This section aims to obtain the statistical characteristics for the influence of manufacture variations on blade aerodynamic performance, and further to obtain that of the systematic and non-systematic variations on blade performance after the variation decomposition and, on this basis, to determine the difference and connection between them.
4.2. Statistic Characteristics of the Influence of Manufacture Variations
Figure 10a presents the profile loss characteristics of the nominal blade and all the measured blades, as well as the mean value and std for the profile losses of the measured blades at each inlet flow angle. The profile loss coefficient was defined as
where
p0,in represents the total pressure evaluated at blade row inlet,
p0,out represents the total pressure evaluated at blade row outlet, and
pin represents the static pressure evaluated at blade row inlet. It can be seen from
Figure 10a that the dispersion (2std) of the profile loss is the minimum at the inlet flow angle with the minimum profile loss. The inlet flow angle with the minimum loss of the nominal blade is defined as the reference inlet flow angle
αref, the corresponding loss as the reference loss
ωreference, and
What is more noteworthy in
Figure 10a is that, under the condition of positive inlet flow angle, not only is the mean loss of the measured blade greater than that of the nominal blade, but almost all the measured blades have greater losses. Similarly, under the condition of negative inlet flow angle, the mean loss and almost all the measured blade profile losses are smaller than the nominal blade. This indicates that the losses of the measured blades are systematically deviated from the nominal blade due to manufacture variations.
Figure 10b presents the loss characteristics of systematic blades, which were decomposed and reconstructed by the corresponding manufacture variations. It presents the influence of systematic variations on the blade profile loss. It can be seen from the figure that the influence of systematic variation is similar to that of the whole manufacture variation, that is, the blade profile loss is systematically increased under the condition of positive inlet flow angle, while it is opposite under the negative inlet flow angle. Associating with the characteristics of the systematic variation in
Table 3, this is possibly because the mean value of systematic variation also deviates significantly from the nominal blade (especially the inlet metal angle). This correlation will be discussed in a later section.
Figure 10c shows the loss characteristics of non-systematic blades, which were obtained by superimposed the corresponding non-systematic variations on the nominal blade profile. It can be seen from the figure that the mean loss of non-systematic blades almost coincides with the nominal blade, but the dispersions (2std) of the positive and negative inlet flow angle are much higher than that of systematic blades.
In order to further illustrate the above characteristics, the loss statistical characteristics of variation blades are exhibited in
Figure 11. The negative/positive range is defined as the condition in which the loss is 1.5 times the reference loss [
17].
Figure 11 illustrates the mean and std of the difference between variation blades and the nominal blade. The Δ
ωrel is defined as:
where
ωrel, variation can be the relative loss of measured, systematic, and non-systematic blades. When Δ
ωrel equals zero, it means that the loss of the variation blade is equal to the nominal blade.
Figure 11a presents the mean Δ
ωrel of variation blades. The following characteristics can be seen from the figure:
When the inlet flow angle αin > 60°, the mean Δωrel of systematic blades is basically consistent with that of measured blades;
When the inlet flow angle αin < 60°, the mean Δωrel of systematic blades deviates from that of measured blades.
When the inlet flow angle αin > 60°, the mean loss of non-systematic blades approximates to that of the nominal blade. When αin < 60°, it deviates from that of the nominal blade.
In summary, it can be considered that the systematic ωrel deviation of measured blades described above is caused by systematic variations in most inlet flow angle conditions.
Figure 11b presents Δ
ωrel std of variation blades. The following characteristics can be seen from the figure:
- 4.
Δωrel std of measured blades in the positive range is approximately coincident with that of non-systematic blades, and is about twice the std of systematic blades.
- 5.
Δωrel std of measured blades in the negative range is closer to that of systematic blades, while the std of non-systematic blades is obviously larger.
To sum up, in the positive range, the non-systematic variation determines the loss dispersion of the variation blades, while in the negative range, the loss dispersion is closer to that caused by systematic variations.
On this basis, in order to verify whether the effects of systematic and non-systematic variations on losses can be linearly superimposed, the loss deviations of each systematic and non-systematic blade were processed as:
where Δ
ωrel, sys, i represents the loss deviation of the systematic blade No.
i, and Δ
ωrel, non, i represents the loss deviation of the corresponding non-systematic blade. Therefore, Δ
ωrel, (sys + non), i represents the linear superposition of the loss deviation between the systematic blade and the corresponding non-systematic blade. The mean value and the std of Δ
ωrel, (sys + non) have been illustrated in
Figure 11 (green line).
