*Article* **Simulation of the Influence of Wing Angle Blades on the Performance of Counter-Rotating Axial Fan**

**Guijun Gao 1,2,3,\* , Qingshan You <sup>4</sup> , Ziming Kou 1,2,3, Xin Zhang 1,2,3 and Xinqi Gao <sup>5</sup>**


**Abstract:** We took the mining counter-rotating fan FBD No.8.0 as the research object, used orthogonal test and numerical simulation to study the influence of wing angle blade on fan performance, and simulated and analyzed its aerodynamic noise. The results show that the pressure distribution of the optimal blade angle blade fan on the pressure surface of the secondary blade is stronger than that of the prototype blade, and the maximum pressure at the blade height of 25%, 50%, and 75% is increased by 2.3%, 9.3%, and 8.1%, respectively, than original blade. Compared with the prototype blade; wing angle blades can effectively reduce the generation of shedding vortices at the trailing edge of the blade, and reduce the strength of shedding vortices, so that the entropy production of the optimal wing angle blade fan is 1.55% lower than that of the prototype fan. Compared with the prototype fan, the full pressure and efficiency of the angle blade fan under the rated flow have increased by 7.24% and 1.76%, and the average increase of 11.32% and 3.88%, respectively, under the full flow condition. Compared with the prototype fan, the maximum sound power of the wing blade fan in the first and second blade trailing edge regions is reduced by 0.17% and 1.62%, respectively.

**Keywords:** counter-rotating axial fan; orthogonal experiment; numerical simulation; wing angle blade

### **1. Introduction**

The blade is the main working component of the counter-rotating axial fan for mining, and its shape and structure will directly affect the overall performance of the fan. However, the traditional counter-rotating fan has the problems of low efficiency and high aerodynamic noise. Therefore, studying the shape of the blade has important reference value for improving the performance of the counter-rotating axial flow fan.

Jin Yongping et al. [1] used response surface method and three-dimensional flow field analysis method to optimize the swept parameters of the two-stage blades of the contra-rotating axial flow fan for mines, which increased the fan efficiency by 1.64% and improved the flow of internal fluid. Chen et al. [2,3] perforated the trailing edge of the primary blade and the leading edge of the secondary blade of a small counter-rotating axial fan, and found that the blade perforation reduced the overall noise of the fan by 6–7 dB (A). Wu et al. [4] used numerical calculation software to simulate three types of counter-rotating fans with a primary impeller hub ratio of 0.72 and a secondary impeller hub ratio of 0.72, 0.67, and 0.62. The highest efficiency was observed when the secondary impeller hub ratio was 0.62. Mistry et al. [5] studied the effect of two-stage impeller spacing on the axial flow fan, and pointed out that when the impeller spacing is 0.9 times the chord length, the performance of the fan is optimal.

In recent years, with the development of biomimetic technology, living organisms with excellent flow field characteristics in nature have attracted more and more attention.

**Citation:** Gao, G.; You, Q.; Kou, Z.; Zhang, X.; Gao, X. Simulation of the Influence of Wing Angle Blades on the Performance of Counter-Rotating Axial Fan. *Appl. Sci.* **2022**, *12*, 1968. https://doi.org/10.3390/app 12041968

Academic Editors: Josep Maria Bergadà and Gabriel Bugeda Castelltort

Received: 5 January 2022 Accepted: 1 February 2022 Published: 14 February 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Since the blade is the main working component of the fan, the bionic research is mostly carried out around the blade. Tian et al. [6] improved the NACA4412 airfoil based on their study of the wing structure of swallows, and the lift coefficient and lift-drag ratio of the bionic blade were improved compared to the original blade. Inspired by the non-smooth leading edge of the long-eared owl's wings, Sun et al. [7] designed a bionic blade with such a leading edge and introduced this type of blade into an axial fan. The noise of the fan was significantly reduced to the range of 500–2000 Hz, and the maximum noise reduction was about 2.52%. Liang et al. [8] improved the performance of the fan by applying a sawtooth structure on the edge of the fan blade based on the silent principle of bird flight. Under ideal conditions, the bionic blade reduced the noise by 2.2 dB (A). The noise reduction rate was about 2.5%, and the fan efficiency was increased by 5.3%. Xu et al. [9,10], based on the low-noise feature of the owl's flight, installed a bionic serrated structure on the trailing edge of the SD 2030 airfoil blade, and explored the aerodynamics of the blade at different angles of attack and with different sizes of the serrated structures. The analysis showed that the blade trail expansion speed increases with the increase of the size of the saw tooth structure.

Studies have shown that the use of bionic methods to optimize the blades can effectively improve the aerodynamic performance of the blades, thereby improving the internal flow field of the fan. However, in the past, the optimal design of counter-rotating fans was mostly the application of conventional methods, while the bionic methods were mostly concentrated on single blade or single-stage axial flow fans, and there were few bionic researches on blades of counter-rotating axial flow fans for mining. Therefore, this paper takes the FBD No.8.0 mine counter-rotating fan as the research object. Inspired by the wing angle structure of migratory birds that will improve the external flow field of the wings during the long-term evolution of migratory birds, the wing angle bionic design of the fan blade was carried out. At the same time, orthogonal experiments and numerical calculations were used to simulate the performance and noise of the modified fan, and a static analysis was carried out. The results were compared with the performance of the prototype fan, providing a design idea and data processing method for the optimal design of similar fans.

### **2. Numerical Calculation Model and Calculation Method**

### *2.1. Numerical Calculation Model*

This article takes a FBD No.8.0 mine counter-rotating fan as the research object. When modeling the wind turbine, in order to facilitate the numerical calculation, its internal structure was appropriately simplified. The final wind turbine model is shown in Figure 1. The entire fan model is composed of six parts: primary and secondary impellers, collector, deflectors and air ducts. The specific parameters of each part are shown in Table 1.

The wind turbine model was imported into ICEM CFD, and the two-stage impeller and the air duct area were meshed separately using a more adaptable unstructured grid. When the impeller was meshed, the flow field in this area was relatively complex and was the main area of the research, so the mesh was encrypted. The mesh size of the blade surface was controlled at 2 mm. Set 5 layers of boundary layer grids on the solid surface of the fan, and the first layer of boundary layer grids was set to 0.05 mm, so that the fan wall grid y<sup>+</sup> = 30. Finally, the two parts were superimposed to form a complete computational domain grid model. The independence of the grid is tested. The results are shown in Table 2. It is found that when the number of cells are 2 million, the efficiency basically remains unchanged, so the number of cells finally selected is 2 million.

**Figure 1.** Three-dimensional model of wind turbine. **Figure 1.** Three-dimensional model of wind turbine.



Mounting Angle of two-stage blade (°) 30 **Table 2.** Grid independence test.


### of the fan, and the first layer of boundary layer grids was set to 0.05 mm, so that the fan wall grid y+ = 30. Finally, the two parts were superimposed to form a complete computa-*2.2. Calculation Method and Solver Settings*

tional domain grid model. The independence of the grid is tested. The results are shown in Table 2. It is found that when the number of cells are 2 million, the efficiency basically Numerical simulation of the wind turbine model was carried out with ANSYS Flunet numerical calculation software. The settings were as follows:


### *2.3. Statics Settings*

During the static analysis of the blade, mesh the hub and blade with an unstructured grid, and ensure that the mesh size on the blade is the same as the mesh size of the first boundary layer of the blade in the fluid calculation, so as to reduce the transmission error in the fluid–solid coupling process. The materials of the hub and blades were steel. Import the blade surface pressure data in the fluid simulation into the ANSYS statics analysis module as the surface pressure load, and apply ±2900 rad/min centrifugal force load and gravity field load to the primary and secondary impellers, respectively.

### *2.4. Experimental Verification*

Since the counter-rotating fan for mining was a press-in fan, the GB/T 1236–2000 Type B device was selected to collect data related to the total pressure Pt and efficiency *η* of the fan, and then the accuracy of the numerical calculation results was experimentally verified.

Figure 2 is a test platform for a counter-rotating axial flow fan. The wind resistance of the fan can be changed by adjusting the distance between the cone-shaped restrictor in Figure 2b and the outlet of the test air tube to achieve the purpose of imposing different loads on the fan, and then it provided conditions for testing the fan under full flow conditions. *Appl. Sci.* **2022**, *12*, x FOR PEER REVIEW 5 of 20

**Figure 2.** Test platform for contra-rotating axial fan. (**a**) Counter-rotating fan and test air duct; (**b**) tapered restrictor. **Figure 2.** Test platform for contra-rotating axial fan. (**a**) Counter-rotating fan and test air duct; (**b**) tapered restrictor.

As shown in Figure 3, the collected results were compared with the numerical calculation results. It can be seen from the figure that the trends of the full pressure *Pt* and efficiency *η* curves of the simulation and experiment were basically the same. Additionally, the average deviations of the total pressure Pt and efficiency *η* of simulation and experiment were 1.75% and 0.60%, respectively; the relative deviations at the rated flow point were about 1.44% and 0.01%, and both were within 5%. This showed that the reliability of modeling, meshing and calculation method settings was high, and the numerical calculation results could reflect the actual operation of the wind turbine. As shown in Figure 3, the collected results were compared with the numerical calculation results. It can be seen from the figure that the trends of the full pressure *Pt* and efficiency *η* curves of the simulation and experiment were basically the same. Additionally, the average deviations of the total pressure Pt and efficiency *η* of simulation and experiment were 1.75% and 0.60%, respectively; the relative deviations at the rated flow point were about 1.44% and 0.01%, and both were within 5%. This showed that the reliability of modeling, meshing and calculation method settings was high, and the numerical calculation results could reflect the actual operation of the wind turbine.

The blade is the main working component of the fan, and its function is to convert the mechanical energy of the rotating blade into fluid pressure energy and kinetic energy. Therefore, the shape and structure of the blades play a decisive role in the performance of

As shown in Figure 4 [14], during the flight of large migratory birds, the leading edge of their wings has wing angle structure, which helps improve the flow field near the wing during long-distance flight. Inspired by this, bionic design of two-stage fan blades was carried out by applying only a certain range of wing angle structure on the blade without changing the parameters of the prototype blade, such as airfoil, twist angle α and blade height H, as shown in Figure 5. The wing angle position s was the distance from the deflection of the upper wing angle to the blade root in the direction of the blade height. The

**Figure 3.** Comparison of fan experiment and simulation data.

**3. Wing Angle Blade Model and Orthogonal Test** 

*3.1. Wing Angle Blade Design* 

the fan.

calculation results could reflect the actual operation of the wind turbine.

**Figure 3.** Comparison of fan experiment and simulation data. **Figure 3.** Comparison of fan experiment and simulation data.

### **3. Wing Angle Blade Model and Orthogonal Test**

### **3. Wing Angle Blade Model and Orthogonal Test**  *3.1. Wing Angle Blade Design*

tapered restrictor.

*3.1. Wing Angle Blade Design*  The blade is the main working component of the fan, and its function is to convert the mechanical energy of the rotating blade into fluid pressure energy and kinetic energy. Therefore, the shape and structure of the blades play a decisive role in the performance of The blade is the main working component of the fan, and its function is to convert the mechanical energy of the rotating blade into fluid pressure energy and kinetic energy. Therefore, the shape and structure of the blades play a decisive role in the performance of the fan.

(**a**) (**b**) **Figure 2.** Test platform for contra-rotating axial fan. (**a**) Counter-rotating fan and test air duct; (**b**)

As shown in Figure 3, the collected results were compared with the numerical calculation results. It can be seen from the figure that the trends of the full pressure *Pt* and efficiency *η* curves of the simulation and experiment were basically the same. Additionally, the average deviations of the total pressure Pt and efficiency *η* of simulation and experiment were 1.75% and 0.60%, respectively; the relative deviations at the rated flow point were about 1.44% and 0.01%, and both were within 5%. This showed that the reliability of modeling, meshing and calculation method settings was high, and the numerical

the fan. As shown in Figure 4 [14], during the flight of large migratory birds, the leading edge of their wings has wing angle structure, which helps improve the flow field near the wing during long-distance flight. Inspired by this, bionic design of two-stage fan blades was carried out by applying only a certain range of wing angle structure on the blade without changing the parameters of the prototype blade, such as airfoil, twist angle α and blade height H, as shown in Figure 5. The wing angle position s was the distance from the deflection of the upper wing angle to the blade root in the direction of the blade height. The As shown in Figure 4 [14], during the flight of large migratory birds, the leading edge of their wings has wing angle structure, which helps improve the flow field near the wing during long-distance flight. Inspired by this, bionic design of two-stage fan blades was carried out by applying only a certain range of wing angle structure on the blade without changing the parameters of the prototype blade, such as airfoil, twist angle α and blade height H, as shown in Figure 5. The wing angle position s was the distance from the deflection of the upper wing angle to the blade root in the direction of the blade height. The tip offset distance a was the offset distance of the wing angle blade relative to the prototype blade at the tip of the blade upward from the chord of the blade. *Appl. Sci.* **2022**, *12*, x FOR PEER REVIEW 6 of 20 tip offset distance a was the offset distance of the wing angle blade relative to the prototype blade at the tip of the blade upward from the chord of the blade.

**Figure 4.** Wing angle structure of bird wing. **Figure 4.** Wing angle structure of bird wing.

According to the key dimensions of the wing angle blades and in order to reduce the number of tests, an orthogonal test was designed to select the wing angle position s and tip offset distance a of the first and second stage blades [15]. Among them, factor A was the position s1 of first-stage blade tip, factor B was the tip offset distance a1 of first-stage blade, factor C was the position s2 of second-stage blade tip, and factor D was the tip offset distance a2 of second-stage blade. According to relevant fan studies, the main work area of the blade was the upper middle part of the blade, and the maximum static pressure coefficient was the largest at the pressure front edge at 75% of the blade height [16]. Therefore, the position s of the wing Angle was selected in this area. The tip offset distance was selected from 10 mm to 20 mm to ensure that the blade strength will not be reduced due to excessive structural deformation. Finally, an orthogonal test with four factors and three

(**a**) Prototype blade (**b**) Wing angle blade

levels was determined, as shown in Table 3.

