## 2 Materials and Methods
### CFD Methodology  
The numerical simulations for the fire turbopump were divided into simulations on the turbine section and pump section under the operating condition of the turbopump. The entire computational domain of fire turbopump is shown in Figure 2, including the turbine suction (containing guide-vane passage), turbine impeller passage, pump impeller passage, pump solute and the extension of inlet and outlet. The polyhedral mesh was generated for each independent domain as shown in Figure 3. The mesh density was also studied to verify the independence of the number of elements for the solution. Turbine shaft power and pump head were used as variables of reference to analyze the mesh independence. Figure 4 presents that the solution independent of the mesh with significant computational saving is provided by the mesh with 1.2 million grid points for turbine section and with 1.1 million grid points for pump section.  
Three-dimensional steady and unsteady flow analyses were performed to study performance characteristics and radial hydraulic forces of the fire turbopump based on the OpenFOAM platform. The steady flow was numerically simulated by using the same computational domains, and the results were used as the initial conditions for the unsteady simulations. The Shear Stress Transport (SST) turbulence model was employed on account of the curved passage and the rotational flow [20]. Turbine impeller and pump impeller are rotational components, while the other parts are considered under stationary. The connection interface between turbine suction and turbine impeller was a Frozen Rotor interface. Similarly, the connection interface between pump impeller and pump volute was a Frozen Rotor interface. For the unsteady simulations, the grids are connected by means of a rotor/stator interface, i.e., they change their relative position through the simulation according to the angular velocity of the impeller. The reference change occurs as the flow crosses the interfaces; the appropriate transformation occurs across the interface without any interface averaging. The boundary conditions used in the numerical simulations are summarized in Table 2, and represents the operating conditions of the fire turbopump.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Location** & **Boundary Condition** & **Option** \\ \hline \multirow{3}{*}{Turbine} & Suction & Inlet & Mass flow rate \\  & Outlet extension & Outlet & Outflow \\  & Solid surfaces & Wall & No slip wall \\ \hline \multirow{3}{*}{Pump} & Inlet extension & Inlet & Velocity \\  & Volute & Outlet & Mass flow rate \\  & Solid surfaces & Wall & No slip wall \\ \hline \hline \end{tabular}
\end{table}
Table 2: Boundary conditions used in the numerical simulation [21; 22].  
Figure 4: Mesh independence analysis. (**a**) Turbine section. (**b**) Pump section.  
Figure 3: Parts of computational grid.  
In the unsteady flow, the discretization in space was of second-order accuracy and the second-order backward Euler scheme was chosen for the discretization in time. The connection interface was set to \"transient rotor-stator\". The convergence criteria for all variables were set as \(10^{-5}\). Unsteady flow simulations with 1 degree, 2 degrees, 3 degrees and 4 degrees of runner rotation were conducted to analyze the effect of the time step on the turbopump performances. There is less difference among simulations with 1 degree, 2 degrees and 3 degrees in the respects of inner flow characteristics and performances of pump and turbine. Considering the simulation power and storage capacity of the computer, the time step was considered as 3 degrees of revolution and the duration time was the length of the runner rotating eight revolutions for the unsteady flow simulation.