The performance of the turbocharger can be improved by optimizing the structural parameters of the compressor impeller. In this work, the existing centrifugal compressor impeller was taken as the case study, and the line shape of the hub was modified in order to study the influence of the line curvature of the hub on the compressor performance. Furthermore, the Navier–Strokes flow equation and the fourth-order Runge–Kutta method were used with Numeca software in order to solve the steady-state RANS equation, and the RANS equation is the averaging of unsteady N-S equations on time to solve the engineeringly required averaging. The numerical simulations and the performance predictions were made for each case, and then, the experiments were carried out to verify the accuracy of the aerodynamic performance that was obtained from the numerical simulations. The research results showed that within the design value range, the curvature of the impeller hub profile increased when the flow range and surge margin of the compressor impeller increased. Moreover, the pressure ratio, efficiency, and post-compression temperature increased when the relative Mach number at the inlet of the impeller decreased, the shock wave intensity that was generated by the gas in the flow channel decreased, and the velocity loss decreased. The obtained influence of the compressor impeller hub linearity on the compressor performance provides an opportunity to further improve the performance of the centrifugal compressor.

## I. INTRODUCTION

Turbocharger technology is very important for the development of vehicle engines. The impeller, as the primary component of the centrifugal compressor, has a significant influence on the performance of the entire compressor. In engineering practice, the basic aerodynamic and thermodynamic relations are often used in order to determine the overall size and the performance of the impeller. However, the design parameters and geometric details are not strictly required. Therefore, the experience of the designer is very important.

A number of past studies have focused on the position of the splitter blades between the main pages, the starting position of the leading edge, and the inclination angle. Moussavi *et al.*^{1} investigated the influence of the position and angle of the front edge of the splitter on the compressor performance without changing the shape and size of the compressor impeller. Omidi *et al.*^{2} studied the influence of parameters such as the angle of the front and rear edge blades and the starting point of the splitter blade on the performance of the compressor. Xu and Amano^{3} studied the position of the splitter blade of the impeller between the two main blades and explored the influence of the splitter port position on the performance of the compressor stage. Malik *et al.*^{4–6} added multiple splitter blades to the centrifugal compressor in order to improve the pressure ratio and efficiency. They found that compared with conventional centrifugal impellers, when the large splitter was close to the pressure surface of the main blade and when the small splitter was close to the suction of the main blade, it reduced the flow separation strength and the pressure loss. Xiang *et al.*^{7} studied the length of the splitter blades corresponding to the optimal number of the blades in the compressor impeller and found that there is a complex coupling relationship between the number of blades and the length of the splitter blades.

Many researchers have conducted a lot of research studies on the geometric parameters and the performance of the compressor impeller, such as the blade shape, backbend angle, and meridian surface. Swain and Engeda^{8,9} pointed out that changing the channel along the meridian area can trim the flow while maintaining the original pressure ratio, and changing the height of the blades at the exit of the impeller can trim the pressure ratio while maintaining the original flow rate. Ahmadabadi *et al.*^{10} solved Euler’s equation on the meridian surface of the impeller and used the ball-spine algorithm in order to iteratively correct the unknown boundary in the numerical domain until the target pressure distribution in the flow channel was satisfied. Wang *et al.*^{11} obtained a centrifugal compressor flow field, which was composed of the parameters such as the pressure ratio, isentropic efficiency, and volume flow rate, and then analyzed the cause of the total pressure loss. Xin *et al.*^{12} used variable guide vanes at the inlet of the centrifugal compressor and added double grooves at the trailing edge of the guide vane. Soghe *et al.*^{13} found that the geometric shape of the recessed tip impeller had a positive effect at a large flow rate, and its performance was reduced under partial loading conditions. Mojaddam *et al.*^{14,15} found that compared with the common arc curve, the elliptic curve was a better initial choice as it can reduce the secondary flow and the related losses. Xie *et al.*^{16} studied the influence of the meridian shape of the steam centrifugal compressor on the aerodynamic performance and analyzed the influence of the combination of a straight line section, a curved section, and the hub line shape that was connected by two straight-line sections on the performance of the entire compressor. Khoshkalam *et al.*^{17} found that by increasing the height of the impeller outlet blade, the impeller inlet blade angle, and the diffuser outlet diameter or reducing the impeller tip diameter at the impeller inlet and outlet blade angle, a higher pressure ratio was obtained. These changes might result in a lower surge margin or a narrower working range. Li *et al.*^{18} studied the effect of pre-swirl on the mass flow function, inlet diameter ratio, and working coefficient and discussed the impeller throat area under the pre-swirl condition and the matching of the pre-swirl impeller and the vane diffuser.