*3.2. Orthogonal Experiment Design* 

**Figure 5.** Blade model.

**Figure 4.** Wing angle structure of bird wing.

(**a**) Prototype blade (**b**) Wing angle blade

#### **Figure 5.** Blade model. **Figure 5.** Blade model.

#### *3.2. Orthogonal Experiment Design 3.2. Orthogonal Experiment Design*

According to the key dimensions of the wing angle blades and in order to reduce the number of tests, an orthogonal test was designed to select the wing angle position s and tip offset distance a of the first and second stage blades [15]. Among them, factor A was the position s1 of first-stage blade tip, factor B was the tip offset distance a1 of first-stage blade, factor C was the position s2 of second-stage blade tip, and factor D was the tip offset distance a2 of second-stage blade. According to relevant fan studies, the main work area of the blade was the upper middle part of the blade, and the maximum static pressure coefficient was the largest at the pressure front edge at 75% of the blade height [16]. Therefore, the position s of the wing Angle was selected in this area. The tip offset distance was selected from 10 mm to 20 mm to ensure that the blade strength will not be reduced due to excessive structural deformation. Finally, an orthogonal test with four factors and three levels was determined, as shown in Table 3. According to the key dimensions of the wing angle blades and in order to reduce the number of tests, an orthogonal test was designed to select the wing angle position s and tip offset distance a of the first and second stage blades [15]. Among them, factor A was the position s1 of first-stage blade tip, factor B was the tip offset distance a1 of first-stage blade, factor C was the position s2 of second-stage blade tip, and factor D was the tip offset distance a2 of second-stage blade. According to relevant fan studies, the main work area of the blade was the upper middle part of the blade, and the maximum static pressure coefficient was the largest at the pressure front edge at 75% of the blade height [16]. Therefore, the position s of the wing Angle was selected in this area. The tip offset distance was selected from 10 mm to 20 mm to ensure that the blade strength will not be reduced due to excessive structural deformation. Finally, an orthogonal test with four factors and three levels was determined, as shown in Table 3.

tip offset distance a was the offset distance of the wing angle blade relative to the proto-

type blade at the tip of the blade upward from the chord of the blade.

**Table 3.** Orthogonal experiment factors level table.


### *3.3. Optimal Wing Angle Blade Fan*

Range analysis was conducted on the efficiency of rated flow points, and the efficiency values of each test were shown in Table 4. According to the principle of orthogonal test, factors with larger range values had greater influence on efficiency. Therefore, the degree of influence of each factor on efficiency was A (s1 of first-stage blade angle), C (s2 of secondstage blade angle), B (a1 of first-stage blade tip offset), and D (a2 of second-stage blade tip offset) in descending order. It could also be concluded that compared with the prototype fan (*η* = 75.18%), the fan efficiency of nine experimental schemes was improved at the rated flow point. The degree of influence of the corresponding factors of the first stage impeller was greater than that of the second stage impeller, which was mainly because in the cyclone machine, the number of blades of the first-stage impeller was more than that of the second-stage impeller, and the flow velocity into the first-stage impeller was lower

than that into the second-stage impeller, and the rotational speed of the two-stage impeller was the same. Therefore, in a unit time, the effect of the first-stage impeller on the flow was stronger than that of the second stage impeller.


**Table 4.** Design flow point efficiency data processing and range analysis.

*ki* in the table represents the average efficiency of test parameters at the level of *i*, and the level corresponding to the maximum *k<sup>i</sup>* is the optimal level of this factor. In conclusion, the optimal level combination of the orthogonal test is A1B1C3D3.

### **4. Analysis of Optimization Results**

In order to further study the influence of wing angle blades on the internal flow field of the fan, the prototype fan was compared with the optimal wing angle blade fan in orthogonal test. For the convenience of the analysis, the orthogonal experiment optimal vane fan is referred to as wing blade fan in the following.

### *4.1. Total Pressure Distribution on Blade Surface*

The pressure distribution on the blade surface can effectively reflect the power capacity of the fan blade. Figure 6 shows the total pressure distribution curves of the first and second blades of the wing angle blade fan and prototype fan at the rated flow point. The prototype blades and wing angle blades are, respectively, analyzed at 25%, 50% and 75% relative blade heights. The relative position R = x/L is defined, where x is the distance from any point on the blade to the leading edge of the blade [16], and L is the chord length of the blade, mm. To the left of the black dotted line is the leading edge of the blade.

In Figure 6a, as the relative blade height increases, the pressure of the pressure surface also increases, which also reflects that the upper half of the blade is the main working area. The total pressure distribution of the wing angle blade and the prototype blade on the suction surface and pressure surface is roughly the same. Additionally, the maximum pressure is located at the front edge of the blade. At a relative blade height of 75%, the maximum pressure of the angle blade is 6082.19 Pa, which is an increase of 22.15% compared to the maximum pressure of the prototype blade of 4979.35 Pa.

**4. Analysis of Optimization Results** 

fan is referred to as wing blade fan in the following.

*4.1. Total Pressure Distribution on Blade Surface* 

In order to further study the influence of wing angle blades on the internal flow field of the fan, the prototype fan was compared with the optimal wing angle blade fan in orthogonal test. For the convenience of the analysis, the orthogonal experiment optimal vane

The pressure distribution on the blade surface can effectively reflect the power capacity of the fan blade. Figure 6 shows the total pressure distribution curves of the first and second blades of the wing angle blade fan and prototype fan at the rated flow point. The prototype blades and wing angle blades are, respectively, analyzed at 25%, 50% and 75% relative blade heights. The relative position R = x/L is defined, where x is the distance from any point on the blade to the leading edge of the blade [16], and L is the chord length of the blade, mm. To the left of the black dotted line is the leading edge of the blade.

From Figure 6b, it can be seen that the total pressure of the secondary stage impeller blade at the pressure surface of the blade is stronger than that of the prototype blade at the same blade height, which indicates that the functional force of the blade angle blade is better than that of the prototype blade, and the maximum total pressure is also located at the pressure surface of the leading edge of the blade. The maximum pressures at 25%, 50%, and 75% of the blade height are 10,289.05 Pa, 12,575.33 Pa, 11,613.50 Pa, respectively, which are 2.3%, 9.3% and 8.1% higher compared to the prototype blade at the same height.

### *4.2. Q Isosurface Analysis*

Due to the relatively complex flow field in the fan wheel area, there are a large number of vortices near the blades, and the vortices will reduce the efficiency of the fan and increase the noise. Therefore, this paper uses the *Q* isosurface method to discriminate the flow field in the two-stage impeller area of the fan, and observe the distribution of the vortex core position.

The research on vortices is in the process of continuous exploration, and a variety of vortex identification technologies have been developed. At present, the *Q* criterion and the Lambda2 criterion are commonly used. Indeed, lambda2 is a very powerful tool in the comparison between the different configurations in terms of vortical structures as done, e.g., in Mariotti et al. [17], Alavi Moghadam et al. [18], and Rocchio et al. [19]. I think *Q* criterion is more suitable for the analysis of fan impeller flow field in this paper.

The definition of *Q* isosurface is:

$$Q = \frac{1}{2} (\Omega\_{\text{ij}} \Omega\_{\text{ij}} - \sigma\_{\text{ij}} \sigma\_{\text{ij}}) \tag{1}$$

Ω*ij* represents the vorticity tensor; *σij* represents the strain rate tensor, and the expressions are as follows:

$$
\Omega\_{ij} = \frac{1}{2} (\frac{\partial u\_i}{\partial \mathbf{x}\_j} - \frac{\partial u\_j}{\partial \mathbf{x}\_i}) \tag{2}
$$

$$
\sigma\_{\rm ij} = \frac{1}{2} (\frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i}) \tag{3}
$$

The equation that reduces Equation (1) to 3D Cartesian coordinates is:

$$\begin{array}{l} Q = -\frac{1}{2} [\left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial y}\right)^2 + \left(\frac{\partial w}{\partial z}\right)^2] \\\ -\frac{\partial u}{\partial y} \cdot \frac{\partial v}{\partial x} - \frac{\partial u}{\partial z} \cdot \frac{\partial w}{\partial x} - \frac{\partial v}{\partial z} \cdot \frac{\partial w}{\partial y} \end{array} \tag{4}$$

In Formula (4): *u*, *v*, *w* are the velocity components of the velocity v in the *x*, *y*, and *z* directions, m/s. When *Q* > 0, it means that the rotation of the fluid mass is dominant.

Figure 7 shows the *Q* isosurface distribution of the blade angle blade fan and the prototype fan in the first-stage and second-stage impeller regions. The *Q* isosurface of the first-stage and second-stage impellers are about 2.5 <sup>×</sup> <sup>10</sup><sup>5</sup> s <sup>−</sup><sup>2</sup> and 8.5 <sup>×</sup> <sup>10</sup><sup>4</sup> s −2 , respectively. It can be seen from the figure that the vortices near the blade are mainly divided into the tip leakage vortex, the tip separation vortex and the blade shedding vortex. From the comparison between the prototype fan and the blade angle blade fan, it is found that compared with the prototype fan blade, although the blade angle structure has no obvious effect on the improvement of the vortex core distribution of the tip separation vortex and the tip leakage vortex, it improves the core distribution of the vortex at the trailing edge of the blade.

The generation mechanism of the blade shedding vortex is the same as that of the Karman vortex street [20]. The formation of the Karman vortex street is due to the interaction of the inertia and viscosity of the fluid on the back of the cylinder after the fluid contacts the solid surface. A periodic vortex is formed, and the vortex will periodically fall off behind the solid with the flow of the airflow. The size of the period is related to the Reynolds number Re of the fluid and the shape of the solid. The blade shedding vortex will also produce periodic vortices in the area of the trailing edge and pressure surface of the blade. When the frequency of the shedding vortex coincides with the natural frequency of the blade rotation, resonance will occur, causing the blade to vibrate violently, which will lead to damage or failure of the blade. Therefore, in the actual design and use of the fan, the generation of blade shedding vortices should be avoided as much as possible.

blade.

**Figure 7.** *Q* isosurface distribution. (**a**) First-stage blade of prototype fan; (**b**) first-stage blade of wing angle fan; (**c**) second-stage blade of the prototype fan; and (**d**) second-stage blade of wing angle fan. **Figure 7.** *Q* isosurface distribution. (**a**) First-stage blade of prototype fan; (**b**) first-stage blade of wing angle fan; (**c**) second-stage blade of the prototype fan; and (**d**) second-stage blade of wing angle fan.

The generation mechanism of the blade shedding vortex is the same as that of the Karman vortex street [20]. The formation of the Karman vortex street is due to the interaction of the inertia and viscosity of the fluid on the back of the cylinder after the fluid

Reynolds number Re of the fluid and the shape of the solid. The blade shedding vortex will also produce periodic vortices in the area of the trailing edge and pressure surface of the blade. When the frequency of the shedding vortex coincides with the natural frequency of the blade rotation, resonance will occur, causing the blade to vibrate violently, which will lead to damage or failure of the blade. Therefore, in the actual design and use of the fan, the generation of blade shedding vortices should be avoided as much as possi-

Since the *Q* isosurface method can only observe the distribution of the core position of the blade vortex, it cannot obtain the strength of the vortex. For this reason, this paper uses the analysis method provided by Inoue [21] to determine the magnitude of the vortex intensity. The method found through experiments that the vortex intensity is inversely related to the static pressure, that is, the lower the static pressure at the trailing edge of the blade, the higher the vortex intensity. Additionally, the lowest area of its static pressure is the core area of the vortex. In order to obtain the distribution of the blade shedding vortex intensity, take the vicinity of the trailing edge of any blade of the two-stage impeller as the starting point, and set three planes A, B, and C at equal intervals from near to far, as shown in Figure 8, where the interval between two adjacent planes is 0.25 L. By observing the static intensity distribution cloud diagrams of these three planes, the strength of

Figure 9 shows the static pressure distribution of planes A, B, and C in the first and second blade trailing edge regions of the prototype fan and the angle blade fan, respectively. The arrows in the figure indicate the direction of rotation of the impeller, and the dashed lines indicate the trajectory of the blade shedding vortex. The contours are discontinuous in both figures, which is caused by errors in the interface surface during data transmission. It can be seen from the figure that the static pressure intensity distribution

Figure 7 shows the *Q* isosurface distribution of the blade angle blade fan and the prototype fan in the first-stage and second-stage impeller regions. The *Q* isosurface of the first-stage and second-stage impellers are about 2.5 × 105 s−2 and 8.5 × 104 s−2, respectively. It can be seen from the figure that the vortices near the blade are mainly divided into the tip leakage vortex, the tip separation vortex and the blade shedding vortex. From the comparison between the prototype fan and the blade angle blade fan, it is found that compared with the prototype fan blade, although the blade angle structure has no obvious effect on the improvement of the vortex core distribution of the tip separation vortex and the tip leakage vortex, it improves the core distribution of the vortex at the trailing edge of the

the shedding vortex at the trailing edge of the blade can be seen.

**Figure 8.** Distribution of plane A, B, and C positions.

ble.

ble.

Since the *Q* isosurface method can only observe the distribution of the core position of the blade vortex, it cannot obtain the strength of the vortex. For this reason, this paper uses the analysis method provided by Inoue [21] to determine the magnitude of the vortex intensity. The method found through experiments that the vortex intensity is inversely related to the static pressure, that is, the lower the static pressure at the trailing edge of the blade, the higher the vortex intensity. Additionally, the lowest area of its static pressure is the core area of the vortex. In order to obtain the distribution of the blade shedding vortex intensity, take the vicinity of the trailing edge of any blade of the two-stage impeller as the starting point, and set three planes A, B, and C at equal intervals from near to far, as shown in Figure 8, where the interval between two adjacent planes is 0.25 L. By observing the static intensity distribution cloud diagrams of these three planes, the strength of the shedding vortex at the trailing edge of the blade can be seen. intensity. The method found through experiments that the vortex intensity is inversely related to the static pressure, that is, the lower the static pressure at the trailing edge of the blade, the higher the vortex intensity. Additionally, the lowest area of its static pressure is the core area of the vortex. In order to obtain the distribution of the blade shedding vortex intensity, take the vicinity of the trailing edge of any blade of the two-stage impeller as the starting point, and set three planes A, B, and C at equal intervals from near to far, as shown in Figure 8, where the interval between two adjacent planes is 0.25 L. By observing the static intensity distribution cloud diagrams of these three planes, the strength of the shedding vortex at the trailing edge of the blade can be seen.

*Appl. Sci.* **2022**, *12*, x FOR PEER REVIEW 11 of 20

(**d**)

**Figure 7.** *Q* isosurface distribution. (**a**) First-stage blade of prototype fan; (**b**) first-stage blade of wing angle fan; (**c**) second-stage blade of the prototype fan; and (**d**) second-stage blade of wing angle fan.

