# Research on the Aerodynamic-Propulsion Coupling Characteristics of a Distributed Propulsion System
## 2 Materials and Methods
### Development of Body Force Model  
The concept of a body force model is that the blades of the rotating turbomachinery are replaced by source term distributions, and the reproduction of the effects of the blades on the work, deflection, and friction of the airflow is achieved [19]. Since there is no need to conduct mesh generation for the actual blades anymore, it can significantly improve the numerical simulation efficiency of turbomachinery and reduce the consumption of computational resources. In this paper, based on the model proposed by [PERSON] [20] and considering the effect of compressibility [21], a circumferential body force model based on local flow field parameters has been developed. The governing equations including the body force source terms are expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla.(\rho\overset{\rightarrow}{V})=-\frac{1 }{b}(\rho\overset{\rightarrow}{V}.\overset{\rightarrow}{\
abla}b) \tag{1}\]  
\[\frac{\partial(\rho\overset{\rightarrow}{V})}{\partial t}+\
abla.(\rho \overset{\rightarrow}{V}\overset{\rightarrow}{V})+\overset{\rightarrow}{ \
abla}(p)=\rho\overset{\rightarrow}{f}-\frac{1}{b}(\rho\overset{ \rightarrow}{V}.\overset{\rightarrow}{\
abla}b)\overset{\rightarrow}{V} \tag{2}\]  
\[\frac{\partial(\rho e_{t})}{\partial t}+\
abla.(\rho h_{t}\overset{ \rightarrow}{V})=\rho r\Omega f_{\theta}-\frac{1}{b}(\rho h_{t}\overset{ \rightarrow}{V}.\overset{\rightarrow}{\
abla}b) \tag{3}\]  
where \(-\frac{1}{b}\left(\rho\overset{\rightarrow}{V}.\
abla b\right)\) is used to simulate the blocking effect caused by the blade thickness and \(b\) denotes the blockage factor, which is the ratio of the axial thickness of the blade \(e_{x}\) to the blade pitch \(s\)[22; 23]. And \(\rho\) denotes the density of air, \(\overset{\rightarrow}{V}\) denotes the absolute velocity and \(\overset{\rightarrow}{V}^{t}\) denotes its transposition, \(p\) denotes the static pressure, \(e\) denotes the total energy per unit mass, \(h_{t}\) denotes the enthalpy of stagnation per unit mass, \(t\) denotes the time, and \(\Omega\) denotes the angular velocity. \(\overset{\rightarrow}{f}\) denotes the body force per unit mass and \(f_{\theta}\) denotes the circumferential component of the body force per unit mass. In this study, the body force \(\overset{\rightarrow}{f}\) is decomposed into two components \(\overset{\rightarrow}{f}=\overset{\rightarrow}{f_{p}}+\overset{ \rightarrow}{f_{n}}\), and the decomposition of body force \(\overset{\rightarrow}{f}\) within the flow plane is illustrated in Figure 1. The force \(f_{p}\) is parallel but opposite to the relative flow direction, which simulates the frictional drag and loss term due to the blade thickness. Meanwhile, the force \(f_{n}\) is perpendicular to the relative flow direction, which simulates the load exerted by the actual blade on the airflow to achieve deflection of the airflow as well as the work performed on the airflow. In this BFM, the source term depends only on the local flow conditions and local geometry of the blade. The expressions of the body force components \(f_{n}\) and \(f_{p}\) are as follows [21; 22]:  
\[f_{n}=K_{\text{Mach}}\frac{(2\pi\delta)(0.5\overset{\rightarrow}{W}^{2}/|n_{ \theta}|)}{sb} \tag{4}\]  
\[f_{p}=\frac{0.5\overset{\rightarrow}{W}^{2}}{sb|n_{\theta}|}[2C_{f}+2\pi K_{ \text{Mach}}(\delta-\delta_{0})^{2}] \tag{5}\]  
in which  
\[K_{\text{mach}}= \begin{cases}\min(\frac{1}{\sqrt{1-M_{\text{ref}}^{2}}},3),&M _{\text{rel}}<1\\ \min(\frac{4}{2\pi\sqrt{M_{\text{ref}}^{2}-1}},3),&M_{\text{rel}}>1\end{cases}  \tag{6}\]  
\[s=\frac{2\pi r}{B} \tag{7}\]  
\[C_{f}=0.0592 Re_{x}^{-0.2} \tag{8}\]  
\[Re_{x}=\frac{\rho Wx}{\mu} \tag{9}\]  
where \(2\pi K_{\text{mach}}(\delta-\delta_{0})^{2}\) denotes the loss terms under off-design conditions. \(K_{\text{mach}}\) denotes the compressibility coefficient, which is used for the correction of compressibility effects caused by the Mach number.