Junying WANG, Botong WANG, Heli YANG, Zhenzhong SUN,Ki ZHOU, Xinqin ZHENG,,*

a State Key Laboratory of Automotive Safety and Energy, School of Vehicle and Mobility, Tsinghua University, Beijing 100084, China

b Department of Aerodynamics and Thermodynamics, Institute for Aero Engine, Tsinghua University, Beijing 100084, China

KEYWORDS Compressor;Geometric uncertainty quantification;Interpretable machine learning;Multiple conditions;Neural network

Abstract Geometric and working condition uncertainties are inevitable in a compressor,deviating the compressor performance from the design value.It’s necessary to explore the influence of geometric uncertainty on performance deviation under different working conditions.In this paper,the geometric uncertainty influences at near stall,peak efficiency, and near choke conditions under design speed and low speed are investigated.Firstly,manufacturing geometric uncertainties are analyzed.Next,correlation models between geometry and performance under different working conditions are constructed based on a neural network.Then the Shapley additive explanations (SHAP)method is introduced to explain the output of the neural network.Results show that under real manufacturing uncertainty,the efficiency deviation range is small under the near stall and peak efficiency conditions.However, under the near choke conditions, efficiency is highly sensitive to flow capacity changes caused by geometric uncertainty, leading to a significant increase in the efficiency deviation amplitude, up to a magnitude of -3.6%.Moreover, the tip leading-edge radius and tip thickness are two main factors affecting efficiency deviation.Therefore, to reduce efficiency uncertainty,a compressor should be avoided working near the choke condition,and the tolerances of the tip leading-edge radius and tip thickness should be strictly controlled.

1.Introduction

As the core component of an aero-engine or gas turbine, the aerodynamic performance of a compressor has an important influence on the engine performance.However, due to manufacturing errors or in-service degradation,1there are a lot of unavoidable geometric variable deviations in a compressor,which have strong uncertainty characteristics.The geometric uncertainties lead to the performance deviating from the design value and presenting a probability distribution.Moreover, with continuous improvement of compressor performance indices, blade loading and therefore the sensitivity of performance to blade geometry are increasing.Therefore, it is becoming more and more important to investigate the influence of geometric uncertainty on performance and find out critical geometric variables2.

The geometric uncertainty quantification problem in compressors has characteristics of high randomness, high dimensionality, and strong nonlinearity.In relative research, it is a core difficulty to evaluate the quantitative relationship between geometry and performance efficiently and accurately.To solve this problem, in recent years, a series of Uncertainty Quantification(UQ)methods,including the direct sampling method,3–5the sensitivity-based method,6,7and the surrogate model-based method,8–11has achieved rapid development and application.Among all the methods, the surrogate model-based method,which uses a fast correlation model as a low computational cost alternative to a high-fidelity model such as Computational Fluid Dynamics (CFD), has the dual advantages of saving computing resources and retaining high accuracy of model results.In terms of the selection of a surrogate model, there exists a trade-off between the complexity and interpretability of the model.12As a highly nonlinear model,a neural network can predict performance accurately.However,due to the opacity of the internal structures of neural networks, usually they can only be used as‘‘black boxes”.To solve this problem,Shapley additive explanations (SHAP), an interpretable machine learning method based on game theory,13,14was firstly introduced to turbomachinery research by Ref.15to explore the multi-geometrical variable coupling effect in a transonic compressor.In this study, the SHAP method was adopted to extract the influence rule and clarify sensitive uncertainty variables from a highly nonlinear neural network.On one hand, results from SHAP analysis can provide guidance for compressor tolerance design and control;on the other hand,they can help explore the influence mechanism of critical geometric uncertainty variables on performance.