It can be seen from
Figure 11 that the mean loss deviation of the linear superposition is consistent with that of the measured blade when the inlet flow angle is greater than 60°. However, the std obtained by the linear superposition is different from that of the measured blade. Therefore, the effects of systematic and non-systematic variations on blade losses have a weak linear additivity, which requires further modification and research.
In conclusion, systematic variations mainly determine the mean loss deviation of blades, while non-systematic variations have a large impact on the loss dispersion. Therefore, the influences of these two decomposition variations will be respectively described below.
4.3. Blade Design Parameter Based Sensitivity Analysis for Systematic Variations
As shown in
Table 3, seven parameters were parameterized for the blade profile when extracting systematic variation. Therefore, systematic variation is further decomposed into variations of these seven independent parameters in this section.
Table 5 shows the range selected when each systematic variation parameter is changed independently. Based on this, the loss of a series of variation blade is calculated. Similar to other researchers’ studies, the effect of these parameters on profile loss has strong linear characteristics [
10,
11,
18,
19]. Therefore, in order to avoid redundancy, only the most remarkable influence parameter of inlet metal angle variation is presented in
Figure 12.
Figure 12a is the schematic diagram of the blade profile with the changing of the inlet metal angle, where +5° represents that the blade profile is bent 5° to the suction surface.
Figure 12b shows the variation rule of Δ
ωrel with Δ
βLE at the selected three inlet flow angles, which
αin = 59.6° means that the inlet flow angle is at the negative incidence limit, and
αin = 63.0° means the minimum loss condition, and
αin = 66.5°, positive incidence limit. It can be seen from
Figure 12b that, as mentioned above, Δ
ωrel has a strong linear effect with the change of Δ
βLE.
On this basis, the sensitivity analysis is conducted on the profile loss to each systematic variation at several inlet flow angles, and the results are shown in
Table 6. The “before regression” line indicates the sensitivity obtained by independently changing the systematic variation parameters. The “post regression” line indicates the sensitivity of correction using linear regression, i.e., by using least square fitting for all the sensitivity coefficients.
After obtaining the sensitivity of the profile loss to each systematic variation parameter, the sensitivity can be used to estimate the profile loss:
where the
ki represents the sensitivity in
Table 6.
Through Equation (13) and the extracted value of the systematic variations, the profile loss of each systematic blade can be estimated, thus obtaining
Figure 13. The abscissa is the calculated value of the loss deviation for each systematic blade, and the ordinate is the estimated value using Equation (13). The red line is the result of the linear fitting, and R
2 is the coefficient of determination. As can be seen from the fitting results in
Figure 13, there is a certain linear relationship between the loss deviation estimated by the sensitivity and that of systematic blades.
For instance, the coefficient R2 = 0.55 when αin = 59.6°, which means that at least 55% of the systematic loss deviation is determined by the linear superposition of parameter sensitivities. However, at the same time, it is worth noting that no matter the inlet flow angle state, the intercept of the fitting line is not zero, nor is the slope one. This indicates that although independent parameter sensitivities can indicate the trend of the systematic loss deviation, there are still some problems in quantitatively estimating the systematic loss deviation according to independent parameter sensitivities. There is a coupling relationship between each systematic variation parameters and therefore the sensitivity needs to be modified.
Because of a partial linear relationship between Δ
ωrel of systematic blades and systematic variation parameters, which has been presented in
Figure 13, based on Formula (12), a linear regression between Δ
ωrel of systematic blades and systematic variation parameters based on Equation (13) is conducted, so as to modify the sensitivity of each parameter. The results of the linear regression are shown in
Figure 14. The ordinate is Δ
ωrel obtained by using linear regression relationship, and the red line is the y = x reference line. Meanwhile, sensitivities of systematic parameters obtained by linear regression are listed as “post regression” results in
Table 6.
As shown in
Figure 14, the estimation of Δ
ωrel obtained by linear regression is not only close to y = x reference line, but also the correlation coefficient R
2 is greater than 0.85. Therefore, regression sensitivities of systematic variation parameters can be used to estimate the statistical characteristics of Δ
ωrel of systematic blades [
35].