Karman vortex street [20]. The formation of the Karman vortex street is due to the interaction of the inertia and viscosity of the fluid on the back of the cylinder after the fluid contacts the solid surface. A periodic vortex is formed, and the vortex will periodically fall off behind the solid with the flow of the airflow. The size of the period is related to the Reynolds number Re of the fluid and the shape of the solid. The blade shedding vortex will also produce periodic vortices in the area of the trailing edge and pressure surface of the blade. When the frequency of the shedding vortex coincides with the natural frequency of the blade rotation, resonance will occur, causing the blade to vibrate violently, which will lead to damage or failure of the blade. Therefore, in the actual design and use of the fan, the generation of blade shedding vortices should be avoided as much as possi-

The generation mechanism of the blade shedding vortex is the same as that of the

Since the *Q* isosurface method can only observe the distribution of the core position

of the blade vortex, it cannot obtain the strength of the vortex. For this reason, this paper uses the analysis method provided by Inoue [21] to determine the magnitude of the vortex

**Figure 8.** Distribution of plane A, B, and C positions. **Figure 8.** Distribution of plane A, B, and C positions.

Figure 9 shows the static pressure distribution of planes A, B, and C in the first and second blade trailing edge regions of the prototype fan and the angle blade fan, respectively. The arrows in the figure indicate the direction of rotation of the impeller, and the dashed lines indicate the trajectory of the blade shedding vortex. The contours are discontinuous in both figures, which is caused by errors in the interface surface during data transmission. It can be seen from the figure that the static pressure intensity distribution Figure 9 shows the static pressure distribution of planes A, B, and C in the first and second blade trailing edge regions of the prototype fan and the angle blade fan, respectively. The arrows in the figure indicate the direction of rotation of the impeller, and the dashed lines indicate the trajectory of the blade shedding vortex. The contours are discontinuous in both figures, which is caused by errors in the interface surface during data transmission. It can be seen from the figure that the static pressure intensity distribution of the prototype blade in section A is significantly smaller than that of the angle blade, and the range of the lowest static pressure zone at section C is also larger than that of the angle blade, which indicates both the strength and range of the shedding vortex of the prototype blade are greater than those of the angle blades. This is because the wing angle structure divides the blade into upper and lower parts at the wing angle position, and the leading edge of the upper half is at a certain angle to the incoming flow direction. This causes the airflow passing through the upper half of the blade to flow toward the tip of the blade after reaching the leading edge of the blade during the rotation of the blade. This destroys the conditions for generating the tail shedding vortex, thereby reducing its vortex intensity and range.

### *4.3. Entropy Production Analysis of Fan*

For axial fans, the existence of a vortex structure in the internal flow field will inevitably increase the entropy production of the system, which in turn will cause the internal energy loss of the fan. Entropy generation theory can evaluate the energy dissipation inside the fan, so more and more scholars use entropy generation analysis to study the internal efficiency of the fan [22–24]. As the internal temperature change of the axial flow fan is very small during operation, the entropy production caused by the temperature change is ignored. Therefore, the total entropy production rate S is composed of two parts, namely, the time-averaged entropy production rate *SPRO,D* produced by the turbulent dissipation of the time-averaged flow field, and the pulsating entropy production rate *SPRO,D'* caused by the pulsating velocity. Among them, the pulsating entropy generation rate *SPRO,D'* cannot be directly calculated due to the use of the RANS equation in the numerical calculation, but Kock found that *SPRO,D'* is related to the turbulent dissipation rate ε through verification [25]. Therefore, the final time average entropy production rate *SPRO,D'* the pulsating entropy production rate *SPRO,D'* and the total entropy production formula are shown as Equations (5)–(7), respectively.

$$\begin{array}{l} S\_{PRO,D} = \frac{\mu}{T} \Big\{ 2\left[\left(\frac{\partial\overline{u}}{\partial x}\right)^2 + \left(\frac{\partial\overline{v}}{\partial y}\right)^2 + \left(\frac{\partial\overline{w}}{\partial z}\right) \right] \\ + \left(\frac{\partial\overline{u}}{\partial y} + \frac{\partial\overline{v}}{\partial x}\right)^2 + \left(\frac{\partial\overline{u}}{\partial z} + \frac{\partial\overline{w}}{\partial x}\right)^2 + \left(\frac{\partial\overline{v}}{\partial z} + \frac{\partial\overline{w}}{\partial y}\right)^2 \end{array} \tag{5}$$

$$S\_{\text{PRO},D'} = \frac{\rho \varepsilon}{\overline{T}} \tag{6}$$

$$\dot{\mathbf{S}} = \int\_{V} \mathbf{S}\_{\text{PRO},D} \mathbf{d}V + \int\_{V} \mathbf{S}\_{\text{PRO},D'}dV\tag{7}$$

In the formula above, *u*, *v*, and *w* are the time-average velocity components of velocity in the *x*, *y*, and *z* directions, m/s; *T* is the time-average temperature, K, because the numerical calculation ignores the influence of temperature changes on the flow field, *T* = 300 K; *ρ* is the density, kg/m<sup>3</sup> ; *ε* is the turbulent dissipation rate, the formula is *ε* = 1.5 (0.16U·Re <sup>−</sup> 0.125) 1.5, m−<sup>2</sup> ·s −3 , where U is the average velocity of the target fluid, m/s, Re is the Reynolds number; V is the volume of the control body, m<sup>3</sup> . structure divides the blade into upper and lower parts at the wing angle position, and the leading edge of the upper half is at a certain angle to the incoming flow direction. This causes the airflow passing through the upper half of the blade to flow toward the tip of the blade after reaching the leading edge of the blade during the rotation of the blade. This destroys the conditions for generating the tail shedding vortex, thereby reducing its vortex intensity and range.

**Figure 9.** Static pressure distribution on plane A, B, and C. (**a**) Static pressure distribution at the section of the first-stage impeller; and (**b**) static pressure distribution at the section of the secondstage impeller. **Figure 9.** Static pressure distribution on plane A, B, and C. (**a**) Static pressure distribution at the section of the first-stage impeller; and (**b**) static pressure distribution at the section of the second-stage impeller.

*4.3. Entropy Production Analysis of Fan*  For axial fans, the existence of a vortex structure in the internal flow field will inevitably increase the entropy production of the system, which in turn will cause the internal Figure 10 shows the distribution of the total entropy production rate S in the primary and secondary blades. From the figure, it can be seen that the total entropy production rate S of the secondary leaves is greater than that of the primary leaves, and the entropy production rate of the tip part is the highest. This is because after the fluid enters the

secondary impeller through the pressurization of the primary impeller, its flow velocity must be greater than that of the primary impeller, and the fluid velocity at the tip of the blade is the largest. It can be seen from Formula (4) and Formula (5) that the time-averaged entropy production rate *SPRO,D'* and the pulsating entropy production rate *SPRO,D'* are both positively correlated with the flow velocity, and the flow field at the tip clearance is relatively complicated. Therefore, under the combined effect of the above factors, the entropy generation rate in the tip area of the secondary blade is higher than that in other positions of the fan. *Appl. Sci.* **2022**, *12*, x FOR PEER REVIEW 14 of 20

(**b**)

**Figure 10.** S distribution of total entropy production on the blade surface. (**a**) S distribution of total entropy yield in the first-stage blade; and (**b**) S distribution of total entropy yield in the second-stage blade. **Figure 10.** S distribution of total entropy production on the blade surface. (**a**) S distribution of total entropy yield in the first-stage blade; and (**b**) S distribution of total entropy yield in the secondstage blade.

From Table 5, it can be seen that the time-average entropy production *SPRO,D* of the prototype fan and the angle blade fan is much smaller than the pulsating entropy production *SPRO,D*′ , which also verifies the conclusion in the literature [22]. Entropy production is arranged in the order of two-stage impeller, first-stage impeller, and air duct. The air duct has the least entropy production because it has a current collector and a rectifier, so it has a relatively favorable aerodynamic shape, which causes minimal loss of entropy production. The total entropy production of the two-stage impeller is greater than that of the firststage impeller, which is similar to the results of the previous analysis. It can also be seen from the table that the total entropy production of the angle blade fan is 1.55% lower than that of the prototype fan. From Table 5, it can be seen that the time-average entropy production . *SPRO*,*<sup>D</sup>* of the prototype fan and the angle blade fan is much smaller than the pulsating entropy production . *SPRO*, *<sup>D</sup>*<sup>0</sup> , which also verifies the conclusion in the literature [22]. Entropy production is arranged in the order of two-stage impeller, first-stage impeller, and air duct. The air duct has the least entropy production because it has a current collector and a rectifier, so it has a relatively favorable aerodynamic shape, which causes minimal loss of entropy production. The total entropy production of the two-stage impeller is greater than that of the first-stage impeller, which is similar to the results of the previous analysis. It can also be seen from the table that the total entropy production of the angle blade fan is 1.55% lower than that of the prototype fan.

**Impeller** 

1.2520 32.0645 45.5913 78.9078

1.2122 31.9316 44.5438 77.6876

*SPRO,D* 0.1150 2.2760 4.2264 6.6174 *SPRO,D*′ 1.0972 29.6556 40.3174 71.0702

**Secondary** 

**Impeller Total** 

**Production Wind Tube Primary** 

**Type Entropy** 

*S*

*S*

Prototype fan

Wing angle blade fan

**Table 5.** Entropy production in each area of the angle blade fan and the prototype fan.


**Table 5.** Entropy production in each area of the angle blade fan and the prototype fan.

#### *4.4. Full Flow Field Analysis 4.4. Full Flow Field Analysis*

In order to analyze the full flow field characteristics of the wing blade fan, the performance of the wing blade fan under full flow conditions was simulated through numerical calculation and compared with the prototype fan. Figure 11 shows the analysis of total pressure *Pt* and efficiency *η*. The optimal wing angle blade fan has a rated flow point of 740 m3/min, and the rated flow point of the prototype fan is 730 m3/min. The efficiency of the angle blade fan at the rated flow point is 77.10%. At the rated flow point of the prototype fan, the total pressure and efficiency of the optimal angle blade fan are increased by 7.24% and 1.76%, respectively. Under full flow conditions, the total pressure and efficiency have increased by 11.32% and 3.88% on average. In order to analyze the full flow field characteristics of the wing blade fan, the performance of the wing blade fan under full flow conditions was simulated through numerical calculation and compared with the prototype fan. Figure 11 shows the analysis of total pressure *Pt* and efficiency *η*. The optimal wing angle blade fan has a rated flow point of 740 m3/min, and the rated flow point of the prototype fan is 730 m3/min. The efficiency of the angle blade fan at the rated flow point is 77.10%. At the rated flow point of the prototype fan, the total pressure and efficiency of the optimal angle blade fan are increased by 7.24% and 1.76%, respectively. Under full flow conditions, the total pressure and efficiency have increased by 11.32% and 3.88% on average.

**Figure 11.** Full flow analysis. **Figure 11.** Full flow analysis.

### *4.5. Noise Estimation 4.5. Noise Estimation*

The noise sources of the counter-rotating axial fans for mines are mainly aerodynamic noise and mechanical noise. Aerodynamic noise is generated by the rotation and eddy currents of the airflow inside the fan [26]. As demonstrated above, the wing blade fan can effectively reduce the strength of the blade shedding vortex, so we analyzed the section of the trailing edge 1 L of the two-stage impeller blade here. Figure 12 shows the sound power distribution in the trailing edge area of the first and second stage blades of the wing blade fan and the prototype fan. Compared to the prototype fan, the sound power distribution of the wing blade fan at the trailing edge of both the first and second stage blades has been reduced. The maximum sound power level of the wing blade fan at the trailing edge of the first-stage blade is 50.95 dB, which is 0.17% lower than that of the prototype fan. The maximum sound power level in the trailing edge area of the two-stage blade is 53.32 dB, which is reduced by 1.62% compared to the prototype fan. The noise sources of the counter-rotating axial fans for mines are mainly aerodynamic noise and mechanical noise. Aerodynamic noise is generated by the rotation and eddy currents of the airflow inside the fan [26]. As demonstrated above, the wing blade fan can effectively reduce the strength of the blade shedding vortex, so we analyzed the section of the trailing edge 1 L of the two-stage impeller blade here. Figure 12 shows the sound power distribution in the trailing edge area of the first and second stage blades of the wing blade fan and the prototype fan. Compared to the prototype fan, the sound power distribution of the wing blade fan at the trailing edge of both the first and second stage blades has been reduced. The maximum sound power level of the wing blade fan at the trailing edge of the first-stage blade is 50.95 dB, which is 0.17% lower than that of the prototype fan. The maximum sound power level in the trailing edge area of the two-stage blade is 53.32 dB, which is reduced by 1.62% compared to the prototype fan.

(**b**)

**Figure 12.** Sound power distribution in the trailing edge area of the first and second stage blades. (**a**) Sound power distribution in the trailing edge area of the first-stage blade; and (**b**) sound power distribution in the trailing edge area of the second-stage blade. **Figure 12.** Sound power distribution in the trailing edge area of the first and second stage blades. (**a**) Sound power distribution in the trailing edge area of the first-stage blade; and (**b**) sound power distribution in the trailing edge area of the second-stage blade.

### **5. Static Analysis**

**5. Static Analysis**  To analyze the partial load distribution of the angle blade fan blade, the finite element software was used to analyze the fluid–structure coupling of the impeller area of the wing angle blade fan and the prototype fan. As shown in Figure 13, the highest stress on the pressure surface of the blade is concentrated at the leading edge blade root. Additionally, as the blade height increases, the equivalent stress gradually decreases. At the suction surface of the blade, the highest equivalent stress is concentrated at the lower half of the leading edge of the blade and the root, and gradually decreases toward the tip of the blade. As summarized in Table 6, the maximum equivalent stress of the first-stage angle blade To analyze the partial load distribution of the angle blade fan blade, the finite element software was used to analyze the fluid–structure coupling of the impeller area of the wing angle blade fan and the prototype fan. As shown in Figure 13, the highest stress on the pressure surface of the blade is concentrated at the leading edge blade root. Additionally, as the blade height increases, the equivalent stress gradually decreases. At the suction surface of the blade, the highest equivalent stress is concentrated at the lower half of the leading edge of the blade and the root, and gradually decreases toward the tip of the blade. As summarized in Table 6, the maximum equivalent stress of the first-stage angle blade is reduced by 13.94% compared to the prototype fan. This is mainly because the air flowing through the upper half of the blade is directed by the angle structure to flow in the direction of the tip, resulting in the reduced airflow at the pressure surface of the leading edge of the blade root, thereby reducing the load in this area. Consequently, the maximum equivalent stress of the angle blade relatively is smaller than that of the prototype blade.

is reduced by 13.94% compared to the prototype fan. This is mainly because the air flowing through the upper half of the blade is directed by the angle structure to flow in the direction of the tip, resulting in the reduced airflow at the pressure surface of the leading edge of the blade root, thereby reducing the load in this area. Consequently, the maximum equivalent stress of the angle blade relatively is smaller than that of the prototype blade. Figure 14 shows the deformation of the prototype fan and the angle blade fan at the first and second blades. Deformation of the first and second blades mainly occurs in the upper middle area of the leading edge of the blade, due to the stronger airflow impact on the leading edge than other parts of the blade, and the largest centrifugal force at this location. As summarized in Table 7, the deformation of the wing angle blade is lower than that of the prototype fan.