From the literature, it is observed that the existing studies have mostly focused on the impeller splitter blades and the geometric dimensions of the impeller. The exploration of the compressor meridian hub line was based on a simple method of connecting straight lines and curves. According to the previous design experience, it is observed that the curvature of the hub and the shroud decreased at the blade inlet and outlet, and the secondary flow loss decreased. However, the influence of the curvature linearity change is still unknown for the passage of the blade. Moreover, during the turbocharger matching test, it is expected to have more interchangeable resources; therefore, it is possible to match different centrifugal impellers for the same volume compressor volutes. Therefore, this study focuses on the shroud unchanging and the impeller hub linearity changing in order to study their impact on the performance.

This study uses the simulation calculation and the experimental verification methods to obtain the influence of the compressor impeller hub linearity on the compressor performance under the condition that the position of the splitter blade remains unchanged, which may show higher performance during the compressor design process.

## II. CASE STUDY

The research object of Case 1 is a 145 series turbocharger that is matched with a 12V190 engine (660 kW-1350 rpm). The centrifugal compressor is equipped with a vaneless diffuser. There are eight main blades and eight splitter blades in the centrifugal compressor impeller. The inlet and outlet diameters of the impeller are 109.8 and 144.2 mm, respectively. The diameter of the impeller inlet hub, the height of the outlet blade, and the bend angle of the impeller are 32, 8.9 mm, and 21°, respectively. Furthermore, the impeller clearance is 0.5 mm, and its design speed and the flow rate are 70 000 rpm and 0.62 kg/s, respectively.

The shape of the compressor impeller hub was changed based on the previous design experience, and the modified scheme is shown in Fig. 1, where the parameter *M* is the meridian length and the parameter *R* is the distance from the hub to the rotating axis. When compared with the original Case 1, it is observed that the Case 2 scheme had moved the hub line inward. Furthermore, the angle between the hub curve and the axis of rotation is found to increase, and the curvature also increases accordingly. Moreover, the Case 3 scheme had moved the impeller hub outward. Furthermore, when the angle between the hub curve and the axis of rotation decreases, the curvature decreases accordingly. Figure 2 shows a schematic diagram of the blade cross-sectional area. In Fig. 2, *R* represents the diameter of the impeller at any point, and *d* represents the impeller passage width. Figure 3 shows the relationship between the impeller diameter and the channel area of the three schemes.

The various schemes of the compressor impeller were modeled, and their performances were predicted using the three-dimensional numerical simulation. Furthermore, in order to reduce the influence of the boundary conditions on the internal flow, straight pipes were added to the inlet and the outlet of the numerical model.

### A. Numerical scheme

The commercial computational fluid dynamics software Numeca (Version 89.1) was used to discretize the computational domain and solve the steady-state RANS equation. The Reynolds averaging simulation (RANS) is the averaging of unsteady N-S equations on time to solve the engineeringly required averaging. The impeller grid was automatically generated using the grid generator IGG/Autogrid 5. The inlet boundary conditions are the total pressure and the total temperature, and the outlet boundary condition is the average static pressure, that is, the compressor backpressure. Furthermore, the air was selected as the flowing medium using the physical model, and the Spalart–Allmaras turbulence model was selected as the mathematical model. Furthermore, the Navier–Strokes flow equation and the fourth-order Runge–Kutta method were used to solve the flow equations. The central spatial discretization scheme was selected using the spatial discretion method, and the simulation CFL value was designated as 2. The turbocharger speed was set to 70 000 rpm.

## III. EXPERIMENTAL VERIFICATION

In order to verify the accuracy of the three-dimensional simulation of the aerodynamic performance of the compressor, the Case 1 scheme was verified using experiments. The experimental data were obtained on the standard turbocharger automatic test bench of Datong North Tianli Supercharging Technology Co., Ltd.

### A. Experimental setup

The turbocharger performance test bench is mainly composed of a controllable gas source, a gas heating system, a lubricating oil system, and a measurement and data acquisition system, and the experimental setup is shown in Fig. 4. The data collected during the test include the speed *N*, the flow rate *G*_{c} of the turbocharger, the pressures *P*_{C1}, *P*_{C2}, *P*_{T1}, and *P*_{T2} and the temperatures *T*_{C1}, *T*_{C2}, *T*_{T1,} and *T*_{T2} at the inlet and outlet of the compressor and turbine, the inlet pressure *P*_{1} and temperature *T*_{1} of the lubricating oil, and the lubrication oil outlet temperature *T*_{2}.

