## 2 Materials and Methodologies
### Development of the Inviscid Blade Force  
In previous research, the inviscid blade force was usually modeled as a term related to time and then solved through the iteration method. The formula proposed by [PERSON] and [PERSON] [17] is:  
\[\frac{\partial(\rho f_{Bu})}{\partial\tau}=C\rho\big{(}{w_{u}}^{\text{target}}- {w_{u}}\big{)} \tag{7}\]where \(\tau\) is the pseudo time; \(C\) is a constant value used in iteration to regulate the convergence of iteration; \(w_{u}\)target is the target tangential velocity; \(w_{u}\) is the tangential velocity determined in the throughflow method. Based on this method, researchers make several improvements upon the inviscid blade force. For example, [PERSON] [18] put forward a model  
\[\frac{\partial(\rho f_{B})}{\partial\tau}=-C\rho(w_{x}n_{Bx}+w_{r}n_{Br}+w_{u} n_{Bu}) \tag{8}\]  
where \(f_{B}\) is the magnitude of the inviscid blade force; \(n_{Bx}n_{Br}\) and \(n_{Bu}\) are components of the unit normal vector of the average stream surface. The modeling of all these inviscid blade force models is based on the assumption that the effect produced by the inviscid blade force is to change the flow direction without entropy production. Therefore, the direction of the inviscid blade force must be orthogonal to the average stream surface. However, these models do not take the effect of the blade geometry into consideration, which is not suitable in the CAM. According to Formula (4), the inviscid blade force takes the form  
\[F_{B}=\frac{N}{2\pi b}\left[ \begin{array}{c}0\\ \left(p_{s}\frac{\partial\varphi_{s}}{\partial x}-p_{p}\frac{\partial\varphi _{p}}{\partial x}\right)\\ \left(p_{s}\frac{\partial\varphi_{s}}{\partial r}-p_{p}\frac{\partial\varphi _{p}}{\partial r}\right)\\ \frac{\left(p_{p}-p_{s}\right)}{r}\\ 0\end{array} \right] \tag{9}\]  
Based on the coordinate definition of the mean camber surface:  
\[\varphi_{M}=\varphi_{M}(x,r) \tag{10}\]  
The circumferential coordinates of the suction side and the pressure side can be displayed as:  
\[ \begin{array}{c}\varphi_{s}=\varphi_{M}+\frac{\tau}{N}(1+b)\\ \varphi_{p}=\varphi_{M}+\frac{\tau}{N}(1-b)\end{array}  \tag{11}\]  
and \(\left(r\frac{\partial\varphi_{M}}{\partial x},r\frac{\partial\varphi_{M}}{ \partial r},-1\right)^{T}\) represents the normal vector of mean camber surface for a blade. Therefore, the inviscid blade force can be transformed as:  
\[F_{B}=\left[ \begin{array}{c}0\\ \frac{N\left(p_{s}-p_{p}\right)}{2\pi b}\left(\frac{\partial\varphi_{M}}{ \partial x}\right)+\frac{p_{s}+p_{p}}{2}\frac{\partial b}{\partial x}\\ \frac{N\left(p_{s}-p_{p}\right)}{2\pi b}\left(\frac{\partial\varphi_{M}}{ \partial r}\right)+\frac{p_{s}+p_{p}}{2}\frac{\partial b}{\partial r}\\ \frac{N}{2\pi b}\left(p_{p}-p_{s}\right)\\ 0\end{array} \right] \tag{12}\]  
That is,  
\[ \begin{array}{c}F_{Bx}=\frac{N\left(p_{s}-p_{p}\right)}{2\pi b}\left(\frac{ \partial\varphi_{M}}{\partial x}\right)+\frac{p_{s}+p_{p}}{2}\frac{\partial b }{\partial x}\\ F_{Bx}=\frac{N\left(p_{s}-p_{p}\right)}{2\pi b}\left(\frac{\partial\varphi_{M} }{\partial r}\right)+\frac{p_{s}+p_{p}}{2}\frac{\partial b}{\partial r}\\ F_{Bu}=\frac{N\left(p_{s}-p_{s}\right)}{2\pi b}\end{array}  \tag{13}\]  
It can be seen that the axial and the radial components of the inviscid blade force include two parts. First, the first item is related to the circumferential inviscid blade force, and it is easy to know:  
\[\left\{ \begin{array}{c}\tan\gamma_{M}=\frac{r\partial\varphi_{M}}{\partial r }\\ \tan\lambda_{M}=\frac{r\partial\varphi_{M}}{\partial x}\end{array} \right. \tag{14}\]where \(\gamma_{M}\) and \(\lambda_{M}\) are the axial flow angle and the radial flow angle, respectively, for the mean camber surface.