> **Minimum Equivalent Stress/MPa**

**Prototype Fan Wing Angle Blade Fan** 

**Minimum Equivalent Stress/MPa** 

**Maximum Equivalent Stress/MPa** 

**Maximum Equivalent Stress/MPa** 

First-stage blade 0.0008 61.8577 0.0011 53.2365 Second-stage blade 0.0048 40.0064 0.0046 40.4835

**Table 6.** Equivalent stress of blade.

**Blade Stage** 

(**a**)

(**b**)

(**c**)

**Figure 13.** Comparison of equivalent distribution of wing angle blades and prototype wind turbines. (**a**) Equivalent stress on pressure surface of the first-stage blade; (**b**) equivalent stress on the suction surface of the first-stage blade; (**c**) equivalent stress on pressure surface of the second-stage blade; and (**d**) equivalent stress on the suction surface of the second-stage blade. **Figure 13.** Comparison of equivalent distribution of wing angle blades and prototype wind turbines. (**a**) Equivalent stress on pressure surface of the first-stage blade; (**b**) equivalent stress on the suction surface of the first-stage blade; (**c**) equivalent stress on pressure surface of the second-stage blade; and (**d**) equivalent stress on the suction surface of the second-stage blade.

Figure 14 shows the deformation of the prototype fan and the angle blade fan at the first and second blades. Deformation of the first and second blades mainly occurs in the upper middle area of the leading edge of the blade, due to the stronger airflow impact on the leading edge than other parts of the blade, and the largest centrifugal force at this location. As summarized in Table 7, the deformation of the wing angle blade is lower than

that of the prototype fan.


**Table 6.** Equivalent stress of blade.

(**a**)

(**b**)

**Figure 14.** Deformation distribution of wing angle blades and prototype blades. (**a**) First-stage blade deformation distribution; and (**b**) second-stage blade deformation distribution. **Figure 14.** Deformation distribution of wing angle blades and prototype blades. (**a**) First-stage blade deformation distribution; and (**b**) second-stage blade deformation distribution.

**Table 7.** Maximum deformation of the blade. **Table 7.** Maximum deformation of the blade.


### **6. Conclusions 6. Conclusions**

flow condition.

The following conclusions can be drawn from this study: The following conclusions can be drawn from this study:


pressure and efficiency are increased by 11.32% and 3.88% on average under the full

second blade trailing edge regions of the optimal wing angle blade fan were reduced

**Author Contributions:** Conceptualization, G.G., Z.K., and Q.Y.; methodology, X.Z.; software, X.Z.; validation, G.G., Z.K. and X.Z.; formal analysis, X.Z.; investigation, G.G., X.Z.; resources, Z.K.; data

by 0.17% and 1.62%, respectively, compared with the prototype fan.

pressure and efficiency are increased by 11.32% and 3.88% on average under the full flow condition.

(4) In the orthogonal experiment, the maximum sound power levels of the first and second blade trailing edge regions of the optimal wing angle blade fan were reduced by 0.17% and 1.62%, respectively, compared with the prototype fan.

**Author Contributions:** Conceptualization, G.G., Z.K. and Q.Y.; methodology, X.Z.; software, X.Z.; validation, G.G., Z.K. and X.Z.; formal analysis, X.Z.; investigation, G.G., X.Z.; resources, Z.K.; data curation, G.G., X.Z.; writing—original draft preparation, G.G., X.Z., X.G.; writing—review and editing, G.G., X.G.; visualization, G.G., X.Z.; supervision, Z.K., Q.Y.; project administration, Z.K., Q.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** This studies do not involve humans or animals.

**Informed Consent Statement:** No involving humans.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors gratefully acknowledge the financial support of Nature fund of Chongqing Science and Technology Bureau (No. cstc2020jcyj-msxmX0793) and the Research plan projectof Chongqing Education Committee (No. KJQN202003402).

**Conflicts of Interest:** The authors declare no conflict of interest.

### **References**


**Xiaobin Xu 1,2, Ruoyu Wang 2,3,\*, Xianjun Yu 2,3 , Guangfeng An 2,3 , Ying Qiu <sup>4</sup> and Baojie Liu 2,3**

<sup>1</sup> School of Energy and Power Engineering, Beihang University, Beijing 100083, China


**Abstract:** Three-dimensional blading is an efficient technique in compressor aerodynamic design, and its function mechanism in the cantilevered stator needs to be addressed. This paper focuses on the sweep and dihedral in the cantilevered stator and seeks to expose their effects through detailed flow field analysis. Results show that the forward sweep could alleviate the corner flow separation by preventing the accumulation of the secondary flow toward the corner region, resulting in stronger flow separation at the blade trailing edge; in summary, forward sweep with appropriate parameters could increase static pressure rise by 14.3%. The positive dihedral will carry the endwall flow to the upper-span sections, thereby reducing blade corner separation; hence, as much as 23.5% improvement in static pressure rise could be obtained with the appropriate dihedral. Moreover, the combination of a relatively large sweep height and a moderate sweep angle with a low dihedral height and a moderate sweep angle provides optimum aerodynamic performance; the static pressure rise coefficient sees an increment of 25.5% at the near stall point. An experiment is then performed to further validate the theory, which shows a 2% improvement in efficiency of 3D blading at small mass flow rates. However, the secondary leakage should be given attention at high mass flow coefficients, while the corner separation needs further elimination at small mass flow rates.

**Keywords:** cantilevered stator; 3D blading; leakage flow; secondary flow

### **1. Introduction**

The cantilevered stator is signified by its simple structure and low weight and hence is a promising configuration in the high-performance compressor. However, the leakage flow created by the radial gap at the stator root substantially complicates the endwall flow, necessitating a thorough understanding and advanced design techniques. Dean stated in the 1950s that the cantilevered stator could obtain better aerodynamic performance than the shrouded stator [1]. His conclusion was then verified experimentally by Lakshminarayana and Horlock, who also pointed out the existence of an optimum clearance size [2–4]. Although the leakage flow could, in a way, benefit endwall flow, it will at the same time introduce flow blockage and mixing loss; thus, studies have been conducted to reveal the flow mechanisms in the corner region. For example, Singh and Ginder, Lee et al., and George et al. believe the leakage flow weakens the corner separation by suppressing the endwall flow [5–7], whereas Gbadebo et al. argue that it is the suppression of the horseshoe vortex in the leading edge that causes removal of the corner separation [8]. Dong et al. state that the suppression of corner separation is mainly caused by the mixing of the high-energy leakage flow with the low-energy corner flow [9]. In terms of the clearance size, Lakshminarayana et al. proposed that the optimum choice is when the strengths of the leakage flow and the secondary flow are identical [2,4], whereas Gbadebo et al. revealed that the clearance flow tends to strengthen the corner separation when the

**Citation:** Xu, X.; Wang, R.; Yu, X.; An, G.; Qiu, Y.; Liu, B. Toward the Utilization of 3D Blading in the Cantilevered Stator from Highly Loaded Compressors. *Appl. Sci.* **2023**, *13*, 3335. https://doi.org/10.3390/ app13053335

Academic Editor: Francesca Scargiali

Received: 21 February 2023 Revised: 3 March 2023 Accepted: 3 March 2023 Published: 6 March 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

clearance size is very small [8]. While George et al. proposed an optimum hub clearance of 1% blade height [5], the optimum stator clearance remains in question [9–13]. Tanwar et al. investigated the hub clearance height and found that the interaction of hub leakage and passage vortex leads to mitigation of overall secondary flow adverse effects [14].

Three-dimensional blading can improve the compressor aerodynamic performance through reorganization of the flow field and hence is widely used in axial compressors. In general, 3D blading can be classified as sweep and dihedral. It is well known that the forward sweep of the rotor blade tip can reduce the local inlet Mach number, thereby weakening the shock wave and reducing the loss [7,13,15–20]. The sweep of the blade can also be used to control the corner flow in the subsonic compressor. As for the dihedral, it is recognized that the positive dihedral can construct a radial pressure gradient in the blade passage, thereby weakening the accumulation of the boundary layer at the corner region and inhibiting the occurrence of flow separation. According to the research of Breugelmans et al. and Weingold et al., the positive dihedral at both ends of the blade can reduce the endwall loss but will increase the loss in the midspan areas [21,22]. Sasaki further notes that the beneficial effect of positive dihedral on the near-wall region is mainly related to the dihedral angle, whereas the negative effect at the midspan is determined by the dihedral height [23]. More information on 3D blading can be found in references [24,25].

Although the application of 3D blading is quite common in conventional rotors/stators, there are few reports about its utilization in the cantilevered stator. Lange et al. attributed the beneficial effect of dihedral to the improvement of the rotor flow according to their experimental measurements [26]. Tweedt et al. found that the forward sweep can draw the high-momentum flow to the corner region of the suction surface, thereby suppressing the thickening of the viscous flow [20]. Lu et al. performed a numerical investigation of the effect of the forward sweep in the cantilevered stator, indicating that a reasonable sweep can not only reduce the shock wave at the stator hub but also reduce the loading near the blade leading edge [27]. Gunn and Hall found that the loss of the non-axisymmetric cantilevered stator with undistorted inflow could be 10% lower than conventional stator [28]. From the above analysis, it can be seen the application of 3D blading in the cantilevered stator is prospective in further improving the compressor aerodynamic performance, and the current attempt is limited to individual sweep or dihedral. To further optimize the cantilevered stator, a comprehensive understanding of the 3D blading mechanism is required, and guidelines for the compound sweep and dihedral design are necessary.

The present paper seeks to shed light on the utilization of 3D blading in the cantilevered stator; it is organized as follows: Section 2 gives an introduction of the investigation methods. Section 3 numerically investigates the effect of the sweep, dihedral, and compound sweep and dihedral, through which the flow mechanisms are revealed, and recommendations of the different parameters are provided. The theory is then validated by an experiment, which consists of the redesign of a cantilevered stator and a detailed comparison of the flow field with the datum scheme, as shown in Section 4. The main conclusions are summarized in Section 5.

### **2. Research Object and Investigation Methodology**

The effects of the 3D blading are investigated both numerically and experimentally in the present work. An introduction of the 3D modeling parameters will be given in this section, followed by details of the research methodology.

### *2.1. Geometric Definition of the 3D Blading*

This paper employs the widely used Sweep-Dihedral Coordinates to define the 3D blading parameters [25,29,30]. As shown in Figure 1, the displacement of the blade section in the chordwise direction is called "sweep". Meanwhile, an obtuse angle between the endwall and leading-edge stacking line in the meridional plane is defined as a forward sweep. On the other hand, the displacement of the blade section in the direction perpendicular to the blade chordwise is called "dihedral". Similarly, an obtuse angle between the

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 3 of 25

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 3 of 25

*2.1. Geometric Definition of the 3D Blading* 

*2.1. Geometric Definition of the 3D Blading* 

endwall and blade stacking line in the meridional plane designates a positive dihedral, and vice versa. endwall and blade stacking line in the meridional plane designates a positive dihedral, and vice versa. and vice versa.

This paper employs the widely used Sweep-Dihedral Coordinates to define the 3D blading parameters [25,29,30]. As shown in Figure 1, the displacement of the blade section in the chordwise direction is called "sweep". Meanwhile, an obtuse angle between the endwall and leading-edge stacking line in the meridional plane is defined as a forward sweep. On the other hand, the displacement of the blade section in the direction perpendicular to the blade chordwise is called "dihedral". Similarly, an obtuse angle between the

This paper employs the widely used Sweep-Dihedral Coordinates to define the 3D blading parameters [25,29,30]. As shown in Figure 1, the displacement of the blade section in the chordwise direction is called "sweep". Meanwhile, an obtuse angle between the endwall and leading-edge stacking line in the meridional plane is defined as a forward sweep. On the other hand, the displacement of the blade section in the direction perpendicular to the blade chordwise is called "dihedral". Similarly, an obtuse angle between the endwall and blade stacking line in the meridional plane designates a positive dihedral,

**Figure 1.** Definitions of 3D blade design in the orthogonal coordinates. **Figure 1.** Definitions of 3D blade design in the orthogonal coordinates.

### *2.2. Numerical Simulation Method 2.2. Numerical Simulation Method 2.2. Numerical Simulation Method*

The cantilevered stator to be investigated comes from the aft stage of a highly loaded high-pressure compressor, whose hub clearance is constant at 1% blade height. As shown in Figure 2, The numerical simulation is performed under the stage environment, with the simulation domain containing three blade rows: the IGV, the rotor, and the stator. The domain inlet is 2.0 times the chord length upstream of the IGV, while the domain outlet is 3.5 times the chord length downstream of the stator blade. The structured grid was generated using NUMECA Autogrid5; the main blade region and the clearances adopt the O4H topology and the H-O topology, respectively. Moreover, the grid is clustered at the near-wall region to satisfy the requirements of the turbulent model; y+ of the first grid off wall is about 2.5 in the region close to the transition position, and the number is smaller near the trailing edge. After grid independence analysis with grid density, the total grid number for the rotor and stator blade rows was 1.28 million and 1.45 million, respectively [31,32]. For a 1.5-times finer mesh, the variation of the loss, the static pressure rise coefficient, and the flow angle compared with the selected mesh is less than 0.1%. The cantilevered stator to be investigated comes from the aft stage of a highly loaded high-pressure compressor, whose hub clearance is constant at 1% blade height. As shown in Figure 2, The numerical simulation is performed under the stage environment, with the simulation domain containing three blade rows: the IGV, the rotor, and the stator. The domain inlet is 2.0 times the chord length upstream of the IGV, while the domain outlet is 3.5 times the chord length downstream of the stator blade. The structured grid was generated using NUMECA Autogrid5; the main blade region and the clearances adopt the O4H topology and the H-O topology, respectively. Moreover, the grid is clustered at the near-wall region to satisfy the requirements of the turbulent model; y+ of the first grid off wall is about 2.5 in the region close to the transition position, and the number is smaller near the trailing edge. After grid independence analysis with grid density, the total grid number for the rotor and stator blade rows was 1.28 million and 1.45 million, respectively [31,32]. For a 1.5-times finer mesh, the variation of the loss, the static pressure rise coefficient, and the flow angle compared with the selected mesh is less than 0.1%. The cantilevered stator to be investigated comes from the aft stage of a highly loaded high-pressure compressor, whose hub clearance is constant at 1% blade height. As shown in Figure 2, The numerical simulation is performed under the stage environment, with the simulation domain containing three blade rows: the IGV, the rotor, and the stator. The domain inlet is 2.0 times the chord length upstream of the IGV, while the domain outlet is 3.5 times the chord length downstream of the stator blade. The structured grid was generated using NUMECA Autogrid5; the main blade region and the clearances adopt the O4H topology and the H-O topology, respectively. Moreover, the grid is clustered at the near-wall region to satisfy the requirements of the turbulent model; y+ of the first grid off wall is about 2.5 in the region close to the transition position, and the number is smaller near the trailing edge. After grid independence analysis with grid density, the total grid number for the rotor and stator blade rows was 1.28 million and 1.45 million, respectively [31,32]. For a 1.5-times finer mesh, the variation of the loss, the static pressure rise coefficient, and the flow angle compared with the selected mesh is less than 0.1%.