### B. Measurement parameters

The parameters such as speed, pressure, temperature, and flow are discussed below.

#### 1. Speed measurement

The speed of the turbocharger was measured using a TOP4450 magnetoelectric speed sensor. The meter range of the sensor is 0–200 000 r/min, and the meter accuracy is ±0.2%. The test speed had to be corrected in time with the temperature change during the test, and the corrected speed is the actual test speed *n*_{tb}, which is given as follows:

where *n*_{tb} and *n*_{db} refer to the actual test speed and the equivalent speed, respectively (r/min), and *T*_{b1} refers to the temperature upstream of the compressor (*K*).

#### 2. Pressure measurement

The pressure measurement was carried out in the inlet and outlet pipes of the compressor. The pipe section and shape were consistent with those of the turbocharger inlet and outlet, and the pipe length was not less than five times the pipe diameter. Furthermore, the static pressure was measured using four through holes with a diameter of φ1.2 mm each. They were evenly distributed along the circumference of the pipe and perpendicular to the wall. The inlet and outlet pressures of the compressor and the turbine were measured using the U-tube pressure gauges with an accuracy of 0.5%. The atmospheric pressure was measured using a barometer, and the measurement accuracy was 0.2%. The oil inlet pressure was measured using a spring pressure gauge, and its accuracy was not lower than level 2.

#### 3. Temperature measurement and flow measurement

The temperature at the inlet and outlet of the compressor was measured using a copper-constantan thermocouple thermometer and a platinum resistance thermometer with an accuracy of ±0.2 °C, respectively. The turbine inlet and the outlet temperature were measured using a nickel–chromium–nickel aluminum thermocouple thermometer with an accuracy of ±1 °C. Furthermore, the position of the temperature measuring hole at the inlet and the outlet of the compressor and turbine was behind the pressure measuring point, and the distance from the pressure measuring point was not less than half of the pipe diameter. The thermometer probe was inserted into the pipe at a depth of one-third to half of the pipe diameter, and the pipe was completely sealed during the measurement. Furthermore, the temperature at the inlet and the outlet of the engine oil was measured using a resistance thermometer with a measurement accuracy of ±2 °C.

The flow measurement was measured using a double-wire flow meter with an accuracy of ±1.5%.

### C. Error transfer and data treating

In the compressor performance test, some results are not measured using the instrument directly. Instead, they are measured using additional parameters and then calculated using the calculation formula indirectly. Therefore, the error of the result parameters is determined using the error of the measured value through the error transfer formula.

There is a relationship between the result parameters *y* and the independent parameters $x1,x2,x3,\u2026,xn$, which are directly measured using the following equation:

By conducting a deformation processing to formula (2), the uncertainty formula is obtained as follows:

The conversion mass flow, pressure ratio, and entropy efficiency of the centrifugal compressor are determined as follows:

where $Gc$ refers to the actual flow for the compressor; $PC1*$ and $PC2*$ refer to the inlet and outlet pressures of the compressor, respectively; $TC1*$ and $TC2*$ refer to the inlet and outlet temperatures of the compressor, respectively; and $k$ refers to the specific heat ratio of air ($k$ = 1.4).

Therefore, the uncertainty of the compressor mass flow and efficiency and the pressure ratio are calculated as follows:

The relationship between the system errors of the measurement accuracy level is given as follows:

where $\gamma $ refers to the maximum relative value of the system error, $\beta $ refers to the sensor accuracy level, *L* refers to the instrument full-scale value, and $Lm$ refers to the measurement reading.

Furthermore, by bringing the accuracy level and range value of each instrument into the appeal relationship calculation, it can be deduced whether the choice of instrument accuracy is within the reasonable error.

## IV. RESULTS AND DISCUSSION

The results of the changes in the compressor, flow field analysis at the surge point and the choke point, and the analysis of experimental verification results are discussed below. In numerical calculations, under the same speed line, the calculation had to be performed from choke to surge. Initially, the pressure conditions of the import and export of the compressor were defined, the pressure ratio was very low, and the compressor was in a choke state. Then, the flow and efficiency of the compressor were calculated, and it was observed that the flow rate did not vary as the pressure ratio increased, and the compressor was still in the choke state.

