## 2 Throughflow Model  
### Governing Equations  
Governing equations for the time-marching throughflow model can be derived from the 3D viscous equations by taking a simple axisymmetric assumption or a circumferential average. For body-fitted curvilinear coordinates \((\xi,\eta,\zeta)\) satisfying Equation (1),  
\[\xi_{\varphi}=\eta_{\varphi}\equiv 0 \tag{1}\]  
their governing equations can be written as Equation (2), which represents the throughflow model adopted in most studies, both for the non-bladed duct and the bladed region. \[\frac{\partial}{\partial t}\left(\frac{r\mathbf{U}}{J}\right)+\frac{\partial} {\partial\xi}\left[\frac{r}{J}(\mathbf{F}\xi_{z}+\mathbf{G}\xi_{r}^{\gamma}) \right]+\frac{\partial}{\partial\eta}\left[\frac{r}{J}(\mathbf{F}\eta_{z}+ \mathbf{G}\eta_{r})\right]=\frac{\mathbf{Q}}{J}+\frac{r}{J}\mathbf{f}_{v}+ \mathbf{Q}_{b} \tag{2}\]  
However, this throughflow model may encounter the computing difficulty in the transonic flow simulation when shock waves exist inside the blade passage, for which the details have been presented in [37]. To overcome this problem, quasi-orthogonal body-fitted coordinates satisfying Equation (3)  
\[ \begin{cases}\xi\cdot\zeta=0\\ \eta\cdot\zeta=0\end{cases}  \tag{3}\]  
were adopted in the present study for the bladed region. This resulted in the normally averaged governing equations of throughflow in the blade passage, as in Equation (4),  
\[\frac{\partial}{\partial t}\left(\frac{r\mathbf{U}}{J}\right)+\frac{\partial} {\partial\xi}\left[\frac{r}{J}\left(\mathbf{F}\xi_{z}+\mathbf{G}\xi_{r}+ \mathbf{H}\frac{1}{r}\xi_{\varphi}\right)\right]+\frac{\partial}{\partial\eta }\left[\frac{r}{J}\left(\mathbf{F}\eta_{z}+\mathbf{G}\eta_{r}+\mathbf{H}\frac{ 1}{r}\eta_{\varphi}\right)\right]=\frac{\mathbf{Q}}{J}+\frac{r}{J}\mathbf{f}_{v }+\mathbf{f}_{b}+\mathbf{Q}_{b} \tag{4}\]where \(\xi_{z},\xi_{r},\xi_{\varphi},\eta_{z},\eta_{r},\eta_{q},\zeta_{z},\zeta_{r},\zeta_{ \varphi}\) are the metrics including the blockage effects of blade thickness. \(\underline{\epsilon}(\xi_{z},\xi_{r},\xi_{\varphi}/r),\eta(\eta_{z},\eta_{r}, \eta_{\varphi}/r)\), and \(\zeta(\zeta_{z},\zeta_{r},\zeta_{\varphi}/r)\) correspond to vectors of three orientations, respectively, and \(J\) is the metric Jacobian. The conservative variables \(\mathbf{U}\) are defined as  
\[\mathbf{U}=\left[\rho,\rho u,\rho v,\rho w,e\right]^{T}\]  
and the flux vectors \(\mathbf{F},\mathbf{G}\) and \(\mathbf{H}\) are  
\[\mathbf{F}=\left[\rho u,\rho u^{2}+p,\rho uv,\rho uw,(e+p)u\right]^{T}\]  
\[\mathbf{G}=\left[\rho v,\rho uv,\rho v^{2}+p,\rho vw,(e+p)v\right]^{T}\]  
\[\mathbf{H}=\left[\rho w,\rho uw,\rho vw,\rho w+p,(e+p)w\right]^{T}\]  
where \(\rho\) is density, \(p\) is pressure, \(e\) is specific energy, and \((u,v,w)\) are the components of the relative velocity vector \(\mathbf{W}\). In the right-hand side (RHS) of Equation (4), the term \(\mathbf{Q}\) contains contributions from the cylindrical-coordinate form of the governing equations, and \(\mathbf{Q}_{b}\) accounts for the variation of the area in the blade passage. Both terms are defined, respectively, as  
\[\mathbf{Q}=\left[0,0,p+\rho(w+\omega r)^{2},\rho v(w+2\omega r),\omega^{2}r^{2 }\rho v\right]^{T}\]  
\[\mathbf{Q}_{b}=\left[0,-p\frac{\partial}{\partial\zeta}\left(\frac{r}{J}\zeta _{z}\right),-p\frac{\partial}{\partial\zeta}\left(\frac{r}{J}\zeta_{r}\right),-p\frac{\partial}{\partial\zeta}\left(\frac{\zeta_{\varphi}}{J}\right),0 \right]^{T}\]  
The inviscid blade force term  
\[\mathbf{f}_{b}=\left[0,-\frac{r\zeta_{z}}{J}\left(\frac{\partial p}{\partial \zeta}\right),-\frac{r\zeta_{r}}{J}\left(\frac{\partial p}{\partial\zeta} \right),-\frac{\zeta_{\varphi}}{J}\left(\frac{\partial p}{\partial\zeta} \right),0\right]^{T}\]  
guarantees a tangential condition between the flow and the mean flow surface inside the blade passage and vanishes in the non-bladed duct. The viscous force term \(\mathbf{f}_{v}=\left[0,f_{vz},f_{vx},f_{vy},0\right]^{T}\) is introduced to simulate viscous losses.