**Figure 2.** Model configuration and simulation grid. **Figure 2.** Model configuration and simulation grid.

**Figure 2.** Model configuration and simulation grid. This paper uses the commercial software ANSYS CFX 18.0 to explore the effects of the 3D blading techniques; previous studies have found that the two-equation eddy-viscosity models could simulate the complex vortex flows in the low-speed compressor with This paper uses the commercial software ANSYS CFX 18.0 to explore the effects of the 3D blading techniques; previous studies have found that the two-equation eddy-viscosity models could simulate the complex vortex flows in the low-speed compressor with This paper uses the commercial software ANSYS CFX 18.0 to explore the effects of the 3D blading techniques; previous studies have found that the two-equation eddyviscosity models could simulate the complex vortex flows in the low-speed compressor with satisfying accuracy [33,34]. As a low-speed compressor (*Ma* ~ 0.2), the atmospheric condition (101,325 Pa, 288.15 K) was imposed at the domain inlet, where the total pressure was specified using a circumferential averaged radial profile obtained from experimental results. The mass flow rate was given at the outlet. Rotational periodic conditions were applied to the side walls, whereas the solid walls were defined as the adiabatic non-slip walls. The rotating speed of the rotor was 1100 rpm, and the interface between the rotor and the stator was modeled as the mixing plane. As for the turbulent model, the standard k-ω model was chosen, as it can capture more accurate 3D flow details than the standard

k-ε model, while obtaining a better convergence than the SST model [19,35]. A combination of the second-order spatial and temporal numerics are selected for the transport equations. tions. To ensure calculation accuracy, in addition to the default parameters, self-defined

satisfying accuracy [33,34]. As a low-speed compressor (*Ma* ~ 0.2), the atmospheric condition (101,325 Pa, 288.15 K) was imposed at the domain inlet, where the total pressure was specified using a circumferential averaged radial profile obtained from experimental results. The mass flow rate was given at the outlet. Rotational periodic conditions were applied to the side walls, whereas the solid walls were defined as the adiabatic non-slip walls. The rotating speed of the rotor was 1100 rpm, and the interface between the rotor and the stator was modeled as the mixing plane. As for the turbulent model, the standard k-ω model was chosen, as it can capture more accurate 3D flow details than the standard k-ε model, while obtaining a better convergence than the SST model [19,35]. A combination of the second-order spatial and temporal numerics are selected for the transport equa-

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 4 of 25

To ensure calculation accuracy, in addition to the default parameters, self-defined parameters, including the compressor pressure ratio, efficiency, and the inlet/outlet mass flow rate, were monitored during the simulation process. The flow field was considered converged when the mass flow discrepancy at the domain inlet and outlet was smaller than 0.1%. parameters, including the compressor pressure ratio, efficiency, and the inlet/outlet mass flow rate, were monitored during the simulation process. The flow field was considered converged when the mass flow discrepancy at the domain inlet and outlet was smaller than 0.1%.

### *2.3. Experimental Method* To check the effect of the 3D blading technique, a cantilevered stator in the existing

*2.3. Experimental Method* 

To check the effect of the 3D blading technique, a cantilevered stator in the existing test facility was redesigned and measured experimentally to reveal the variations of the flow field. The experiment was conducted in the low-speed large-scale axial compressor (LSLSAC) test facility at Beihang University. As shown in Figure 3a, the LSLSAC, whose hub-to-tip ratio is 0.75, adopts the 1.5 stage configuration, with a row of inlet guide vanes. The rotor and stator blades are nearly radially stacked by the controlled diffusion airfoil (CDA). The rotating speed is 1100 rpm, which is the same as the numerical simulation. At the design point, the stage loading coefficient is approximately 0.46 (based on midspan velocity), whereas the nominal rotor tip clearance and stator hub clearance are 1.5% and 1.0% blade height, respectively. test facility was redesigned and measured experimentally to reveal the variations of the flow field. The experiment was conducted in the low-speed large-scale axial compressor (LSLSAC) test facility at Beihang University. As shown in Figure 3a, the LSLSAC, whose hub-to-tip ratio is 0.75, adopts the 1.5 stage configuration, with a row of inlet guide vanes. The rotor and stator blades are nearly radially stacked by the controlled diffusion airfoil (CDA). The rotating speed is 1100 rpm, which is the same as the numerical simulation. At the design point, the stage loading coefficient is approximately 0.46 (based on midspan velocity), whereas the nominal rotor tip clearance and stator hub clearance are 1.5% and 1.0% blade height, respectively.

**Figure 3.** Schematics of the LSLSAC. (**a**) Geometry configuration; (**b**) measurement locations. **Figure 3.** Schematics of the LSLSAC. (**a**) Geometry configuration; (**b**) measurement locations.

As shown in Figure 3b, five measurement planes are arranged along the axial direction, where multiple static pressure taps are installed on the casing wall. The mass flow coefficient is monitored by the four circumferential static pressure taps at Plane 0, while the static pressure rise of the compressor is measured by circumferential static pressure taps at Plane 1 and Plane 5. The outlet total pressure is measured by the pitot probes at Plane 5. It should be mentioned that the inlet total pressure is the ambient pressure, which is measured by an atmospheric pressure gauge. Moreover, a torque meter is used to measure the input shaft power to the compressor, which is utilized to calculate the efficiency of the compressor. The torque efficiency is calculated as follows: As shown in Figure 3b, five measurement planes are arranged along the axial direction, where multiple static pressure taps are installed on the casing wall. The mass flow coefficient is monitored by the four circumferential static pressure taps at Plane 0, while the static pressure rise of the compressor is measured by circumferential static pressure taps at Plane 1 and Plane 5. The outlet total pressure is measured by the pitot probes at Plane 5. It should be mentioned that the inlet total pressure is the ambient pressure, which is measured by an atmospheric pressure gauge. Moreover, a torque meter is used to measure the input shaft power to the compressor, which is utilized to calculate the efficiency of the compressor. The torque efficiency is calculated as follows:

$$\eta = \frac{30}{\pi} \frac{kR}{k-1} \frac{m\_0 T\_0^\* \left[ \left( p\_5^\* / p\_2^\* \right)^{(k-1)/k} - 1 \right]}{Mn} \tag{1}$$

where *M* denotes the torque, *n* is the rotating speed, and *m*<sup>0</sup> and *T* ∗ 0 are the mass flow rate and the total temperature at the compressor inlet (Plane 0), respectively.

To obtain the 3D velocity and pressure profiles at the stator inlet and outlet, measurement at Planes 3 and 4 was executed using an L-shaped five-hole probe. In the radial direction, the nearest measurement point to the hub and the shroud wall was 2.0% and 2.5% blade height, respectively. Moreover, a novel zonal method was utilized to process the pressure data, through which the measurement angle range was extended to ±60◦ [36]. The pressure was acquired by the Rosemount pressure transducers, whose measurement range and uncertainty were ±6.22 KPa and 0.025% FS, respectively. Error analysis demonstrated

that the measurement uncertainties of the five-hole probe were 0.5◦ for the flow angles, 1% (normalized by the flow dynamic pressure) for the total pressure, 2% (normalized by the flow dynamic pressure) for the static pressure, and 1% for the flow velocity [36]. *3.1. Effects of the Blade Sweep*  In the present work, the forward sweep was employed to control the corner flow. To

**3. Effects of 3D Blading on the Cantilevered Stator** 

ployed to evaluate the various design schemes.

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 5 of 25

and the total temperature at the compressor inlet (Plane 0), respectively.

where *M* denotes the torque, *n* is the rotating speed, and *m*0 and *T*0\* are the mass flow rate

urement at Planes 3 and 4 was executed using an L-shaped five-hole probe. In the radial direction, the nearest measurement point to the hub and the shroud wall was 2.0% and 2.5% blade height, respectively. Moreover, a novel zonal method was utilized to process the pressure data, through which the measurement angle range was extended to ±60° [36]. The pressure was acquired by the Rosemount pressure transducers, whose measurement range and uncertainty were ±6.22 KPa and 0.025% FS, respectively. Error analysis demonstrated that the measurement uncertainties of the five-hole probe were 0.5° for the flow angles, 1% (normalized by the flow dynamic pressure) for the total pressure, 2% (normalized by the flow dynamic pressure) for the static pressure, and 1% for the flow velocity

To obtain the 3D velocity and pressure profiles at the stator inlet and outlet, meas-

In the present study, oil-flow visualization tests were conducted to exhibit the flow

To reveal the effect of 3D blading on the cantilevered stator, a parametric investiga-

tion of the 3D modeling parameters was first conducted. Numerical simulation was em-

patterns in the stator blade passage. The material used to make the skin-friction lines was a mixture of industrial silicone. The running time of each test was between 5 and 10 min.

In the present study, oil-flow visualization tests were conducted to exhibit the flow patterns in the stator blade passage. The material used to make the skin-friction lines was a mixture of industrial silicone. The running time of each test was between 5 and 10 min. determine the sweep height and the sweep angle, the effect of these two parameters are discussed. As specified in Table 1, the sweep height is varied between 30% and 70% of the

### **3. Effects of 3D Blading on the Cantilevered Stator** total blade height, whereas the sweep angle is between 120° and 150°. The modeling

To reveal the effect of 3D blading on the cantilevered stator, a parametric investigation of the 3D modeling parameters was first conducted. Numerical simulation was employed to evaluate the various design schemes. schemes are named by the following rule: "parameter type + index + parameter type + index". For example, scheme "A1B1" designates a sweep starting from 30% blade span

### *3.1. Effects of the Blade Sweep* with a sweep angle of 120°.

[36].

In the present work, the forward sweep was employed to control the corner flow. To determine the sweep height and the sweep angle, the effect of these two parameters are discussed. As specified in Table 1, the sweep height is varied between 30% and 70% of the total blade height, whereas the sweep angle is between 120◦ and 150◦ . The modeling schemes are named by the following rule: "parameter type + index + parameter type + index". For example, scheme "A1B1" designates a sweep starting from 30% blade span with a sweep angle of 120◦ . **Table 1.** Modeling scheme for blade sweep. **Parameter Type 1 2 3**  A: Sweep height 30% span 60% span 70% span

**Table 1.** Modeling scheme for blade sweep. B: Sweep angle 120° 135° 150°


### 3.1.1. Effects of the Sweep Height shown in Figure 4, the design schemes are A1B2, A2B2, and A3B2. A slight forward sweep

The influence of sweep height is first compared using the control variate method. As shown in Figure 4, the design schemes are A1B2, A2B2, and A3B2. A slight forward sweep is also adopted at the tip region to balance the pressure gradient in the radial direction (135◦ , 70% span). Results are compared to the orthogonal/straight blade (Orth.). Note that no dihedral is utilized in this section. is also adopted at the tip region to balance the pressure gradient in the radial direction (135°, 70% span). Results are compared to the orthogonal/straight blade (Orth.). Note that no dihedral is utilized in this section.

**Figure 4.** Radial distribution of blade sweep for the cases with different sweep heights. **Figure 4.** Radial distribution of blade sweep for the cases with different sweep heights.

The pressure rise and loss characteristics of the cantilevered stators are demonstrated in Figure 5. Results show that, except for the near-stall condition of A1B2, the utilization of the forward sweep could always improve the stator aerodynamic performance in comparison to the baseline case. Moreover, the comparison of different schemes indicates that the 60% sweep height (A2B2) outperforms the other designs. At the near-stall condition, the static pressure rise coefficient and the total pressure loss coefficient in A2B2 are increased and decreased by 14.3% and 5.4%, respectively.

creased and decreased by 14.3% and 5.4%, respectively.

creased and decreased by 14.3% and 5.4%, respectively.

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 6 of 25

The pressure rise and loss characteristics of the cantilevered stators are demonstrated in Figure 5. Results show that, except for the near-stall condition of A1B2, the utilization of the forward sweep could always improve the stator aerodynamic performance in comparison to the baseline case. Moreover, the comparison of different schemes indicates that the 60% sweep height (A2B2) outperforms the other designs. At the near-stall condition, the static pressure rise coefficient and the total pressure loss coefficient in A2B2 are in-

The pressure rise and loss characteristics of the cantilevered stators are demonstrated in Figure 5. Results show that, except for the near-stall condition of A1B2, the utilization of the forward sweep could always improve the stator aerodynamic performance in comparison to the baseline case. Moreover, the comparison of different schemes indicates that the 60% sweep height (A2B2) outperforms the other designs. At the near-stall condition, the static pressure rise coefficient and the total pressure loss coefficient in A2B2 are in-

**Figure 5.** Pressure rise and loss characteristics for the stator with different sweep heights. (**a**) Pressure rise and (**b**) loss. **Figure 5.** Pressure rise and loss characteristics for the stator with different sweep heights. (**a**) Pressure rise and (**b**) loss. **Figure 5.** Pressure rise and loss characteristics for the stator with different sweep heights. (**a**) Pressure rise and (**b**) loss.

