# Research on Dynamic Modeling and Transition Flight Strategy of VTOL UAV
## 2 Aerodynamic Expression
### Aerodynamic Model of Lift Fan  
Compared with the isolated propeller, the ducted propeller produces larger tension at the same power and the same diameter, that is, the diameter required to produce the same tension at the same power is smaller. The mechanism of ducted propeller causing this difference is as follows:  
1. The presence of duct changes the slipstream state of the propeller downstream, such as increasing the slipstream area, and reducing the slipstream velocity and slipstream energy loss, so that more kinetic energy behind the propeller disc was converted into pressure energy. 2. The duct wall improves the turbulence characteristics of the propeller tip and reduces the flow loss around it. 3. A large zone with low pressure is formed at the leading edge of the inlet of the duct, generating additional tension. The coaxial contra rotating lift fans are a pair of lift fans installed in tandem and rotating around the same axis with opposite direction. The coaxial contra-rotating lift fan has the following advantages:  
1. Due to coaxial rotation, the reaction torque from the engine to the aircraft is reduced to zero. 2. Under the condition of V\({}_{\mathrm{max}}\), the total efficiency of coaxial propeller is higher than that of non-coaxial propeller with the same number of blades with the same flight conditions (V\({}_{0}\), H, n, N and D). 3. In high-speed flight, the efficiency of coaxial propeller is higher than that of the two isolated propellers with the same task. 4. The take-off performance (in-situ tension) of a coaxial propeller is better than that of a single propeller. At present, there is no coaxial dual-rotor aerodynamic theory with duct for reference. Based on the mathematical model of coaxial dual-rotors, the method of duct tension factor \(q\) is introduced to give the dynamic model of coaxial dual-rotor ducted fan. The Blade Element Moment Theory (BEMT) [12; 13] is used to calculate the lift, power and moment of the coaxial double-rotor. The dimensionless form proposed by NASA [14] is adopted to calculate the formula:  
\[ \begin{array}{l}T=\rho A(\Omega R)^{2}C_{T}\\ P=\rho A(\Omega R)^{3}C_{P}\\ M=\rho AR(\Omega R)^{2}C_{M}\\ \end{array} , \tag{1}\]  
Figure 1: Body coordinate system. The flow model of the coaxial dual rotor system is shown in the Figure 2. The basic assumption is that the area of the propeller disk of the lower rotor works in the wake area of the upper rotor, and the slipstream velocity of the upper rotor has a maximum change in the vena contracta before the wake of the upper rotor enters the rotor disk. The contraction of the wake is based on the ideal fluid flow hypothesis or supported by specific experimental data [4; 5; 15; 16; 17]. The BEMT model assumes that the slipstream (outflow) of the upper rotor affects the slipstream (inflow) of the lower rotor, while the lower rotor does not affect inflow of the upper rotor. The theory of blade element momentum can be generalized with the axial free inflow velocity, which is generally denoted as \(V_{\infty}\). For the upper rotor of the coaxial system, momentum theory principles can be used to find incremental thrust from the mass flow rate through the annulus and the induced velocity there. Mass flow velocity through rotor annulus is expressed as:  
\[d\dot{m}=\rho(V_{\infty}+v_{u})dA=2\pi\rho(V_{\infty}+v_{u})ydy, \tag{2}\]  
where \(dA=2\pi ydy\), so the increase in thrust over the annular area is as follows:  
\[dT_{u}=2\rho(V_{\infty}+v_{u})v_{u}dA=4\pi\rho(V_{\infty}+v_{u})v_{u}ydy. \tag{3}\]  
To convert the thrust into a compiled form, the above equation can be written as  
\[dC_{T_{u}}=\frac{dT_{u}}{\rho(\pi R^{2})(\Omega R)^{2}}=\frac{2\rho(V_{\infty }+v_{u})v_{u}dA}{\rho\pi R^{2}(\Omega R)^{2}}=\frac{2\rho(V_{\infty}+v_{u})v_ {u}(2\pi ydy)}{\rho\pi R^{2}(\Omega R)^{2}}=4\lambda\lambda_{u}rdr, \tag{4}\]  
Figure 2: Flow model used for the BEMT (Blade Element Moment Theory) analysis of a coaxial rotor system.