg{(}\frac{1-r}{r\phi}\bigg{)}, \tag{8}\]  
where \(\phi=\lambda(r)/r\) is the inflow angle under the assumption of small angle. Then the tension coefficient of the annular area modified by Prandtl propeller tip loss equation is expressed as  
Figure 2.0  
According to the classical blade element theory, the tension generated by the same annular area on the paddle disc is expressed as  
Figure 2.1  
where, \(\theta_{u}\) refers to the blade pitch of the upper rotor, which may include the zero lift angle of attack of the blade section, and \(\sigma_{u}\) is the degree of disc consolidation. By comparing the tension coefficients obtained by the classical blade element theory and the blade element momentum theory, it can be obtained to be  
Figure 2.2  
Convert this into a quadratic equation of \(\lambda\)  
Figure 2.3  
This equation has the solution  
Figure 2.4  
where \(\lambda_{u}=\lambda-\lambda_{\infty}\). The above equation can be solved numerically on a series of discrete elements of the paddle disk. The same mathematical principle can be applied to the analysis of the lower rotor as well. According to the previous hypothesis, a part of the lower rotor works in the slipstream area of the upper rotor. In an ideal state, the vena contracta can be obtained according to the contraction of the flow tube \(A_{c}=\pi a^{2}R^{2}\), where \(a=0.707\), \(A_{c}/A=0.5\). In this particular case, the inner area of the lower rotor encounters incoming contracted differential streamtubes with velocity \(V_{\infty}+v_{u}/a^{2}\). With the same calculation method as that of the upper rotor, the velocity of the lower rotor in the area (\(r\leq a\)) affected by the slipstream of the upper rotor can be obtained  
Figure 2.5  
where \(\theta_{l}\) refers to the blade pitch of the lower rotor, \(\lambda_{l}=\lambda-\lambda_{\infty}\). For the area (\(r>a\)) that is not affected by the slipstream of the upper rotor, the velocity is calculated as follows  
Figure 2.6  
The above three equations give the approximate calculation method of velocity of each discrete radial position of upper and lower rotor at any blade pitch, blade torsion and airfoil section. According to the above calculation formula of tension coefficient and flow velocity, the tension, power and torque of the rotor can be obtained by numerical integration of each rotor disc. The tension coefficient of the upper rotor is expressed as follows  
Figure 2.7  
The power coefficient is expressed as follows  
Figure 2.8  
where \(C_{d}\) is the sectional profile drag coefficient. In approximate calculation, \(C_{d}\) can be replaced by the zero-rise resistance coefficient \(C_{d0}\). In general, for a specific airfoil blade, the drag coefficient needs to be obtained by looking up the table.