There has been a series of studies on compressor geometric uncertainty quantification based on different UQ methods.Based on the Latin Hypercube Sampling (LHS) method,Lange et al.16explored the influences of 18 manufacturing geometric uncertainty variables of a 1.5-stage high-pressure compressor blade on performance.Results showed that the leading-edge thickness was the most important geometric parameter affecting efficiency.Based on the same method,similar studies were carried out on asymmetric compressors17and multistage axial compressors.18Based on the adjoint method,Zhang et al.19evaluated the influences of geometric uncertainties on the compressor performance.Moreover, Ju and Zhang20used the Support Vector Regression (SVR) model as a surrogate model to rapidly predict the cascade performance under pollution conditions.Most of the existing studies are quantitative analysis under a single working condition.However,due to different matching criteria or operating environments, a compressor may be used under different working conditions at different rotational speeds.Under different working conditions,compressor mass flow rates,loading characteristics, and internal flow field structures are different, so the influence of geometry on performance may also have significant differences.Schnell et al.21studied the distribution of the outflow angle β2caused by geometric uncertainties under different incident angles β1based on a 2-dimensional blade profile.Results showed that, with the same geometric uncertainty, the deviation range of β2showed an obvious growth trend with β1.At present, research on geometric uncertainty quantification under different working conditions is still very limited, and it is not very clear how geometry and different flow field structures interact to influence performance.

The motivation of this paper is to systematically explore the differences in the influences of real manufacturing uncertainties of geometric variables at different speeds and under different working conditions.Based on the results, guidance for compressor design and tolerance design and control can be provided.The rest of the paper is organized as follows.Firstly,the distribution characteristics of geometric variables with manufacturing uncertainty are extracted from measured data.Then, the uncertainty quantification method adopted in this paper is introduced, including CFD, fully connected neural network, and the SHAP method.Afterward, the influences of geometric uncertainties are analyzed and discussed, while performance distribution and parameter sensitivity are compared under different working conditions.Finally, the flow mechanism behind the influence of the critical uncertainty variable is discussed based on CFD simulation results.

2.Case overview

2.1.Compressor geometry and parameterization

The research object of this study is the first-stage rotor of a three-stage compressor, and the illustration of the computation domain is presented in Fig.1.

To study the influences of different geometric uncertainty variables on performance, the rotor blade geometry should be parameterized first, as shown in Fig.2.Firstly, the number of blades and the meridional profile of hub and shroud are parameterized.Secondly, five spanwise sections with 0, 0.19,0.51,0.83,and 1.00 spanwise are parameterized,and the stacking lines between different sections are defined.Finally, each spanwise section of the blade is defined by parameters including stagger angle G, leading-edge radius Rle, trailing-edge radius Rte, camber curve shape, and blade thickness distribution T1-T5.

2.2.Compressor geometric manufacturing uncertainty

Fig.1 Illustration of compressor rotor computation domain.

Fig.2 Schematic of geometrical uncertainty variables of compressor rotor.

For the compressor rotor,the three-coordinate measuring data of more than 100 new blades manufactured are statistically analyzed.For each blade, the profiles in 0.19, 0.51, and 0.83 spanwise sections are measured and represented by the coordinates of blade profile points.An automatic parameterization program is written to extract the geometric variables (Rle,Rte, T, and G) from coordinate points.The thickness T,leading-edge radius Rle, and trailing-edge radius Rteare the radii of the corresponding inner circles of the blade profile.To calculate the stagger angle G, firstly, the center points of the inner circles are connected as the camber curve.Then,two intersection points of the camber curve and the blade profile are defined as the leading/trailing edge points.Finally, the angle between the line formed by the leading/trailing points and the axial direction is defined as the stagger angle G.

The absolute deviations of the measured data from the design values are marked as ΔRle,ΔRte,ΔT,and ΔG(note that the range of ΔT is the union set of the ranges of ΔT1-ΔT5).Table 1 lists the variation ranges of the deviation values at 0.19, 0.51, and 0.83 spanwise sections (marked as L, M, and H sections, respectively).It can be seen that: the relative deviations of Rleand Rteare very obvious and on the order of ±50% relative to the design values; the relative deviations of T are smaller and on the order of ±6% relative to the design values; the absolute deviation of the stagger angle is on the order of ±0.1°.

In the following study, the variation ranges of the 0.19,0.51, and 0.83 spanwise sections are given directly based onmeasured results, while in other sections,geometric deviations are given by B-spline interpolation.This interpolation can be considered reasonable referring to Refs.,16,22which present smooth distributions of geometric deviations along blade heights.

Table 1 Geometric variable deviation ranges of rotor due to manufacturing uncertainty.