The calculations are repeated continuously until the flow rate varies as the pressure ratio increases. Moreover, the compressor was observed to be at a near-choke point, and therefore, the previous calculation point is considered its choke point.

When the calculations are carried out at near surge conditions, the pressure ratio change was observed to be small. Furthermore, when the pressure ratio reached the maximum value, the compressor might be in a stall state. During this time, the calculation results are already very close to surge. When the calculation results appeared to diverge, it showed that the surge occurred.

### A. Changes in the compressor performance curve

Figure 5 shows the effect of the different hub profiles on the compressor performance at a speed of 70 000 rpm. As a single centrifugal compressor was considered in this study, the kinetic energy at the outlet of the compressor was lost. Therefore, the pressure ratio and the efficiency obtained in this study are the total-to-static types. In Fig. 5, *π* represents the total-to-static pressure ratio and *η* represents the total-to-static efficiency.

From Fig. 5, it is observed that the compressor flow range is affected by the impeller hub profile. Furthermore, the flow range of the Case 2 scheme is found to be larger than that of the Case 1 scheme. Moreover, at the same time, the surge margin of the Case 2 scheme is also found to be greater than that of the Case 1 scheme, and the flow range and surge margin of the Case 3 scheme are the smallest. Furthermore, it is observed that the compressor pressure ratio and efficiency are also affected by the impeller hub profile. Therefore, it is concluded that the Case 2 scheme has the highest pressure ratio. At a flow rate of 0.656 kg/s, the pressure is found to be 2.89% higher than that of the Case 1 scheme. However, in the blocking conditions, the pressure ratio in Case 2 is found to be lower than that in the Case 1 scheme, and the Case 3 pressure ratio is found to be the lowest when the flow rate is 0.486 kg/s. Furthermore, the pressure is observed to be 4.38% lower than that in the Case 1 scheme. Case 2 has the lowest efficiency, and Case 1 and Case 3 schemes are similar under the small flow conditions. The efficiency of Case 1 is found to be in the range of 7.9%–12.8% higher than that of Case 3 under the high-flow conditions. The high-efficiency area of Case 1 and Case 2 is found near the large-flow condition, which is in the blocked area. Furthermore, the high-efficiency area of Case 3 is found in the center of its flow range. The pressure ratio and the efficiency curves of the three schemes show that the pressure ratio and the efficiency of the compressor decrease rapidly under the large-flow conditions. Figures 5(c) and 5(d) show the absolute total temperature ratio and the relative Mach number at the impeller inlet with the flow rate, respectively. It is observed that the Case 2 scheme has the highest absolute total temperature ratio. Furthermore, it is observed that when the flow rate is 0.45 kg/s, the absolute total temperature of the Case 2 scheme is 6.96% more than that of the Case 1 scheme. Moreover, the Case 3 scheme has the lowest absolute total temperature ratio. At a flow rate of 0.49 kg/s, the absolute total temperature of the Case 3 scheme is found to be 3.3% lower than that of the Case 1 scheme. Furthermore, it is observed that the Case 2 scheme has the highest temperature at the compressor outlet. Case 2 has the lowest relative Mach number at the impeller inlet, and Case 3 has the highest relative Mach number at the impeller inlet. This indicates that Case 3 has the highest gas flow velocity in the impeller. Therefore, it is concluded that when the curvature of the impeller hub profile is less and the cross-sectional area of the channel is narrow, the flow range of the impeller is less, and the prone to surge is more. Furthermore, in this condition, the pressure ratio and efficiency and the temperature after compression decrease, and the relative Mach number at the impeller inlet increases.