The sensitivities of systematic variation parameters before and after regression are compared in
Table 6. The sensitivity changes before and after regression represent the effect of the parameter coupling relationship. The remarkable characteristic of the coupling relationship is that the sensitivities of the inlet metal angle and the stagger angle are obviously reduced, and the sensitivities of the chord length, the radii of the LE and the TE, and the maximum thickness are improved.
Figure 15 presents the contribution of each systematic variation parameter to the mean value and std of Δ
ωrel using the sensitivity obtained by linear regression. As can be seen from
Figure 15a, the effect of the inlet metal angle on the mean Δ
ωrel is much higher than that of other parameters. This is mainly because, as shown in
Table 3, the mean variation of the inlet metal angle is very large, which can be seen from its |mean/2std| being much higher than other parameters.
Meanwhile, it can be seen from
Figure 15b that in the condition of
αin = 63.0°, the radius of the TE makes the largest contribution to Δ
ωrel std. In other conditions, the contributions of the outlet metal angle, the chord length, and the radius of the TE to std are small, while the contributions of the inlet metal angle, the radius of the LE, and the maximum thickness are obviously larger than other parameters. It is also worth noting that when
αin = 59.6°, the contribution of the radius of the LE to std is even greater than that of the inlet metal angle.
Obviously, the contribution of systematic variations to the mean value and std of Δ
ωrel is not only related to their sensitivities, but also directly related to their manufacture statistical characteristics. Thus,
Figure 15 does not clearly indicate the comparison of the sensitivity between each of the systematic variation parameters.
At the same time, due to the different units of each systematic variation parameter, the sensitivity values in
Table 6 have no comparative significance with each other. Therefore, in order to compare the sensitivity between each systematic variation parameter, the upper and lower limits that can be allowed in manufacture processing for each parameter are selected to evaluate the effect of each parameter on Δ
ωrel std. Thus,
Figure 16 is obtained.
Figure 16 shows that the radius of the TE is the most sensitive parameter when
αin = 63.0°. In other conditions, the inlet metal angle is the most sensitive parameter, especially in the condition of positive inlet angle of
αin = 66.5°. Its sensitivity is far greater than other parameters. Meanwhile, the sensitivities of the radius of the LE and the maximum thickness are obviously greater than other parameters. The sensitivity of the radius of the LE is relatively higher at
αin = 59.6°, i.e., at negative inlet flow angle condition.
4.4. Region Decomposition Based Sensitivity Analysis for Non-Systematic Variations
Previous studies show that the manufacture variation of the LE has the most significant effect on the profile loss. In this section, the blade surface is further decomposed into different regions to reveal the effect of non-systematic variations on the profile loss at different locations. As shown in
Figure 8a, the std of non-systematic variations has several “nodes” close to zero. Therefore, these “nodes” are used to divide the blade into eight regions as shown in
Figure 17.
The multipliers in Equation (14) were used to extract the non-systematic variations in different regions. The variations in each region were superimposed on the nominal blade profile, so as to obtain eight groups of variation blades. Each group of blades represents the non-systematic variations in its own region.
Figure 18 shows the calculation results of the effect of non-systematic variations on the profile loss in different regions. And
Figure 18a shows the effect of LE variations on the profile loss. It can be seen that, similar to the effect of the whole non-systematic variations, the mean loss caused by LE variations is approximately coincident with the loss of the nominal blade, and the std of the loss caused by the LE variation is large.
In addition to the LE region, the mean loss of non-systematic variations in other regions is also consistent with the loss of the nominal blade. Δ
ωrel std due to non-systematic variations in each region is illustrated in
Figure 18b. It can be seen that the std caused by LE variations is significantly greater than that of other regions. Thus, the profile loss is most sensitive to non-systematic variations in the LE region. It should be noted that, according to the std of non-systematic variations obtained in
Figure 8a, the most sensitive LE region only accounts for a geometric variation of about 0.45%.
As shown in
Figure 18b, Δ
ωrel std generated by non-systematic variations in other regions is close to zero. On this basis, Δ
ωrel std caused by non-systematic variations superimposed on the LE only and on the whole blade is compared in
Figure 19. It can be seen that in the positive range, Δ
ωrel std caused by the LE variations and the whole non-systematic variations approximately coincides, but in the negative range, it deviates greatly. Thus, the coupling of non-systematic variations in different regions is stronger under the condition of negative inlet flow angle.