3.Uncertainty quantification method

3.1.Method overview

Based on the statistical analysis results of compressor geometric uncertainty, the influence of uncertainty is quantified and analyzed through the following three main modules:

I Design of Experiment (DoE).3In this study, it is assumed that geometric uncertainty variables conform to uniform distribution within the variation range, and the LHS3method is adopted to generate 365 blade samples with different geometric variable values.Note that for each sample, ΔT1-ΔT5are assumed the same and marked as ΔT.Therefore, there are totally 12 independent geometric uncertainty variables in this study.The total number of samples is more than 30 times the total number of geometric uncertainties, which ensures the validity of sampling.For each sample, its performance is obtained through CFD simulation, and the detailed settings of the CFD method will be described in Section 3.2.

II Surrogate model construction.Based on the database generated in the DoE process, the fully connected neural network23is trained to map the correlation between geometric variables and compressor performance, and is used as the surrogate model of 3-dimensional CFD simulation to rapidly predict compressor performance.The detailed setting of the neural network will be described in Section 3.3.

III Sensitivity analysis.Based on the neural network surrogate model, the interpretable machine learning method SHAP can be used to extract the influence rule and clarify sensitive uncertainty variables from the highly nonlinear neural network.The principles of the SHAP method will be described in Section 3.4.

Based on Modules I and II, the distribution of compressor performance due to uncertainty can be obtained.This part of study aims at providing guidance for compressor design.Based on Module III, critical geometric variables can be extracted from a large number of variables.This part of study aims at providing guidance for compressor tolerance design and control.

3.2.CFD method and validation

FINE/Turbo (Version 15.1) software is used for CFD simulation.The computation is set as steady and compressible.The multi-grid procedure is adopted to improve the convergence rate of simulation,and the Spalart-Allmaras turbulence model is adopted.24The computational domain is a single rotor passage with periodic boundary conditions.At the inlet, the total pressure (101325 kPa), total temperature (288.15 K), and velocity direction (given according to the mean outlet flow angle of the inlet guide vane of the 3-stage compressor) are defined as the boundary condition.At the outlet, under complete choking conditions, the static pressure is defined, while under other conditions, the mass flow rate is defined as the boundary condition.Moreover, all the solid walls are set as non-slip wall boundaries.The blade and hub surfaces are set as rotating, while the shroud surface is set as stationary.

The software AutoGrid is used to discretize the computational domain.To quantify the discrete error, mesh independence analysis is conducted.The number of grid nodes in all dimensions increases to about 1.2–1.5 times uniformly when the mesh is refined.For all the meshes,the thickness of the first layer grid is fixed as 0.003 mm, corresponding to an average y+value below 3 and a maximum y+value below 10.The details of the five meshes are summarized in Table 2.

CFD simulations under 100 % design speed are conducted using the five sets of meshes,and Fig.3 shows how the normalized choke mass flow rate mc,Nand normalized peak efficiency ηp,Nvary with the total number of grid points ng.Based on Fig.3, the Fine grid is selected in this study considering the trade-off between computational accuracy and cost.Compared with the UltraFine results,the relative error of the choke mass flow rate and the absolute error of the peak efficiency are -0.01% and -0.11%, respectively.

CFD simulation validation is conducted based on the experimental results of the 3-stage compressor.It needs to be noted that, in the simulation of the 3-stage compressor, the above simulation settings are adopted, and the mesh grid settings of each blade row are basically consistent with that of the above Fine grid.Table 3 presents the compressor performance errors of CFD simulation compared with experiment results, which shows that the simulation can underestimate the choke mass flow rate (mc) and peak efficiency (ηp) of the 3-stage compressor.The relative error of the choke mass flow is -0.62% and the absolute error of the peak efficiency is -0.41%, which meet the numerical simulation accuracy requirement in this study.

Fig.3 Compressor performance simulated with different mesh sizes.

Table 3 Comparison of 3-stage compressor performance between experimental and CFD results.

It is worth noting that 0.41%is the CFD error of the whole 3-stage compressor instead of the single rotor alone.However,in the subsequent UQ simulation and analysis, the research object is only a single rotor blade (R1).Generally speaking,the fact that CFD errors for multistage compressors are significantly higher than that of a single blade row25,26is mainly resulted from the following two reasons: (A) errors induced by the mixing plane setting at the rotor/stator interface27; (B)stage-by-stage amplification from upstream to downstream blade rows.27,28In addition,CFD should be used on a comparative basis, especially in the situation of UQ or optimization study.27Refs.29,30have demonstrated the ability of CFD to capture the relative deviation of performance change, bycomparing the CFD and experimental results of compressors with geometric deviations.