### B. Flow field analysis of the surge point

Figure 6 shows the gas flow velocity magnitude of the compressor impeller at 10%, 50%, and 90% of the height of the blade near the surge point when the other boundary conditions in the calculation area were the same at the speed of 70 000 rpm. From Fig. 6, it is observed that the trend of the velocity magnitude change of the three schemes was almost the same. It is also observed that the gas velocity at the top of the blade is greater than the velocity at the root of the blade. This implies that the speed at the height of 90% of the blade is greater than the speed at the height of 10% of the blade. At the bottom and middle of the blade among the Case 1 and Case 2 solutions, the maximum gas velocity is found to appear at the exit of the impeller, which is followed by the inlet of the splitter blade [Figs. 6(a), 6(b), 6(d), and 6(e)]. Moreover, at the top of the blade, the maximum gas velocity is found to appear in the flow path between the long and short blades of the impeller, which is near the position of the long blade suction surface [Figs. 6(g) and 6(h)]. Furthermore, in Case 3, it is observed that the velocity in the flow channel at 90% of the height of the blade is significantly higher than that in Case 1 and Case 2 [Fig. 6(i)]. This could be due to that the cross-sectional area of the channel for Case 3 is the narrowest. It might also be due to that the shock waves occur easily in the following positions: the leading edge of the main blade [position I in Fig. 6(i)], upstream of the channel of the splitter blade [position II in Fig. 6(i)], and the channel position III formed by the suction surface of the main blade and the pressure surface of the splitter blade. Moreover, there might be a strong interference between the shock wave near the top of the blade and the boundary layer at the operating point of Case 3, resulting in increased velocity loss. It can also be seen from this figure that the gas flow rate at the blade exit of the middle and the root of the Case 3 solution is larger than that of the other two solutions. Furthermore, the trend is found to be similar to the other two solutions. The high-speed area of the gas velocity is observed to appear at the impeller outlet position and the suction surface of the long blade near the splitter blade inlet.

Although the average mass flow rate through each channel of the impeller is the same in one impeller rotation period at the stable operating point, due to the instability of each impeller channel, the transient flow rate through each impeller channel is not the same. Moreover, the flow is unsteady in the time domain, which causes the mass flow to be unevenly distributed in the space domain. This uneven mass flow causes more flow losses and reduces the efficiency of the impeller.

Figure 7 shows the loading of the blade of the compressor impeller at 10%, 50%, and 90% of the height of the blade near the surge point when the other boundary conditions in the calculation area are the same at the speed of 70 000 rpm. In Fig. 7, the X-axis is the meridian length of the main blade and splitter blade, where 0 is for the import, 1 is for the exit, *ps* is for the blade pressure surface, and *ss* is for the blade suction surface. From Fig. 7, it is observed that at the inlet of the main blade, at 10% of the blade height, the loading of the blade on the pressure surface of the three schemes is significantly different. The loading of the blade on the pressure side fluctuates at the inlet of the splitter blade, and as it approaches the impeller outlet position, the pressure surface loading decreases rapidly after it is raised. Furthermore, at 50% of the blade height, the Case 2 scheme is observed to have the lowest loading of the blade on the pressure side at the import and the largest loading of the blade at the exit. Moreover, the Case 3 scheme has the largest loading of the blade on the pressure side at the import and the lowest blade loading at the exit. At 90% of the blade height, Case 3 is found to have the lowest loading of the blade on the pressure side at the exit of the impeller. Furthermore, the loading of the blade of Case1 and Case 2 is found to be similar. It is observed that near the surge point, when the curvature of the impeller hub profile increases, the impeller channel area increases, the loading of the blade on the pressure surface at the inlet of the impeller decreases, and the loading of the blade on the suction on the pressure surfaces at the outlet increases.

### C. Flow field analysis of the choke point

Figure 8 shows the magnitude of the gas flow velocity of the compressor impeller at 10%, 50%, and 90% of the height of the blade of the compressor impeller at the near-choking operating point under the conditions of 70 000 rpm.

The other boundary conditions in the calculation area are the same. The velocity of the gas at the top of the blade in the three schemes is found to be greater than that at the root of the blade. In Case 1 and Case 2 solutions, the maximum gas velocity at the bottom and middle of the blade is found to appear at the exit of the impeller and then at the inlet of the splitter blade near the suction surface of the long blade [Figs. 8(a), 8(b), 8(d), and 8(e)]. This means that the position of the splitter blades will also affect the speed distribution. Furthermore, it is observed that the Case 1 and Case 2 schemes have the maximum gas velocity at the outlet of the impeller and the downstream position of the passage at 90% of the height of the blade [Figs. 8(g) and 8(h)], and the gas velocity of the Case 1 scheme is higher than that of the Case 2 scheme. The velocity of Case 3 in the flow channel at 50% and 90% of the height of the blade is significantly higher than that of Case 1 and Case 2. The maximum gas flow rate is found to occur at the upstream of the splitter blade and the channel consisting of the main blade suction surface and the splitter blade pressure surface [Figs. 8(f) and 8(i)], indicating that the shock wave that is generated here is the strongest, leading to its maximum velocity loss, and causing a decrease in the efficiency. Furthermore, it is observed that at the point of near-choking conditions, when the curvature of the impeller hub decreases, the impeller channel area decreases, the shock wave intensity generated by the gas at the inlet of the splitter blade, the velocity loss increases, and the efficiency decreases.