Figure 6 illustrates the flow field distribution for the cases with different sweep heights. Both the leakage streamlines and the surface streamlines are depicted. Results show that the forward sweep moves the hub leakage flow upstream, thus enhancing the hindrance to the secondary flow and attenuating the accumulation of the low-energy fluid toward the corner region. Consequently, the blockage at the corner region of the suction surface witnesses a remarkable shrink. However, the forward sweep will incur the radial expansion of the suction surface flow separation; hence, the wake is broadened in the upper span areas. Moreover, under the same mass flow ratio, the sweep height exhibits little effect on the leakage flow but will influence the trailing edge separation significantly. At the sweep height of 60% blade span (A2B2), the trailing edge separation tends to be uniform along the radial direction, thus bringing optimum aerodynamic performance. Figure 6 illustrates the flow field distribution for the cases with different sweep heights. Both the leakage streamlines and the surface streamlines are depicted. Results show that the forward sweep moves the hub leakage flow upstream, thus enhancing the hindrance to the secondary flow and attenuating the accumulation of the low-energy fluid toward the corner region. Consequently, the blockage at the corner region of the suction surface witnesses a remarkable shrink. However, the forward sweep will incur the radial expansion of the suction surface flow separation; hence, the wake is broadened in the upper span areas. Moreover, under the same mass flow ratio, the sweep height exhibits little effect on the leakage flow but will influence the trailing edge separation significantly. At the sweep height of 60% blade span (A2B2), the trailing edge separation tends to be uniform along the radial direction, thus bringing optimum aerodynamic performance. Figure 6 illustrates the flow field distribution for the cases with different sweep heights. Both the leakage streamlines and the surface streamlines are depicted. Results show that the forward sweep moves the hub leakage flow upstream, thus enhancing the hindrance to the secondary flow and attenuating the accumulation of the low-energy fluid toward the corner region. Consequently, the blockage at the corner region of the suction surface witnesses a remarkable shrink. However, the forward sweep will incur the radial expansion of the suction surface flow separation; hence, the wake is broadened in the upper span areas. Moreover, under the same mass flow ratio, the sweep height exhibits little effect on the leakage flow but will influence the trailing edge separation significantly. At the sweep height of 60% blade span (A2B2), the trailing edge separation tends to be uniform along the radial direction, thus bringing optimum aerodynamic performance.

**Figure 6.** Flow field distribution for the cases with different sweep heights. **Figure 6.** Flow field distribution for the cases with different sweep heights. **Figure 6.** Flow field distribution for the cases with different sweep heights.

To evaluate the aerodynamic performance of the cantilevered stator quantitatively, the radial distribution of the aerodynamic parameters is given in Figure 7. The variation of the mass flow coefficient suggests that the forward sweep is able to improve the flow capacity in the hub region, whereas the alleviation of corner separation flow brings a reduction in the deviation angle. The effect of the forward sweep is more pronounced at small mass flow ratios (i.e., the conditions with higher loading). By comparing different blading schemes, it can be seen that the larger sweep height adds to the beneficial effect. Nevertheless, when the sweep height is greater than 60% (A3B2), the performance improvement at the

0.0

0.2

0.4

0.6

*Normalized span*

0.8

1.0

corner region by further increasing the sweep height becomes less significant, yet the upper span performance starts to deteriorate; hence, the 60% sweep height is suitable for the present case. provement at the corner region by further increasing the sweep height becomes less significant, yet the upper span performance starts to deteriorate; hence, the 60% sweep height is suitable for the present case. 0.2 *Normalized span*

0.4

0.6

0.8

1.0

To evaluate the aerodynamic performance of the cantilevered stator quantitatively, the radial distribution of the aerodynamic parameters is given in Figure 7. The variation of the mass flow coefficient suggests that the forward sweep is able to improve the flow capacity in the hub region, whereas the alleviation of corner separation flow brings a reduction in the deviation angle. The effect of the forward sweep is more pronounced at small mass flow ratios (i.e., the conditions with higher loading). By comparing different blading schemes, it can be seen that the larger sweep height adds to the beneficial effect. Nevertheless, when the sweep height is greater than 60% (A3B2), the performance im-

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 7 of 25

To evaluate the aerodynamic performance of the cantilevered stator quantitatively,

Orth. A1B2 A2B2 A3B2

o ) σ ( o )

the radial distribution of the aerodynamic parameters is given in Figure 7. The variation of the mass flow coefficient suggests that the forward sweep is able to improve the flow capacity in the hub region, whereas the alleviation of corner separation flow brings a reduction in the deviation angle. The effect of the forward sweep is more pronounced at small mass flow ratios (i.e., the conditions with higher loading). By comparing different blading schemes, it can be seen that the larger sweep height adds to the beneficial effect. Nevertheless, when the sweep height is greater than 60% (A3B2), the performance improvement at the corner region by further increasing the sweep height becomes less significant, yet the upper span performance starts to deteriorate; hence, the 60% sweep height

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 7 of 25

is suitable for the present case.

Orth. A1B2 A2B2 A3B2

**Figure 7.** Radial distribution of aerodynamic performance for the cases with different sweep heights. (**a**) *φ* = 0.52; (**b**) *φ* = 0.46. **Figure 7.** Radial distribution of aerodynamic performance for the cases with different sweep heights. (**a**) *ϕ* = 0.52; (**b**) *ϕ* = 0.46. flow rate first decreases in the areas between 0~20% blade chord and then continues to increase toward the trailing edge. The increase in the mass flow rate of the leakage flow

To reveal the effect of sweep height on the leakage flow, the variation of the mass flow rate of the leakage flow is presented in Figure 8. Results indicate that the leakage flow rate first decreases in the areas between 0~20% blade chord and then continues to increase toward the trailing edge. The increase in the mass flow rate of the leakage flow will enhance the removal of low-energy fluid in the corner region of the blade suction surface, thus confirming the former analysis. Additionally, although increasing the sweep will enhance the 3D blading effect, the leakage characteristics of the A2B2 case and the A3B2 case exhibit similar patterns; hence, further increasing the sweep height will result To reveal the effect of sweep height on the leakage flow, the variation of the mass flow rate of the leakage flow is presented in Figure 8. Results indicate that the leakage flow rate first decreases in the areas between 0~20% blade chord and then continues to increase toward the trailing edge. The increase in the mass flow rate of the leakage flow will enhance the removal of low-energy fluid in the corner region of the blade suction surface, thus confirming the former analysis. Additionally, although increasing the sweep will enhance the 3D blading effect, the leakage characteristics of the A2B2 case and the A3B2 case exhibit similar patterns; hence, further increasing the sweep height will result in less benefit. will enhance the removal of low-energy fluid in the corner region of the blade suction surface, thus confirming the former analysis. Additionally, although increasing the sweep will enhance the 3D blading effect, the leakage characteristics of the A2B2 case and the A3B2 case exhibit similar patterns; hence, further increasing the sweep height will result in less benefit.

**Figure 8.** The comparison of the streamwise leakage mass flow rate under different sweep heights (*φ* = 0.46). **Figure 8.** The comparison of the streamwise leakage mass flow rate under different sweep heights (*ϕ* = 0.46).

### 3.1.2. Effects of the Sweep Angle

To investigate the influence of the sweep angle, the design schemes A2B1, A2B2, and A2B3 are compared in this section, as illustrated in Figure 9. A tip region of each case employs the forward sweep at the 70% span with the same sweep angle as the hub. The orthogonal/straight blade works as the benchmark of comparison.

To investigate the influence of the sweep angle, the design schemes A2B1, A2B2, and

orthogonal/straight blade works as the benchmark of comparison.

To investigate the influence of the sweep angle, the design schemes A2B1, A2B2, and

A2B3 are compared in this section, as illustrated in Figure 9. A tip region of each case

employs the forward sweep at the 70% span with the same sweep angle as the hub. The

(**a**) (**b**)

0.45 0.50 0.55 0.60

ϕ

The flow field distributions for the design schemes with different sweep angles are

given in Figure 11. Similar to the previous conclusions, the forward sweep will weaken

the flow separation in the blade corner region, but at the same time enhance the flow sep-

aration at the midspan. With the increase of the sweep angle, the leakage flow tends to

move toward the pressure surface of the adjacent blade, thus increasing the traveling dis-

tance to the outlet. Consequently, the mixing of the leakage flow with the corner flow is

improved, and the radial dimension of the low-speed area at the blade outlet shrinks. It

should be noted that excessively large sweep angle will lead to a significant increase in

the suction surface flow separation (A2B3), thereby weakening the aerodynamic perfor-

mance gains brought by the forward sweep; thus, the sweep angle should be appropri-

**Figure 9.** Radial distribution of blade sweep for the cases with different sweep angles. **Figure 9.** Radial distribution of blade sweep for the cases with different sweep angles. Figure 10 presents the pressure rise and loss characteristics of the cantilevered sta-

3.1.2. Effects of the Sweep Angle

3.1.2. Effects of the Sweep Angle

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 8 of 25

Figure 10 presents the pressure rise and loss characteristics of the cantilevered stators. It can be seen that the aerodynamic performances of different design schemes share similar trends; except for the near-stall condition, the pressure rise coefficients are en-Figure 10 presents the pressure rise and loss characteristics of the cantilevered stators. It can be seen that the aerodynamic performances of different design schemes share similar trends; except for the near-stall condition, the pressure rise coefficients are enhanced significantly in comparison with the orthogonal blade. Observation of the near-stall condition indicates that the sweep angle should be controlled within a proper range, as a too-large sweep angle (A2B3, 150◦ ) will deteriorate the blade pressure rise coefficient; however, the performance remains better than that of the straight blade. tors. It can be seen that the aerodynamic performances of different design schemes share similar trends; except for the near-stall condition, the pressure rise coefficients are enhanced significantly in comparison with the orthogonal blade. Observation of the nearstall condition indicates that the sweep angle should be controlled within a proper range, as a too-large sweep angle (A2B3, 150°) will deteriorate the blade pressure rise coefficient; however, the performance remains better than that of the straight blade.

**Figure 10.** Pressure rise and loss characteristics for the stator with different sweep angles. (**a**) Pressure rise and (**b**) loss. **Figure 10.** Pressure rise and loss characteristics for the stator with different sweep angles. (**a**) Pressure rise and (**b**) loss.

0.25 0.30 0.35 0.04 0.08 0.12 *Cps* ω The flow field distributions for the design schemes with different sweep angles are given in Figure 11. Similar to the previous conclusions, the forward sweep will weaken the flow separation in the blade corner region, but at the same time enhance the flow separation at the midspan. With the increase of the sweep angle, the leakage flow tends to move toward the pressure surface of the adjacent blade, thus increasing the traveling distance to the outlet. Consequently, the mixing of the leakage flow with the corner flow is improved, and the radial dimension of the low-speed area at the blade outlet shrinks. It should be noted that excessively large sweep angle will lead to a significant increase in the suction surface flow separation (A2B3), thereby weakening the aerodynamic performance gains brought by the forward sweep; thus, the sweep angle should be appropriately selected when at the design stage. The flow field distributions for the design schemes with different sweep angles are given in Figure 11. Similar to the previous conclusions, the forward sweep will weaken the flow separation in the blade corner region, but at the same time enhance the flow separation at the midspan. With the increase of the sweep angle, the leakage flow tends to move toward the pressure surface of the adjacent blade, thus increasing the traveling distance to the outlet. Consequently, the mixing of the leakage flow with the corner flow is improved, and the radial dimension of the low-speed area at the blade outlet shrinks. It should be noted that excessively large sweep angle will lead to a significant increase in the suction surface flow separation (A2B3), thereby weakening the aerodynamic performance gains brought by the forward sweep; thus, the sweep angle should be appropriately selected when at the design stage.

0.45 0.50 0.55 0.60

ϕ

ately selected when at the design stage.

sure rise and (**b**) loss.

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 9 of 25

**Figure 11.** Flow field distribution for the cases with different sweep angles. **Figure 11.** Flow field distribution for the cases with different sweep angles.

The radial distribution of the aerodynamic parameter for the cases with different sweep angles is given in Figure 12. Results of both the *φ* = 0.52 and the *φ* = 0.46 conditions are provided. With the increase of sweep angle, the flow coefficient at the hub region increases, resulting in the improvement of the flow capacity. On the contrary, the flow capacity at the upper span parts is decreased, corresponding to the widening of the blade wake in Figure 11. The influence of the sweep angle on the radial flow of the blade is more significant at small mass flow rates. Considering the influence of the sweep angle on the corner flow and the blade separation flow, a moderate sweep angle (approximately 135° for the present study) is appropriate for the cantilevered stator. The radial distribution of the aerodynamic parameter for the cases with different sweep angles is given in Figure 12. Results of both the *ϕ* = 0.52 and the *ϕ* = 0.46 conditions are provided. With the increase of sweep angle, the flow coefficient at the hub region increases, resulting in the improvement of the flow capacity. On the contrary, the flow capacity at the upper span parts is decreased, corresponding to the widening of the blade wake in Figure 11. The influence of the sweep angle on the radial flow of the blade is more significant at small mass flow rates. Considering the influence of the sweep angle on the corner flow and the blade separation flow, a moderate sweep angle (approximately 135◦ for the present study) is appropriate for the cantilevered stator. The radial distribution of the aerodynamic parameter for the cases with different sweep angles is given in Figure 12. Results of both the *φ* = 0.52 and the *φ* = 0.46 conditions are provided. With the increase of sweep angle, the flow coefficient at the hub region increases, resulting in the improvement of the flow capacity. On the contrary, the flow capacity at the upper span parts is decreased, corresponding to the widening of the blade wake in Figure 11. The influence of the sweep angle on the radial flow of the blade is more significant at small mass flow rates. Considering the influence of the sweep angle on the corner flow and the blade separation flow, a moderate sweep angle (approximately 135° for the present study) is appropriate for the cantilevered stator.

gles. (**a**) *φ* = 0.52; (**b**) *φ* = 0.46.

(**a**) (**b**) **Figure 12.** Radial distribution of aerodynamic performance for the cases with different sweep an-**Figure 12.** Radial distribution of aerodynamic performance for the cases with different sweep angles. (**a**) *φ* = 0.52; (**b**) *φ* = 0.46. **Figure 12.** Radial distribution of aerodynamic performance for the cases with different sweep angles. (**a**) *ϕ* = 0.52; (**b**) *ϕ* = 0.46.

Figure 13 presents the variation of the mass flow rate for the stator leakage flow along the streamwise direction. Increasing the sweep angle tends to reduce the leakage flow at the blade leading edge, yet it will enhance the leakage flow in the other regions. The total mass flow rate of the leakage flow will be increased upon the utilization of the forward Figure 13 presents the variation of the mass flow rate for the stator leakage flow along the streamwise direction. Increasing the sweep angle tends to reduce the leakage flow at the blade leading edge, yet it will enhance the leakage flow in the other regions. The total mass flow rate of the leakage flow will be increased upon the utilization of the forward sweep, thus strengthening the interaction of different corner flow structures. Figure 13 presents the variation of the mass flow rate for the stator leakage flow along the streamwise direction. Increasing the sweep angle tends to reduce the leakage flow at the blade leading edge, yet it will enhance the leakage flow in the other regions. The total mass flow rate of the leakage flow will be increased upon the utilization of the forward sweep, thus strengthening the interaction of different corner flow structures.

sweep, thus strengthening the interaction of different corner flow structures.