Finally, the performance map of the compressor rotor,including the pressure ratio-mass flow rate and efficiencymass flow rate performance curves at 100% and 70% design rotational speeds, is obtained through CFD simulation, as shown in Fig.4.In the figure,the mass flow rate(m),pressure ratio(π),and efficiency (η)are normalized.The normalization methods are as follows:

In the following part of this study, on one hand, at 100%design speed with supersonic/transonic flow fields,three working conditions are selected:near stall point(A),peak efficiency point (B), and near choke point (C).On the other hand, at 70 % design speed with subsonic flow fields, three working conditions are selected: near stall point (D), highest efficiency point(E),and near choke point(F).For each of the 6 working conditions, the influence of geometric uncertainty on performance will be quantitatively analyzed.

3.3.Fully connected neural network

The fully connected neural network is a widely used model to extract nonlinear mapping from a large amount of data.Generally,a neural network consists of an input layer,several hidden layers, and an output layer.Each layer has several neurons, which are connected to the neurons of the upstream and downstream layers in a linear combination.The nonlinearity of the model is introduced by the activation function of each intermediate neuron.The structure of the neural network model mimics that of the human brain.

In this study,the neural network establishment and training process is based on the PyTorch framework.The neural network consists of one input layer, two hidden layers, and one output layer, as shown in Fig.5.The number of nodes in the input layer is 12 (i.e., nin= 12), where each node corresponds to a geometric uncertainty variable.There are 6 nodes in the output layer (i.e., nout= 6), where each node corresponds to the efficiency of a working condition.The hyperbolic tangent function is chosen as the activation function.The root mean square error is used as the loss function.For all 365 samples,300 samples are used as the training set and 65 samples are used as the validation set.The training process of the neural network is driven by the adaptive moment estimation optimizer, and the number of iterative steps is set to 20000, which is sufficient to find the optimal parameters of the neural network to make the model with the highest accuracy.More details on the structure and algorithms of neural networks can be found in Ref.31.

The node numbers of two hidden layers (i.e., nh1and nh2)are the hyperparameters, which need to be specified before neural network training.To determine the optimal hyperparameters,a grid search is performed for nh1and nh2.Moreover,the K-fold cross-validation(K=6 to find the balance between bias and variance) and determination coefficients (R2) are adopted to evaluate the performance of neural networks.The concrete steps are:

Step 1.The original training set is partitioned into K subsets(folds) of size 365/K.

Step 2.For each of the K fold, a neural network is trained on the union of the other K - 1 folds (training set).Then the error of its output is estimated using the fold (validation set).In this study, the determination coefficient (R2) between the ηNvalues predicted by CFD and the neural network is chosen as the evaluation index of the network error.

Step 3.After Step 2, K neural networks are trained and R2on the validation set is calculated K times.Then the average R2of all K folds, marked as mean(R2), is used to measure the accuracy of the neural network.

Fig.6 presents the contour of the mean(R2) as nh1and nh2change.According to Fig.6, the point with nh1= 8 and nh2=10 is chosen as the optimal hyperparameter point,which is marked as a yellow star in the figure.Using the hidden layer nodes above, the mean(R2) of the neural network on the validation set is higher than 0.98, indicating that the neural network model has established an effective mapping between geometric uncertainty variables and the compressor efficiency.

Fig.4 Rotor performances at 100% and 70% design rotational speeds.

Fig.5 Schematic of fully connected neural network structure.

In addition, when nh1= 8 and nh2= 10, the neural network of Fold 3 has the highest R2, therefore it is selected for the subsequent result analysis.To provide more information about the precision of the neural network, the Root Mean Squared Error (RMSE) on the validation set is calculated,and the RMSE values are 0.01%, 0.01%, 0.08%, 0.01%,0.01%, and 0.05% at Conditions A - F, respectively.This result further shows that the prediction accuracy of the neural network can meet the needs of this study.

