4.1. Static Thrust Evaluation
The static thrust is defined as zero forward speed related to the hovering or takeoff phases. Hover performance is critical to analyze since it usually determines the aircraft’s maximum payload, especially in vertical flight operations. Additionally, the maximum thrust available in the case of proprotors is generally encountered for the flight condition of zero velocity. Further details about this last statement are described in later sections of this project.
The evaluated bench-tests [
25] were conducted outdoors, without aerodynamic interference from other elements, and out of ground effect. The test rig and its supporting structure provide negligible blockage of the rotor wake. It has been reported that the proprotor is sufficiently separated from the ground to be considered an out-of-ground effect (OGE).
The thrust balance is accurate within ±0.1% error up to 50 kN, with no significant interactions caused by other forces or moments. The instrumented drive shaft torque is accurate to within ±0.3% error up to the maximum capacity of 28.5 kNm. The tests have been performed with winds equal to or less than 1.5 m/s. The measured rotor torque has been corrected for the effect of wind using an empirical momentum theory-based correction procedure, for which further detail is available in the referred Technical Note [
25]. The magnitude of the correction during the performed tests, at less than 1.5 m/s, is below 3%. This correction percentage is denominated in this document as WACP: Wind Adjusted Power Coefficient.
The experimental bench tests have been grouped into RPM sets with 511, 553, 565, 586, and 624 values. The data has been grouped so that the maximum deviation is 4 revolutions, which is considered to have a negligible effect on the results. The number of data points for each RPM set is 7, 4, 4, 168, and 8, respectively. Therefore, the population is considered enough for statistical analysis, just for the RPM set 586. The results of the Stahlhut solver for the RPM 586 are given in
Figure 6.
Considering an average air density value of 1.235 , an equivalent pressure altitude (ZP) of -80 m has been configured for all the model computations. The progression of and is achieved by increases in the collective, which is aligned between the experimental and the model configuration in the case of the Stahlhut solver. For the BEMT solver, higher collective angles have been needed to match the values.
A second-order polynomial equation, graphically represented by the name “Fitting Curve”, adjusts the error of throughout the range. Fitting the deviation of With such a methodology, the physical meaning of the errors between the model and the experiments might not be captured. However, once the experimental data is known, this fitting curve is considered useful for a smooth adjustment of the model results. General parameters, such as the absolute average, standard deviation, and variance, are calculated for the statistical evaluation purposes of these deviations.
Table 2 presents the main statistical parameters: the absolute average, standard deviation, and variance for each RPM setting and selected solver. The absolute average is presented to account for the positive and negative deviations of
.
The analysis of the static thrust evaluation reveals intriguing relationships between solver choice, RPM settings, and key statistical parameters such as average, variance, and deviations. Across both BEMT and Stahlhut solvers, noticeable trends emerge. For instance, examining the average values, it’s noticeable that the Stahlhut solver consistently yields significantly lower deviations than BEMT across all RPM settings. This suggests a potentially superior accuracy of the Stahlhut model in predicting thrust performance. Furthermore, as RPM increases, there appears to be a slight decrease in average for both solvers, albeit with some variability. This trend could be attributed to an improved model convergence at higher RPMs and Reynolds numbers.
The relationship between RPM and variance cannot be clearly defined. While the variance tends to fluctuate across different RPM settings, it is interesting to note that for the BEMT solver, there’s a general increase in variance as RPM rises. This could indicate increased dispersion or variability in model predictions at higher rotational speeds. In contrast, the variance for the Stahlhut solver shows less consistent patterns across RPM settings, suggesting potentially different sources of variability or model behavior.
Additionally, examining alongside variance provides insights into the consistency and reliability of model predictions. Generally, higher variance values may imply more significant uncertainty or inconsistency in the model’s performance, potentially leading to larger deviations from experimental data. Interestingly, while the Stahlhut solver exhibits lower deviations overall, its variance values are not consistently lower than BEMT, indicating a nuanced relationship between accuracy and variability.
Considering the varying sample sizes across RPM settings adds a layer of complexity to the analysis of statistical parameters. The smaller sample sizes for specific RPM settings, such as 511, 553, and 565, could potentially introduce more significant variability and uncertainty in the calculated averages, variances, and deviations. For instance, the relatively small sample size for the 553 and 565 RPM settings might contribute to the observed higher variances compared to other settings. With fewer data points available for analysis, there may be less precision in estimating the true variability of the data. Conversely, the larger sample size for the 586 RPM setting allows for more robust statistical analysis, potentially resulting in more stable average, variance, and deviation estimates.
4.2. Sensitivity Analysis
The probability density distribution (PDD) becomes a valuable method to analyze the likelihood of a particular
, to be found at each RPM setting. The area enclosed by each probability density curve, the maximum values, and the relative position with respect to
are presented in
Figure 7 for the BEMT and the Stahlhut solver, respectively.
The similarity in the shape of the PDD curves for the BEMT solver suggests that the performance of this solver is relatively consistent across different RPM settings. The wider range of
values for the BEMT solver, from 20 to 88% (RMS 68%), compared to the Stahlhut solver, from -3 to 4% (RMS 7%), clearly represent the error ranges between the small and the large inflow angle modeling approaches. The deviations order of magnitude are similar to previous studies, also based in the Stahlhut methodology, for the same proprotor, with RMS errors of 6.16% between the
and
for Mach 0.6 (RPM = 586) [
30]. The other RPM settings have not been validated by the previously mentioned reference. Additionally, the algorithm configuration was not disclosed, what difficult the comparison between the approach of this paper, and the mentioned reference.
These findings suggest that the BEMT solver tends to be more precise than the Stahlhut solver, but this second one is notoriously more accurate. Looking toward the implementations of these numerical-analytical methodologies, BEMT could be recommended to analyze trends in the design changes of proprotors. In contrast, the Stahlhut solver could be more suitable for quantifying the performance values of each design in the subsequent design validation loops.
Overall, the Stahlhut solver is more conservative than the conventional BEMT, especially closer to the tip and at higher collective angles. In other words, higher and lower tend to be found when considering the allowance for the large inflow angles approach. Except for the induced velocity, the rest of the parameters tend to maintain a certain parallelism between the BEMT and the Stahlhut results. The Reynolds throughout the blade span for both solvers and collective angles remain practically unchanged, as the primary contributor to this parameter is the angular velocity, which is fixed for this analysis.
Conventional BEMT solvers tend to have a linear progression of deltas between experimental and model results. These models are not recommended in the case of tiltrotor configurations, which possess great twist and speed gradients from root to blade tip. These conventional methodologies might be considered attractive if low computational costs are desired and analysis of trends-differences between different proprotor geometrical configurations and operational conditions are desired. The absolute values of the proprotor performance would lead to misleading conclusions in the case of proprotor configurations such as the ones considered in this study.
Expanding the validation loops to encompass a broader range of proprotors’ geometries is recommended. This would serve to verify the reliability and robustness of the proprotor performance tool across various geometrical configurations and operating regimes. Incorporating additional proprotors into the validation process will further enhance the tool’s applicability and build confidence in its accuracy in architectures with not so accentuated pitch differences or greater RPMs.