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 10 of 25

Orth. A2B1 A2B2 A2B3

**Figure 13.** The comparison of the streamwise leakage mass flow rate under different sweep angles (*φ* = 0.46). **Figure 13.** The comparison of the streamwise leakage mass flow rate under different sweep angles (*ϕ* = 0.46). Hence, the forward sweep could inhibit the transverse flow near the hub endwall

and alleviate the flow separation at the corner region. Increasing the sweep height facili-

Hence, the forward sweep could inhibit the transverse flow near the hub endwall and alleviate the flow separation at the corner region. Increasing the sweep height facilitates a uniform separation along the span without changing the endwall flow signifi-Hence, the forward sweep could inhibit the transverse flow near the hub endwall andalleviate the flow separation at the corner region. Increasing the sweep height facilitates a uniform separation along the span without changing the endwall flow significantly. tates a uniform separation along the span without changing the endwall flow significantly.

#### cantly. *3.2. Effects of the Blade Dihedral 3.2. Effects of the Blade Dihedral*

#### *3.2. Effects of the Blade Dihedral*  3.2.1. Effects of the Dihedral Height 3.2.1. Effects of the Dihedral Height

3.2.1. Effects of the Dihedral Height Except for the forward sweep, the positive dihedral is also adopted in the present study to optimize the stator performance. Therefore, the effects of the dihedral height and dihedral angle need to be clarified. As shown in Table 2, the dihedral height is varied between 20% and 60% of the total blade height, whereas the dihedral angle is between 120° and 150°. The modeling schemes are named following the same rule as that of the sweep (e.g., scheme "C1D1" corresponds to a dihedral starting from 30% blade span with Except for the forward sweep, the positive dihedral is also adopted in the present study to optimize the stator performance. Therefore, the effects of the dihedral height and dihedral angle need to be clarified. As shown in Table 2, the dihedral height is varied between 20% and 60% of the total blade height, whereas the dihedral angle is between 120◦ and 150◦ . The modeling schemes are named following the same rule as that of the sweep (e.g., scheme "C1D1" corresponds to a dihedral starting from 30% blade span with a dihedral angle of 120◦ ). Except for the forward sweep, the positive dihedral is also adopted in the present study to optimize the stator performance. Therefore, the effects of the dihedral height and dihedral angle need to be clarified. As shown in Table 2, the dihedral height is varied between 20% and 60% of the total blade height, whereas the dihedral angle is between 120° and 150°. The modeling schemes are named following the same rule as that of the sweep (e.g., scheme "C1D1" corresponds to a dihedral starting from 30% blade span with a dihedral angle of 120°).

a dihedral angle of 120°). **Table 2.** Modeling scheme for blade dihedral. **Table 2.** Modeling scheme for blade dihedral.


D: Dihedral angle 120° 135° 150°

Likewise, the influence of dihedral height is first compared using the control variate method. As shown in Figure 14, the design schemes are C1D3, C2D3, and C3D3. A slight positive sweep is also adopted at the tip region (150°, 90% span). Simulation results are compared to the orthogonal/straight blade (Orth.); note that no sweep is utilized in this Likewise, the influence of dihedral height is first compared using the control variate method. As shown in Figure 14, the design schemes are C1D3, C2D3, and C3D3. A slight positive sweep is also adopted at the tip region (150◦ , 90% span). Simulation results are compared to the orthogonal/straight blade (Orth.); note that no sweep is utilized in this section. Likewise, the influence of dihedral height is first compared using the control variate method. As shown in Figure 14, the design schemes are C1D3, C2D3, and C3D3. A slight positive sweep is also adopted at the tip region (150°, 90% span). Simulation results are compared to the orthogonal/straight blade (Orth.); note that no sweep is utilized in this section.

0.0 0.2 0.4 0.6 0.8 1.0 *Normalized span* **Figure 14.** Radial distribution of blade dihedral for the cases with different dihedral heights. **Figure 14.** Radial distribution of blade dihedral for the cases with different dihedral heights.

**Figure 14.** Radial distribution of blade dihedral for the cases with different dihedral heights. Figure 15 presents the pressure rise and loss characteristics for the design schemes with different dihedral heights. Compared with the orthogonal blade, the scheme with a small dihedral height (C1D1) could improve the diffusing capacity of the cantilevered Figure 15 presents the pressure rise and loss characteristics for the design schemes with different dihedral heights. Compared with the orthogonal blade, the scheme with a small dihedral height (C1D1) could improve the diffusing capacity of the cantilevered Figure 15 presents the pressure rise and loss characteristics for the design schemes with different dihedral heights. Compared with the orthogonal blade, the scheme with a small dihedral height (C1D1) could improve the diffusing capacity of the cantilevered stator without increasing its total pressure loss, thus improving the aerodynamic performance of the cantilevered stator. However, with the increase of the dihedral height (C2D3 and

C3D3), the blade loss will start to rise, and the pressure rise capacity is reduced remarkably, thus eliminating the advantages of the positive dihedral. and C3D3), the blade loss will start to rise, and the pressure rise capacity is reduced remarkably, thus eliminating the advantages of the positive dihedral. and C3D3), the blade loss will start to rise, and the pressure rise capacity is reduced remarkably, thus eliminating the advantages of the positive dihedral.

stator without increasing its total pressure loss, thus improving the aerodynamic performance of the cantilevered stator. However, with the increase of the dihedral height (C2D3

stator without increasing its total pressure loss, thus improving the aerodynamic performance of the cantilevered stator. However, with the increase of the dihedral height (C2D3

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 11 of 25

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 11 of 25

**Figure 15.** Pressure rise and loss characteristics for the stator with different dihedral heights. (**a**) Pressure rise and (**b**) loss. **Figure 15.** Pressure rise and loss characteristics for the stator with different dihedral heights. (**a**) Pressure rise and (**b**) loss. **Figure 15.** Pressure rise and loss characteristics for the stator with different dihedral heights. (**a**) Pressure rise and (**b**) loss.

To determine the reason why increasing the dihedral height will reduce the stator performance, the flow field distribution was established for the cases with different dihedral heights, as presented in Figure 16. Results at the mass flow coefficient of 0.52 demonstrate that the positive dihedral will not only push the trajectory of the leakage vortex away from the blade suction surface but also promote the radial migration of the lowenergy fluid. In scheme C1D3, the dihedral height is relatively low, and the accumulation of low-energy fluid at the corner region of the blade suction surface is reduced by the circumferential migration of the leakage flow. Therefore, the blockage at the blade outlet is alleviated significantly. With the increase of the dihedral height (C2D3), although the corner separation at the stator hub is weakened effectively, the wake in the lower and middle parts of the blade is elongated and widened remarkably, which is detrimental to the comprehensive aerodynamic performance of the cantilevered stator. Moreover, if the dihedral height is further increased to 60% (C3D3), the leakage flow will undergo an obvious radial migration under the strong blade force. As a result, the separation at the blade trailing edge will be significantly enhanced, and the performance of the cantilever stator will further deteriorate. Note that with the decrease of the mass flow coefficient, the influence of the dihedral amplifies substantially. To determine the reason why increasing the dihedral height will reduce the stator performance, the flow field distribution was established for the cases with different dihedral heights, as presented in Figure 16. Results at the mass flow coefficient of 0.52 demonstrate that the positive dihedral will not only push the trajectory of the leakage vortex away from the blade suction surface but also promote the radial migration of the low-energy fluid. In scheme C1D3, the dihedral height is relatively low, and the accumulation of low-energy fluid at the corner region of the blade suction surface is reduced by the circumferential migration of the leakage flow. Therefore, the blockage at the blade outlet is alleviated significantly. With the increase of the dihedral height (C2D3), although the corner separation at the stator hub is weakened effectively, the wake in the lower and middle parts of the blade is elongated and widened remarkably, which is detrimental to the comprehensive aerodynamic performance of the cantilevered stator. Moreover, if the dihedral height is further increased to 60% (C3D3), the leakage flow will undergo an obvious radial migration under the strong blade force. As a result, the separation at the blade trailing edge will be significantly enhanced, and the performance of the cantilever stator will further deteriorate. Note that with the decrease of the mass flow coefficient, the influence of the dihedral amplifies substantially. To determine the reason why increasing the dihedral height will reduce the stator performance, the flow field distribution was established for the cases with different dihedral heights, as presented in Figure 16. Results at the mass flow coefficient of 0.52 demonstrate that the positive dihedral will not only push the trajectory of the leakage vortex away from the blade suction surface but also promote the radial migration of the lowenergy fluid. In scheme C1D3, the dihedral height is relatively low, and the accumulation of low-energy fluid at the corner region of the blade suction surface is reduced by the circumferential migration of the leakage flow. Therefore, the blockage at the blade outlet is alleviated significantly. With the increase of the dihedral height (C2D3), although the corner separation at the stator hub is weakened effectively, the wake in the lower and middle parts of the blade is elongated and widened remarkably, which is detrimental to the comprehensive aerodynamic performance of the cantilevered stator. Moreover, if the dihedral height is further increased to 60% (C3D3), the leakage flow will undergo an obvious radial migration under the strong blade force. As a result, the separation at the blade trailing edge will be significantly enhanced, and the performance of the cantilever stator will further deteriorate. Note that with the decrease of the mass flow coefficient, the influence of the dihedral amplifies substantially.

**Figure 16.** Flow field distribution for the cases with different dihedral heights. **Figure 16.** Flow field distribution for the cases with different dihedral heights.

The radial distribution of the aerodynamic parameter for the cases with different di-

strict the flow capacity at the blade inlet, while increasing the dihedral height will amplify the effect. On the other hand, the mass flow coefficient at the outlet of the cantilevered stator distributes differently with the variation of the dihedral height. For scheme C1D3, the flow capacity below 20% blade height is increased significantly due to the weakening of the corner separation flow, Meanwhile, the mass flow coefficient in the areas above 20% span witnesses a slight reduction owing to the enhancement of the trailing edge separation, the deviation angle also rises correspondingly. With the increase of the dihedral height (C2D3 and C3D3), the flow capacity above 20% blade height suffers from a significant reduction because of the strengthening of the radial flow migration in the blade channel, thus bringing adverse effects to the aerodynamic performance of the cantilevered

**Figure 17.** Radial distribution of aerodynamic performances for the cases with different dihedral

0.2 0.4 0.6

ϕ

*out i* (

ϕ*in* 0.2 0.4 0.6 0 10 20 0 10 20 30

o ) σ ( o )

Orth. C1D3 C2D3 C3D3

0.0

0.2

0.4

0.6

*Normalized span*

0.8

1.0

Figure 18 presents the variation of the leakage mass flow rate. An interesting phenomenon is that increasing the dihedral height will first increase and then decrease the leakage mass flow rate at the blade leading edge. Consequently, the uniformity of the leakage flow characteristic along the blade chord is first decreased and then increased. Moreover, although the slight positive dihedral (C1D3) could increase the mass flow rate of the leakage flow along the axial direction, increasing the dihedral height will incur a significant reduction of leakage flow at the blade leading edge, thus weakening the effect

 (**a**) (**b**)

heights. (**a**) *φ* = 0.52; (**b**) *φ* = 0.46.

σ ( o )

0.2 0.4 0.6 0 10 20 0 10 20 30

o )

of the 3D blading technique.

stator.

Orth. C1D3 C2D3 C3D3

0.2 0.4 0.6

ϕ

*out i* (

ϕ*in*

0.0

0.2

0.4

0.6

*Normalized span*

0.8

1.0

*φ* = 0.46

The radial distribution of the aerodynamic parameter for the cases with different dihedral heights is given in Figure 17 to evaluate the stator performance quantitively. The increase of incidence angle at the hub region indicates that the positive dihedral will restrict the flow capacity at the blade inlet, while increasing the dihedral height will amplify the effect. On the other hand, the mass flow coefficient at the outlet of the cantilevered stator distributes differently with the variation of the dihedral height. For scheme C1D3, the flow capacity below 20% blade height is increased significantly due to the weakening of the corner separation flow, Meanwhile, the mass flow coefficient in the areas above 20% span witnesses a slight reduction owing to the enhancement of the trailing edge separation, the deviation angle also rises correspondingly. With the increase of the dihedral height (C2D3 and C3D3), the flow capacity above 20% blade height suffers from a significant reduction because of the strengthening of the radial flow migration in the blade channel, thus bringing adverse effects to the aerodynamic performance of the cantilevered stator. hedral heights is given in Figure 17 to evaluate the stator performance quantitively. The increase of incidence angle at the hub region indicates that the positive dihedral will restrict the flow capacity at the blade inlet, while increasing the dihedral height will amplify the effect. On the other hand, the mass flow coefficient at the outlet of the cantilevered stator distributes differently with the variation of the dihedral height. For scheme C1D3, the flow capacity below 20% blade height is increased significantly due to the weakening of the corner separation flow, Meanwhile, the mass flow coefficient in the areas above 20% span witnesses a slight reduction owing to the enhancement of the trailing edge separation, the deviation angle also rises correspondingly. With the increase of the dihedral height (C2D3 and C3D3), the flow capacity above 20% blade height suffers from a significant reduction because of the strengthening of the radial flow migration in the blade channel, thus bringing adverse effects to the aerodynamic performance of the cantilevered stator.

The radial distribution of the aerodynamic parameter for the cases with different di-

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 12 of 25

**Figure 16.** Flow field distribution for the cases with different dihedral heights.

**Figure 17.** Radial distribution of aerodynamic performances for the cases with different dihedral heights. (**a**) *φ* = 0.52; (**b**) *φ* = 0.46. **Figure 17.** Radial distribution of aerodynamic performances for the cases with different dihedral heights. (**a**) *ϕ* = 0.52; (**b**) *ϕ* = 0.46.

Figure 18 presents the variation of the leakage mass flow rate. An interesting phenomenon is that increasing the dihedral height will first increase and then decrease the leakage mass flow rate at the blade leading edge. Consequently, the uniformity of the leakage flow characteristic along the blade chord is first decreased and then increased. Moreover, although the slight positive dihedral (C1D3) could increase the mass flow rate of the leakage flow along the axial direction, increasing the dihedral height will incur a significant reduction of leakage flow at the blade leading edge, thus weakening the effect of the 3D blading technique. Figure 18 presents the variation of the leakage mass flow rate. An interesting phenomenon is that increasing the dihedral height will first increase and then decrease the leakage mass flow rate at the blade leading edge. Consequently, the uniformity of the leakage flow characteristic along the blade chord is first decreased and then increased. Moreover, although the slight positive dihedral (C1D3) could increase the mass flow rate of the leakage flow along the axial direction, increasing the dihedral height will incur a significant reduction of leakage flow at the blade leading edge, thus weakening the effect of the 3D blading technique. *Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 13 of 25

**Figure 18.** Comparison of the streamwise leakage mass flow rate under different dihedral heights (*φ* = 0.46). **Figure 18.** Comparison of the streamwise leakage mass flow rate under different dihedral heights (*ϕ* = 0.46).

height, the stator performances with different dihedral angles are inspected. As illustrated in Figure 19, the cases to be investigated are C1D1, C1D2, and C1D3; note that the blade tip also adopts positive dihedrals (at 80% span with the same dihedral angle as the hub)

**Figure 19.** Radial distribution of blade dihedral for the cases with different dihedral angles.

where the value of *Cps* is improved by 23.5% at the near-stall condition.