3.4.Shapley additive explanations

To extract the critical geometric variables from a large number of variables and provide guidance for compressor tolerance design and control, sensitivity analysis is needed, as the geometric uncertainty quantification problem in compressors has the characteristics of high dimensionality and strong nonlinearity.In general, the sensitivity of a geometric variable is not uniform at different points in the uncertainty space.Moreover,the impact of one geometric variable on performance can be significantly affected by other geometric variables15.

To provide accurate sensitivity analysis results and effective guidance for tolerance control, both local and global sensitivity indices are needed.Additionally, for each sensitivity index,both univariate and multivariable effects should be taken into consideration accurately.Facing such demands, traditional sensitivity analysis methods, such as using the coefficients of linear/polynomial regression as a sensitivity index,cannot provide comprehensive results which satisfy the above requirements.

Therefore, in this study, an interpretable machine learning method named SHAP is introduced.The principle and mathematical derivation of the SHAP method are as follows:

(1) Local interpretable model-agnostic explanations.

There is a tradeoff between model complexity and model interpretability.On one hand,some interpretable models,such as linear regression models,decision trees,or rule lists,are simple in structure, but have insufficient prediction accuracy for complex problems.On the other hand, a well-trained neural network can provide sufficient predictions, but it’s difficult to interpret prediction results due to the structural complexity of the model.

Fig.6 Contours of mean(R2) scores with different numbers of hidden layer nodes under Conditions A - F.

That is, the neural network is usually used as a black box.For example, given any combination of geometric uncertainty variables,the neural network obtained in the above section can predict the corresponding performance.However, the contribution of each geometric uncertainty variable to performance uncertainty cannot be obtained directly.Facing such contradiction, a Local Interpretable Modelagnostic Explanations (LIME) method is proposed, and its main idea is using simple local surrogate models to explain the predictions of single instances of complex black box models.32A local surrogate model can be trained to approximate the predictions of an underlying black box model in the neighborhood of an instance to be explained.Mathematically, a local surrogate model with explanatory constraints can be expressed as follows:

(2) SHAP method: local explanation for a single instance.

SHAP13,15is a method used to explain the output of machine learning models based on the game theory.The goal of SHAP is to explain the prediction of a certain instance by calculating the contribution of each feature to the prediction.

The SHAP method combines ideas from the game theory and the LIME method.Based on the game theory,the Shapley value of each input feature can be calculated and used as a measure of its contribution to the output for an instance.Shapley values tell us how to evenly distribute the prediction between features.An important feature of this method is that Shapley values are additive.Therefore, a linear model can be constructed between the output value and the Shapley values of each input features.Based on the idea of the LIME method,this linear model can be used as the local surrogate model g to explain the output of the black box model.

In the SHAP method, the explanation of X is specified as

where EYk(S ) is the prediction for feature values in the subset that are marginalized over features that are not included in S,i.e.,

As can be seen from Eq.(7),the difference between the output prediction for a subset S with and without feature {xi} is used as a measurement of its contribution to the output.Therefore,for all subsets S,the corresponding δi(S )can be calculated.Then Eq.(6), the Shapley value calculation formula,can be perfectly derived by weighting and summing up all possible combinations of feature values.Finally, the obtained Shapley value can be used to measure the contributions of input features to the output.

Since φiis obtained through the calculation of the marginal contribution of variable {xi}, φihas both univariate and multivariable effects.More details about how to extract the multivariable coupling effect from φican be found in Ref.15.

To sum up,by adopting the SHAP method,both local and global sensitivity indices can be calculated accurately.Moreover,for each sensitivity index, both univariate and multivariable effects can be taken into consideration.Therefore, it is chosen in this study.

4.Results and analysis

4.1.Performance distribution analysis

To obtain the performance distribution due to manufacturing geometry uncertainty, all geometric variables are assumed to be gaussian distributions, which are close to the distributions obtained by the real measured data.For each geometric variable,its mean value is the midpoint of the corresponding measured variation range in Table 1, and it is assumed that the measured variation range in Table 1 corresponds to ± 3σ(standard deviation).

In addition, for any two geometric variables, the coefficients R2are calculated based on the measured data of more than 100 samples.Results show that for most groups,the coefficients R2are smaller than 0.5,indicating that the correlations between different geometric variables are not significant.Therefore, for these groups, the variables are assumed to be independent on each other.However, the coefficient R2between ΔGMand ΔGHis 0.52,and the coefficient R2between ΔTMand ΔTHis 0.51.Therefore,for each of these two groups,a multivariate Gaussian distribution is adopted as the joint probability distribution based on the real covariance matrix.Based on the above distribution, 20000 samples are generated based on the LHS method and gaussian distribution in the geometric uncertain space.