Figure 9 shows the loading of the blade of the compressor impeller at 10%, 50%, and 90% of the height of the blade near the choke point when the other boundary conditions in the calculation area are the same at 70 000 rpm. From Fig. 9, it is observed that at the impeller inlet, the loading of the blade on the pressure side of the Case 2 scheme is the lowest, and the loading of the blade on the pressure side of the Case 3 scheme is the highest. Furthermore, at the outlet of the impeller, the loading of the blade on the pressure side of the Case 2 scheme is also the lowest. Moreover, it is observed that Case 1 and Case 3 have almost the same loading of the blade on the pressure side. Furthermore, it is seen that in the blocking area, when the curvature of the impeller hub profile increases, the impeller channel area increases, and the loading of the blade of the impeller decreases.

### D. Analysis of the experimental verification results

In order to verify the correctness of the results obtained in the numerical analysis, a compressor performance test was performed on the Case 1 turbocharger with a speed of 70 000 rpm. The comparison between the results of the test and the numerical simulation is shown in Fig. 10. From Fig. 10, it is observed that the compressor pressure ratio and the efficiency curve of the numerical simulation are relatively close to the experimental value in both trends and quantity. Furthermore, the error is found to increase at the surge point and the choke point. The compressor inlet flow at the near-surge operating point that was obtained by the test and the calculation is 0.42 and 0.4405 kg/s, respectively, with an error of 4.88%. Furthermore, the compressor inlet flow at the near-choke operating point obtained from the test and the calculation is 0.68 and 0.6987 kg/s, respectively, and the error is 2.75%. Under large flow conditions, the compressor efficiency curve obtained from the test has a quicker downward trend than the calculated value.

As mentioned above, when the compressor approaches the stall operating point and the choke operating point, the factors such as separation, drainage, congestion, and other unstable factors appear in the flow field. Furthermore, there may be interference between the shock wave and the boundary layer, the circumferential unevenness of the flow at the rim, etc. All these complex flow phenomena may affect the accuracy of the calculation, resulting in a difference between the calculated value and the test value of the performance curve when it is close to the surge operating point and the choke operating point. Moreover, on the turbocharger performance test bench, a long pipe is usually installed at the inlet of the compressor in order to measure pressure, temperature, and flow, and the measurement position is different from that of the calculation model. Furthermore, this section of the pipe may cause flow loss. In addition, the ambient temperature during the test is uncontrollable and does not match the initial field settings of the calculation model, which may also lead to an increase in error in the results of the numerical simulation and the testing. Furthermore, there are artificial response errors during the test, and the true limit of surge cannot be accurately captured. Therefore, in this paper, the stall deviation predicted by the test is not discussed. In summary, the experimental data can be used to verify the correctness of the CFD method prediction results.

## V. CONCLUSIONS

In this work, it is found that the linearity of the compressor impeller hub had a vital influence on compressor performance. Furthermore, in this work, the linear curvature of the compressor impeller hub is modified, and using the numerical simulation calculations and the experimental verification of each scheme, the following conclusions are obtained.

From the results, it was observed that when the curvature of the impeller hub increased, the impeller channel area increased, the flow range and surge margin of the impeller increased, and the pressure ratio, efficiency, and post-press temperature increased. Furthermore, the relative Mach number at the impeller inlet decreased. When the curvature of the impeller hub increased, the impeller channel area increased, the shock wave decreased, and the speed loss decreased, and therefore, the efficiency increased. Furthermore, the loading of the blade on the pressure surface also decreased.

Therefore, according to the influence law of the compressor impeller hub linearity on the compressor performance, the compressor impeller can be optimized to obtain better performance.

## ACKNOWLEDGMENTS

This research was funded by the Natural Science Foundation of Liaoning Province (China), Grant No. 2019-MS-160, the Xingliao Yingcai Program of Liaoning Province (Grant No. XLYC1807150), and the Liaoning Province Department of Education Project (Grant No. LJKZ0367). The authors would like to acknowledge the experimental platform provided by Datong North Tianli Supercharging Technology Co., Ltd.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors declare no conflict of interest.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.