## 4 Validation and Applications
### Low-Speed Centrifugal Compressor (LSCC)  
The LSCC is a low-speed centrifugal compressor with a semi-open and backswept impeller that was designed by the NASA Lewis research center, and it has a design performance of 30 kg/s in the mass flow rate and 1.166 in terms of the total pressure ratio. As a validation for the developed throughflow method, the numerical simulation of the LSCC case is compared with the experimental data by [PERSON] et al. [54]. The computations were performed on a 365 \(\times\) 40 grid, which is sufficient for the Euler-based time-marching throughflow computation, as can be seen in the grid-independence validation shown in Figure 2b. For clarity, a coarser grid is depicted in Figure 2a instead of the actual grid. With the grid, a design-point throughflow simulation costs less than 2 min on an R5-5600H processor to achieve convergence. The convergence histories enforcing different CBCs, i.e., the Riemann-invariant-based method and the developed CBCs based on [PERSON]'s method in this study, are compared in Figure 3. It can be observed that, except for a slight saving of two iterations, the CBCs based on [PERSON]'s method and the Riemann-invariant obtained almost the same convergence curves. This indicates that there exists little difference between these two kinds of CBCs when they are adopted into the computation of the low-speed centrifugal compressor. The computed overall performances at the design mass flow rate are compared with the experimental data in Table 2. The results are in good agreement, with an error of 1.8% for the total pressure ratio and \(-\)0.3% for the isentropic efficiency. This shows the adequacy of the present throughflow method of design-point simulation in predicting the overall performance of the LSCC. Figure 3: Convergence histories of LSCC simulation. Figure 2: Computational grid of the LSCC. (**a**) Coarser grid; (**b**) grid-independence validation. A further investigation of the off-design performance of the LSCC was implemented. It was reported by [PERSON] et al. [40] that the recirculation flow model has a great influence on the off-design predictions. Therefore, both the recirculation loss models developed by [PERSON] et al. [40] and [PERSON] et al. [48] were validated in this case. Figure 4 shows the comparison between different settings of the recirculation loss model in terms of both the total pressure ratio and the isentropic efficiency. In Figure 4a, it seems that the recirculation loss has little effect on the total pressure ratio. This is reasonable because as one kind of parasitic loss, the recirculation loss only leads to a change in enthalpy and a corresponding change in efficiency. It can also be seen that the discrepancy between the throughflow results and the experimental data increased with the mass flow rate. Nonetheless, the maximum error was less than 2.5% within all operating ranges. Unlike the total pressure ratio, the off-design performance of the isentropic efficiency was greatly influenced by the recirculation loss model setting, as shown in Figure 4b. When the mass flow rate was larger than the design value, [PERSON]'s recirculation model was in good agreement with the experiment, whereas it seems that no significant loss was produced by [PERSON]'s model and thus resulted in higher isentropic efficiency. For the small mass flow rate, both [PERSON]'s and [PERSON]'s models showed dramatically increased discrepancy compared to the experiment as the mass flow rate decreased. Nevertheless, the trend of the rapid efficiency decrement was predicted by both models. The comparison shown in Figure 4b indicates that a more accurate recirculation flow loss model needs to be developed. Table 3 further reports the detailed contributions of each loss to the total loss in the throughflow simulation at different operating conditions when [PERSON]'s recirculation model was adopted.