For each of the samples, the corresponding performance is predicted by the neural network obtained in Section 3.3.The distributions of normalized efficiencies ηNof all samples are shown as the red distribution histograms in Fig.7.

It can be indicated from Fig.7 that:

(1) Under most of the conditions, the efficiencies of most samples are lower than the design value, and the efficiency decrease amplitude is greater than or equal to the efficiency increase amplitude, indicating that uncertainty will lead to the decline of average performance.

(2) At 100% design rotational speed, the efficiency deviation range is small under the near stall and peak efficiency conditions caused by manufacturing geometric uncertainty,with a maximum efficiency decline amplitude of -0.3% and -0.4%,respectively, while under the near choke condition, the efficiency is highly sensitive to flow capacity changes caused by geometric uncertainty, leading to a significant increase in the efficiency deviation range, with a maximum decrease of -3.9%.Therefore, to reduce the efficiency deviation range caused by geometric uncertainty, the compressor should be avoided working near the choke condition.

(3)At 70%design rotational speed,the efficiency deviation range is small under the near stall and peak efficiency conditions caused by manufacturing geometric uncertainty, with a maximum efficiency decline amplitude of -0.2%and -0.2%, respectively, while under the near choke condition, the efficiency is highly sensitive to flow capacity changes caused by geometric uncertainty, leading to a significant increase in the efficiency deviation range, with a maximum decrease of -2.5%.Therefore, to reduce the efficiency deviation range caused by geometric uncertainty, the compressor should be avoided working near the choke condition.

(4) A comparison is conducted between the efficiency deviations at 100% design rotational speed with mainly supersonic/transonic flow field and at 70% design rotational speed with mainly subsonic flow fields in the compressor.Results show that despite there exist more complex flow structures such as shock wave in the supersonic/transonic flow field,which do not exist in the subsonic flow fields,there is no significant difference between the efficiency deviation ranges under high and low speeds.

4.2.Sensitivity analysis

Under all the 6 working conditions, for each of the 365 samples,the Shapley value of each geometric variable is calculated and used as a measurement of its contribution to the efficiency deviation.Fig.8 presents the Shapley value information of all samples under each working condition.The bar chart on the left is the SHAP feature importance figure, and the horizontal axis represents the mean|Shapley| value, which is the absolute mean of Shapley values for all samples and can be used to measure the average contribution of the geometrical variables on the total efficiency deviation.In addition, the scatter plot on the right is the Shapley value summary figure for 365 samples.Each point in the scatter plot represents a sample, with the color of the point representing the geometric feature value and the axial position of the point representing the corresponding Shapley value.

As can be seen from the SHAP feature importance plots on the left of the 6 working conditions in Fig.8, under near stall Conditions A and D and peak efficiency Conditions B and E,the blade tip leading-edge radius Rle,Hplays a dominant role in efficiency deviation.In working Conditions A and D,the average contribution rate of Rle,Hto total efficiency deviation can reach 34%and 41%,respectively,while in working Conditions B and E, the average contribution rate of Rle,Hto total efficiency deviation can reach 49% and 54%, respectively.More importantly, in the near choke Conditions C and F, the sensitivity of the blade tip thickness THincreases significantly,which together with the leading-edge radius Rle,Hbecomes the two main factors affecting efficiency deviation.In working Conditions C and F,the sum of the average contribution rates of THand Rle,Hto total efficiency deviation can reach 36%and 25%, respectively.In addition, the influences of the stagger angle and thickness are also important under near stall Conditions A and D and near choke Conditions C and F, but not obvious under peak efficiency Conditions B and E.However,the trailing-edge radius of all spanwise sections has almost zero influence on efficiency under all working conditions.Therefore, among a variety of geometric variables, the tip leadingedge radius and tip thickness should be strictly controlled in terms of manufacturing tolerances.

Fig.7 Distribution of compressor efficiency deviation considering geometric uncertainty under Conditions A - F.

