## 3 Numerical Methods  
In this study, a commercial software, Numeca FineTM/Turbo [14], is applied to the impeller performance analysis based on a 3-D steady and unsteady compressible finite volume scheme. It is assumed in the simulation that internal flow in impeller is relatively stable three-dimensional viscous compressible turbulent flow. The governing equations are continuous equation including Reynolds-averaged Navier-stokes equation, energy conservation equation, the ideal gas state equation [15]. The governing equations written in a Cartesian frame can be expressed as follows:  
\[\left\{ \begin{array}{l}\dfrac{\partial\rho}{\partial t}+\
abla\bullet\left( \rho\overset{\rightarrow}{V}\right)=0\\ \dfrac{\partial}{\partial t}\rho\overset{\rightarrow}{V}+\
abla\cdot\left( \rho\overset{\rightarrow}{V}\right)=\rho\overset{\rightarrow}{F}-\
abla \cdotp+\
abla\cdot\overset{\leftrightarrow}{\tau}\\ \dfrac{\partial E}{\partial t}+\
abla\cdot\left(\left(E+P\right)\overset{ \rightarrow}{V}\right)=\rho\overset{\rightarrow}{F}\cdot\overset{ \rightarrow}{V}+\
abla\cdot\left(\overset{\leftrightarrow}{\tau}\cdot \overset{\rightarrow}{V}\right)+\
abla\cdot\left(k\
abla T\right)\end{array}  \right., \tag{1}\]  
where, \(\rho\) is flow density, \(t\) is time, \(\overset{\rightarrow}{V}\) is velocity vector, \(\overset{\rightarrow}{F}\) is external force, \(p\) is external pressure, \(\overset{\leftrightarrow}{\tau}\) is viscous stress, \(E\) is internal energy, \(P\) is kinetic energy, \(k\) is adiabatic index and \(T\) is temperature. The ideal gas state equation is:  
\[p=\rho RT, \tag{2}\]  
where, \(R\) is gas constant. Supplementary equations are expressed as:  
\[\left\{ \begin{array}{l}e=C_{v}T\\ h=e+pV\\ TdS=de+pdV\end{array} \right., \tag{3}\]  
where, \(e\) is total energy, \(C_{v}\) is the specific heat at constant volume, \(h\) is enthalpy of fluid and \(S\) is entropy. The employed turbulence modelling to consider the compressibility is the Spalart-Allmaras model [16]. The Spalart-Allmaras (S-A) turbulence model contains a set of relatively new single equation, which uses an eddy-viscosity variable \(\overset{\sim}{v}\), and is combined with speed, precision and stability, the turbulent eddy transport equation. This can be expressed as follows [17]:  
\[\dfrac{\partial}{\partial t}\left(\rho\overset{\sim}{v}\right)+\dfrac{\partial }{\partial x_{i}}\left(\rho\overset{\sim}{v}u_{i}\right)=G_{v}+\dfrac{1}{ \sigma_{\overset{\sim}{v}}}\left[\dfrac{\partial}{\partial x_{j}}\left\{ \left(\mu+\rho\overset{\sim}{v}\right)\dfrac{\partial\overset{\sim}{v}}{ \partial x_{j}}\right\}+C_{b2}\rho\left(\dfrac{\partial\overset{\sim}{v}}{ \partial x_{j}}\right)^{2}\right]-Y_{v}, \tag{4}\]  
where, \(G_{v}\) is production of turbulent viscosity, \(\sigma_{\overset{\sim}{v}}\) and \(C_{b2}\) are model constants, \(v\) is molecular kinematic viscosity, \(\overset{\sim}{v}\) is turbulent viscosity, \(\mu\) is dynamic viscosity, \(x\) is cartesian coordinate and \(Y_{v}\) is reduction of turbulent viscosity, which happens near the wall. In the upstream and downstream, the computational domain of single flow passage is extended as shown in Figure 4a. The number of grid points for a single passage is around 1,250,000, which is large enough according to the grid independence investigation. y+ is small enough for the S-A turbulence model. The enlarged detail mesh figure around blade leading edge is shown in Figure 4c. A single passage model with periodical boundary conditions is used in steady simulation. Velocity direction, absolute total pressure and total temperature are imposed at the inlet boundary, outlet mass-flow and initial pressure on the outlet boundary. In this study, the working point close to stall condition is figured out by the lowest mass flow, with which the numerical simulation can reach convergent results. The LL(both inlet line and middle line changed model) impeller is chosen to analyze the steady and unsteady performance with the help of the full passage model.