# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 4 Results and Discussion
### Uncertainty Quantification of the Turboshaft Engine Performance
#### 4.3.2 Comparison of Effects of the Independent and Correlated Uncertain Parameters  
To illustrate the rationality of the independence assumption, two specific scenarios are compared in this section. The first one is that six uncertain parameters are mutually independent, and the other one is that correlations between some uncertain parameters are considered. For the first scenario, six uncertain parameters are assumed to be independent identically distributed. As for the second scenario, a strong correlation between the flow capacity and the adiabatic efficiency of the compressor is additionally assumed, whereas  
Figure 4: (**a**) Mean of \(T_{45}\) versus sample size; (**b**) variance of \(T_{45}\) versus sample size; (**c**) mean of \(P_{e}\) versus sample size; (**d**) variance of \(P_{e}\) versus sample size; (**e**) mean of SFC versus sample size; (**f**) variance of SFC versus sample size.  
the flow capacity and adiabatic efficiency of the HPT and the LPT are still independent. In each scenario, the uncertain parameters follow the truncated Gaussian distribution with zero mean. Three different standard deviations, i.e., 0.002, 0.003, and 0.004, are considered to investigate the impact of the uncertain parameters. In addition, a correlation efficiency of 60% is specified to represent the correlation between the flow capacity and adiabatic efficiency of the compressor in the second scenario. Thus, a joint Gaussian distribution regarding the two uncertain parameters is generated. Under these assumptions, the distributions of three PMs in each case are predicted through the validated GPM-based LHS method. The results are presented in Figure 5. The blue solid lines represent the distributions of the PMs obtained under the assumption of independence, while the red dashed lines represent those obtained by considering the correlation of partial parameters. It should be noted that the mean and variance of the distribution are predicted by their mean estimates. It can be observed that there is no significant difference in mean estimates between the two cases and the difference is negligible. This can also be found from the summarized results in Table 5. The maximum relative deviation is only 0.003923%. However, the correlation between flow capacity and adiabatic efficiency of the compressor has an impact on the standard deviations of the PMs, especially for the output power \(P_{e}\). The standard deviation of \(P_{e}\) changes by around 6.6-8.7% as the standard deviation of the uncertain parameters varies within given ranges. As for the other two PMs, the change is about 2.3-4.2% for \(T_{45}\) and 3.0-3.7% for SFC. Moreover, it can be observed in Table 5 that the correlation has a negative effect on the standard deviations of \(T_{45}\) and \(P_{e}\), but a positive effect on that of SFC. In this situation, if the effect of correlation is ignored, the performance scatter of \(T_{45}\) and \(P_{e}\) may be somewhat overestimated and that of SFC be underestimated to some extent. In practice, the correlation largely depends on the design of the compressor. Thus, it is difficult to obtain a specific correlation between the flow capacity and adiabatic efficiency of the compressor in the preliminary design stage of the engine. In this case, one may preliminarily make an assumption about the correlation based on the engineering experience or directly assume that the parameters are independent of each other. Since the effect of correlation appears weak, the assumption of independence can be acceptable in the early design of the engine.