Fig.8 SHAP feature importance (left) and Shapley value summary (right) under Conditions A - F.

According to the Shapley value summary plots on the right of the 6 working conditions in Fig.8, the Shapley value range of geometric features with greater significance is wider than that of insensitive geometric features.In addition, by investigating the relationship between feature values and Shapley values,most geometric features show an obvious monotone effect on efficiency.For example, for the leading-edge radii Rle,H,Rle,M, and Rle,L, the Shapley value decreases with an increase of the radius value in almost all working conditions, which means that when the leading-edge radius increases, the efficiency will be reduced.What’s more, in the choke conditions,efficiency also presents a significant trend of decline with an increase of the stagger angle (GH, GM, GL) or the thickness(TH, TM, TL).For a specific compressor case with unqualified performance,the local Shapley value can be used to guide how to repair and make it qualify.

4.3.Flow mechanism analysis

4.3.1.Comparison of influence mechanisms under different working conditions

To explore the physical mechanism behind the influence of the critical geometric uncertainty variable on efficiency, two samples with a maximum and a minimum value of Rle,Hare added,with other geometric variables kept as the design values.Simulation results of the two samples under different working conditions at 100%and 70%design rotational speeds are shown in Fig.9.It can be indicated that at both 100 % and 70 %design rotational speeds, an increase of Rle,Hwill lead to a decrease of the choke flow rate and efficiencies, and the efficiency reduction amplitude in the near choke condition is significantly higher than those in the peak efficiency and near stall conditions.This result corresponds to the Shapley value of Rle,H under different working conditions in Fig.8.

Based on the CFD results, a flow field analysis is conducted.To explore the mechanisms behind the efficiency decline, the entropy loss coefficient ζsis adopted33, which is defined as

Fig.9 Influence of blade tip leading-edge radius on compressor efficiency with 100% and 70% rotational speeds.

Fig.10 shows the distribution of the efficiency deviation coefficient ΔηMmax{Rle,H}-min {Rle,H} along the streamwise direction.The streamwise locations LEhuband TEhubare the leading-edge and trailing-edge locations at the hub section of the rotor blade, respectively.The black, red, and blue curves in the figure correspond to working Conditions A, B, and C,respectively.Through Fig.10, the streamwise position which is obviously affected by a change of Rle,Hcan be located.It can be seen that in Conditions A and B, the efficiency reduction mainly occurs around the leading edge of the blade, and the position with the largest efficiency deviation is denoted as Section I, while in working Condition C, on one hand,the efficiency decrease occurs around the leading edge of the blade; on the other hand, it occurs near the rear part of the blade.The endpoint of the region with the largest increase in the efficiency deviation coefficient is marked as Section II.

To further investigate the mechanism of efficiency deviation under different working conditions, the detailed flow field at the 0.80 spanwise section is analyzed.Fig.12 presents the contour of the relative Mach number (Ma) at the 0.80 spanwise section.It’s indicated that when Rle,Hincreases, the strength of the detached shock wave near the leading edge is enhanced significantly, resulting in a higher shock wave loss, which corresponds to the increase of the efficiency deviation near the leading edge in Fig.10.

Fig.13 presents the distribution of the static pressure rise coefficient Cpalong the blade surface at the 0.80 spanwise section, and the definition of Cpis

where p01and p1are the average total and static pressure at the compressor inlet, respectively.

In Fig.13, M’is the normalized coordinate of the streamwise position from the blade inlet to outlet at the 0.80 spanwise section.It can be indicated that under Conditions A,B,and C,when Rle,Hchanges, the pressure distribution near the leading edge changes significantly, corresponding to loss changes caused by acceleration and deceleration process changes near the pressure surface.In addition, under Conditions B and C,when Rle,Hincreases,the intensity of the spike near the suction surface increases, leading to an increase of the boundary layer loss.34More importantly, it can be seen from Fig.12 and Fig.13 that for Condition C, when Rle,Hincreases, the shock wave moves downstream, and the Ma after the passage shock wave increases obviously.From the general point of view of transonic rotor design, when the leading-edge radius changes,the area distribution along the streamwise direction changes,and then the flow acceleration and deceleration process in the passage changes, resulting in a change of the shock wave strength and a flow loss.Meanwhile, the wake region near the blade trailing edge increases due to a change of Ma after the shock wave,leading to an increase of the blade profile loss.

Fig.11 Efficiency deviation distributions along spanwise direction at Sections I and II under Conditions A, B, and C.

Fig.12 Relative Mach number contours at 0.80 spanwise section under Conditions A, B, and C.

Fig.13 Cp distribution along blade surface at 0.80 spanwise section under Conditions A, B, and C.

Fig.14 Efficiency deviation distribution along streamwise direction under Conditions B and E.

Fig.15 Efficiency deviation distribution along spanwise direction at Section III under Conditions B and E.

4.3.2.Comparison of influence mechanisms under different rotational speeds tion.The black and red curves in the figure correspond to working Conditions B and E respectively.Through Fig.14, the streamwise position which is obviously affected by a change of Rle,Hcan be located.It can be seen that for Condition B at 100% rotational speed, the efficiency reduction starts from a long distance upstream from the blade leading edge and lasts to a short distance downstream from the leading edge.For Condition E at 70% rotational speed,the efficiency reduction starts relatively downstream and only occurs in a short distance upstream and downstream from the leading edge.The position where the efficiency deviation is almost the largest in both conditions is marked as Section III.

Figs.16(a) and 16(b) present the distributions of the static pressure rise coefficient Cpalong the blade surface at the 0.80 spanwise section under Conditions B and E, respectively.In the figure, M’is the normalized coordinate of the streamwise position from the blade inlet to outlet at the 0.80 spanwise section.It can be indicated that under Conditions B and E, when Rle,Hchanges, the pressure distribution near the leading edge changes significantly, corresponding to loss changes caused by acceleration and deceleration process changes near the pressure surface.Fig.16(c) presents the contour of the relative Mach number (Ma) at the 0.80 spanwise section under Condition E.Results show that different from working Condition B at 100% rotational speed, there is no shock wave near the leading edge under Condition E at 70% rotational speed.It can be seen from Fig.12 that in Condition B, there is a detached shock wave near the leading edge, and the shock wave strength varies significantly with Rle,H, which could explain the difference between the starting points of efficiency reduction under Conditions B and E in Fig.14.

Fig.16 Cp distribution along blade surface and Ma contours at 0.80 spanwise section under Conditions B and E.

5.Conclusions

In this paper,the geometric uncertainty influences at near stall,peak efficiency, and near choke conditions under design speed and low speed are investigated, based on the real manufacturing geometric uncertainty of a compressor rotor blade.Conclusions are shown as follows:

(1) At design speed and low speed, the efficiency deviation ranges caused by manufacturing geometric uncertainties are small under the near stall and peak efficiency conditions.For the case studied in this paper, the maximum efficiency decline amplitude is on the order of -0.4%.Meanwhile, under the near choke condition, the efficiency is highly sensitive to flow capacity changes caused by geometric uncertainty, leading to a significant increase in the efficiency deviation range.For the case studied in this paper, the maximum efficiency decline amplitude is on the order of -3.9%.Therefore, to reduce the efficiency deviation range caused by geometric uncertainty,the compressor should be avoided working near the choke condition.

(2) Under the near stall and peak efficiency conditions, the blade tip leading-edge radius plays a dominant role in efficiency deviation.For the case studied in this paper,the average contribution rate of the tip leading-edge radius to total efficiency deviation can reach 34%-54%.Meanwhile,in the near choke conditions, the sensitivity of the blade tip thickness increases significantly, which together with the tip leadingedge radius becomes the two main factors affecting efficiency deviation.For the case studied in this paper, the sum of the average contribution rates of the tip leading-edge radius and blade tip thickness to total efficiency deviation can reach 25%-36%.Therefore,among a variety of geometric variables,the tip leading-edge radius and tip thickness should be strictly controlled in terms of manufacturing tolerances.

(3) A comparison is conducted between the efficiency deviations at 100% design rotational speed with mainly supersonic/transonic flow field and at 70% design rotational speed with mainly subsonic flow fields in the compressor.Results show that despite there exist more complex flow structures such as shock wave in the supersonic/transonic flow field,which do not exist in the subsonic flow fields,there is no significant difference between the efficiency deviation ranges under high and low speeds.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This research was supported by the National Science and Technology Major Project, China (No.2017-II-0004-0016).