Orth. C1D1 C1D2 C1D3

The pressure rise and loss characteristics for the design schemes with different dihedral heights are demonstrated in Figure 20. The influence of the dihedral is more pronounced in terms of pressure rise coefficient and at small mass flow ratios. In comparison to the orthogonal blade, the positive dihedral will always improve the blade static pressure coefficient, whereas the increment will first increase and then decrease with the increase of the dihedral angle. The optimum dihedral angle in the present study is 135°,

(**a**) (**b**)

0.06

0.09

ω

0.12

0.15

0.45 0.50 0.55 0.60

ϕ

**Figure 20.** Pressure rise and loss characteristics for the stator with different dihedral angles. (**a**) Pres-

to balance the pressure gradient in the radial direction.

Orth. C1D1 C1D2 C1D3

0.0 0.2 0.4 0.6 0.8 1.0

*Normalized span*

0.45 0.50 0.55 0.60

ϕ

3.2.2. Effects of the Dihedral Angle

0.0

0.1

*Nomalized dihedral*

sure rise and (**b**) loss.

0.25

0.30

0.35

*Cps*

0.40

### 3.2.2. Effects of the Dihedral Angle height, the stator performances with different dihedral angles are inspected. As illustrated **Figure 18.** Comparison of the streamwise leakage mass flow rate under different dihedral heights (*φ* = 0.46).

3.2.2. Effects of the Dihedral Angle

0.0 0.2 0.4 0.6 0.8 1.0

*Normalized chord*

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 13 of 25

Orth. C1D3 C2D3 C3D3

(*φ* = 0.46).

0.0002

0.0004

0.0006

*m*leak (kg/s)

0.0008

0.0010

0.0002

0.0004

0.0006

*m*leak (kg/s)

0.0008

0.0010

To investigate the influence of the dihedral angle, under the optimum dihedral height, the stator performances with different dihedral angles are inspected. As illustrated in Figure 19, the cases to be investigated are C1D1, C1D2, and C1D3; note that the blade tip also adopts positive dihedrals (at 80% span with the same dihedral angle as the hub) to balance the pressure gradient in the radial direction. in Figure 19, the cases to be investigated are C1D1, C1D2, and C1D3; note that the blade tip also adopts positive dihedrals (at 80% span with the same dihedral angle as the hub) to balance the pressure gradient in the radial direction. 3.2.2. Effects of the Dihedral Angle To investigate the influence of the dihedral angle, under the optimum dihedral height, the stator performances with different dihedral angles are inspected. As illustrated in Figure 19, the cases to be investigated are C1D1, C1D2, and C1D3; note that the blade

0.0 0.2 0.4 0.6 0.8 1.0

*Normalized chord*

**Figure 18.** Comparison of the streamwise leakage mass flow rate under different dihedral heights

To investigate the influence of the dihedral angle, under the optimum dihedral

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 13 of 25

Orth. C1D3 C2D3 C3D3

**Figure 19.** Radial distribution of blade dihedral for the cases with different dihedral angles. **Figure 19.** Radial distribution of blade dihedral for the cases with different dihedral angles.

**Figure 19.** Radial distribution of blade dihedral for the cases with different dihedral angles.

The pressure rise and loss characteristics for the design schemes with different dihedral heights are demonstrated in Figure 20. The influence of the dihedral is more pronounced in terms of pressure rise coefficient and at small mass flow ratios. In comparison to the orthogonal blade, the positive dihedral will always improve the blade static pres-The pressure rise and loss characteristics for the design schemes with different dihedral heights are demonstrated in Figure 20. The influence of the dihedral is more pronounced in terms of pressure rise coefficient and at small mass flow ratios. In comparison to the orthogonal blade, the positive dihedral will always improve the blade static pressure coefficient, whereas the increment will first increase and then decrease with the increase of the dihedral angle. The optimum dihedral angle in the present study is 135◦ , where the value of *Cps* is improved by 23.5% at the near-stall condition. The pressure rise and loss characteristics for the design schemes with different dihedral heights are demonstrated in Figure 20. The influence of the dihedral is more pronounced in terms of pressure rise coefficient and at small mass flow ratios. In comparison to the orthogonal blade, the positive dihedral will always improve the blade static pressure coefficient, whereas the increment will first increase and then decrease with the increase of the dihedral angle. The optimum dihedral angle in the present study is 135°, where the value of *Cps* is improved by 23.5% at the near-stall condition.

0.09 *Cps* ω **Figure 20.** Pressure rise and loss characteristics for the stator with different dihedral angles. (**a**) Pressure rise and (**b**) loss. **Figure 20.** Pressure rise and loss characteristics for the stator with different dihedral angles. (**a**) Pressure rise and (**b**) loss.

(**a**) (**b**) 0.45 0.50 0.55 0.60 0.25 0.30 0.45 0.50 0.55 0.60 0.06 ϕ The flow field distributions at the mass flow coefficients of 0.52 and 0.46 are given in Figure 21. It is apparent that the leakage flow will deviate from the blade suction surface with the increase of the dihedral angle. As a result, the mixing of the leakage flow with the mainstream is more sufficient, and the blockage at the outlet alleviates. However, the increase of the dihedral angle will also enhance the radial migration of the blade corner flow, thus strengthening the flow separation on the suction surface. Of all the design schemes, C1D2 could not only suppress the secondary flow at the hub endwall but also avoid excessive flow separation on the blade suction surface, hence obtaining the optimum aerodynamic performance.

**Figure 20.** Pressure rise and loss characteristics for the stator with different dihedral angles. (**a**) Pres-

ϕ

sure rise and (**b**) loss.

mum aerodynamic performance.

**Figure 21.** Flow field distribution for the cases with different dihedral angles. **Figure 21.** Flow field distribution for the cases with different dihedral angles.

As shown in Figure 22, the influence of the dihedral angle on the performance of the cantilevered stator is illustrated clearly. Results show that the current dihedral angles barely influence the distribution of the mass flow coefficient at the stator inlet; however, the mass flow distribution at the stator outlet is altered. To be specific, at the *φ* = 0.52 condition, the mass flow coefficient in the lower and upper regions of the 20% blade height is increased and decreased, respectively, which echoes the flow field characteristics in Figure 21. The comparison of different design schemes indicates that with the increase of the dihedral angle, the enhancement of flow capacity at the stator hub will become inconspicuous, yet the worsening of aerodynamic performance at the upper span becomes more significant, which consequently weakens the total beneficial effect. The above phenomenon is more significant at small mass flow ratios. As shown in Figure 22, the influence of the dihedral angle on the performance of the cantilevered stator is illustrated clearly. Results show that the current dihedral angles barely influence the distribution of the mass flow coefficient at the stator inlet; however, the mass flow distribution at the stator outlet is altered. To be specific, at the *ϕ* = 0.52 condition, the mass flow coefficient in the lower and upper regions of the 20% blade height is increased and decreased, respectively, which echoes the flow field characteristics in Figure 21. The comparison of different design schemes indicates that with the increase of the dihedral angle, the enhancement of flow capacity at the stator hub will become inconspicuous, yet the worsening of aerodynamic performance at the upper span becomes more significant, which consequently weakens the total beneficial effect. The above phenomenon is more significant at small mass flow ratios. *Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 15 of 25

The flow field distributions at the mass flow coefficients of 0.52 and 0.46 are given in Figure 21. It is apparent that the leakage flow will deviate from the blade suction surface with the increase of the dihedral angle. As a result, the mixing of the leakage flow with the mainstream is more sufficient, and the blockage at the outlet alleviates. However, the increase of the dihedral angle will also enhance the radial migration of the blade corner flow, thus strengthening the flow separation on the suction surface. Of all the design schemes, C1D2 could not only suppress the secondary flow at the hub endwall but also avoid excessive flow separation on the blade suction surface, hence obtaining the opti-

**Figure 22.** Radial distribution of aerodynamic performances for the cases with different dihedral angles. (**a**) *φ* = 0.52; (**b**) *φ* = 0.46. **Figure 22.** Radial distribution of aerodynamic performances for the cases with different dihedral angles. (**a**) *ϕ* = 0.52; (**b**) *ϕ* = 0.46.

0.0006 0.0008 0.0010 Orth. C1D1 C1D2 C1D3 *m*leak (kg/s) Figure 23 presents the variation of the leakage mass flow rate. Results demonstrate that although the mass flow rate of the leakage flow is increased slightly in comparison with the orthogonal blade, the current range of the dihedral angle barely influences the axial distribution of the leakage flow, and the absolute mass flow rates for different blading schemes are approximately identical.

According to the former analysis, the mechanism of 3D blading in the cantilevered stator is summarized in Figure 24. The forward sweep can enhance the hindrance of the leakage flow on the low-energy fluid near the endwall, while flow separation on the blade suction side is exaggerated slightly. Increasing the sweep height facilitates a uniform separation along the span without changing the endwall flow significantly, whereas excessively large sweep angles lead to a large-scale separation on the blade suction surface and harm the total effect. On the other hand, as shown in Figure 24b, the positive dihedral will not only push the trajectory of the leakage vortex away from the blade suction surface but also promote the radial migration of the low-energy fluid at the hub corner. The utilization of large dihedral heights will elongate the blade wake and induce the radial transportation of the leakage flow, while the excessive dihedral angle will damage the performance at

In consideration of the effects above, a novel cantilevered stator with comprehensive sweep and dihedral is designed, whose aerodynamic performance is evaluated via numerical simulation. As shown in Figure 25, the 3D cantilevered stator employs a relatively high sweep height with a moderate sweep angle, and the dihedral is designed to have a low height and a moderate angle. Note that the blade tip also adopts a forward sweep to

0.0 0.2 0.4 0.6 0.8 1.0

*Normalized chord*

*3.3. Effects of the Compound Sweep and Dihedral* 

balance the radial pressure gradient.

(*φ* = 0.46).

0.0002 0.0004

the midspan.

0.0

0.2

0.4

0.6

*Normalized span*

0.8

1.0

0.2 0.4 0.6

ϕ

*out i* (

ϕ*in*

angles. (**a**) *φ* = 0.52; (**b**) *φ* = 0.46.

0.2 0.4 0.6 0 10 20 0 10 20 30

o )

Orth. C1D1 C1D2 C1D3

 (**a**) (**b**)

σ ( o )

**Figure 23.** Comparison of the streamwise leakage mass flow rate under different dihedral angles (*φ* = 0.46). **Figure 23.** Comparison of the streamwise leakage mass flow rate under different dihedral angles (*ϕ* = 0.46).

**Figure 22.** Radial distribution of aerodynamic performances for the cases with different dihedral

0.2 0.4 0.6

ϕ

*out i* (

ϕ*in* 0.2 0.4 0.6 0 10 20 0 10 20 30

o ) σ ( o )

Orth. C1D1 C1D2 C1D3

0.0

0.2

0.4

0.6

*Normalized span*

0.8

1.0

*3.3. Effects of the Compound Sweep and Dihedral*  According to the former analysis, the mechanism of 3D blading in the cantilevered stator is summarized in Figure 24. The forward sweep can enhance the hindrance of the leakage flow on the low-energy fluid near the endwall, while flow separation on the blade Hence, the positive dihedral will not only push the trajectory of the leakage vortex away from the blade suction surface but also promote the radial migration of the lowenergy fluid at the hub corner. A large dihedral height will elongate the blade wake and induce the radial transportation of the leakage flow.

### suction side is exaggerated slightly. Increasing the sweep height facilitates a uniform sep-*3.3. Effects of the Compound Sweep and Dihedral*

aration along the span without changing the endwall flow significantly, whereas excessively large sweep angles lead to a large-scale separation on the blade suction surface and harm the total effect. On the other hand, as shown in Figure 24b, the positive dihedral will not only push the trajectory of the leakage vortex away from the blade suction surface but also promote the radial migration of the low-energy fluid at the hub corner. The utilization of large dihedral heights will elongate the blade wake and induce the radial transportation of the leakage flow, while the excessive dihedral angle will damage the performance at the midspan. In consideration of the effects above, a novel cantilevered stator with comprehensive sweep and dihedral is designed, whose aerodynamic performance is evaluated via numerical simulation. As shown in Figure 25, the 3D cantilevered stator employs a relatively high sweep height with a moderate sweep angle, and the dihedral is designed to have a low height and a moderate angle. Note that the blade tip also adopts a forward sweep to According to the former analysis, the mechanism of 3D blading in the cantilevered stator is summarized in Figure 24. The forward sweep can enhance the hindrance of the leakage flow on the low-energy fluid near the endwall, while flow separation on the blade suction side is exaggerated slightly. Increasing the sweep height facilitates a uniform separation along the span without changing the endwall flow significantly, whereas excessively large sweep angles lead to a large-scale separation on the blade suction surface and harm the total effect. On the other hand, as shown in Figure 24b, the positive dihedral will not only push the trajectory of the leakage vortex away from the blade suction surface but also promote the radial migration of the low-energy fluid at the hub corner. The utilization of large dihedral heights will elongate the blade wake and induce the radial transportation of the leakage flow, while the excessive dihedral angle will damage the performance at the midspan. *Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 16 of 25

**Figure 24.** The mechanisms of sweep and dihedral on the cantilevered stator: (**a**) sweep; (**b**) dihedral. **Figure 24.** The mechanisms of sweep and dihedral on the cantilevered stator: (**a**) sweep; (**b**) dihedral.

0.0 0.1 0.2 0.0 0.2 0.4 0.6 0.8 1.0 Sweep Dihedral *sweep*/*dihedral* In consideration of the effects above, a novel cantilevered stator with comprehensive sweep and dihedral is designed, whose aerodynamic performance is evaluated via numerical simulation. As shown in Figure 25, the 3D cantilevered stator employs a relatively high sweep height with a moderate sweep angle, and the dihedral is designed to have a low height and a moderate angle. Note that the blade tip also adopts a forward sweep to balance the radial pressure gradient.

dihedral scheme (no sweep) are also provided for the convenience of comparison. Figure 26 implies that the combination of the sweep and dihedral will intensify the beneficial effects, as the "sweep + dihedral" scheme has the highest static pressure ratio and the lowest total pressure loss over the whole operating range. At the near-stall point, the static pressure rise coefficient and the total pressure loss coefficient are increased and decreased by 25.5% and 11.1%, respectively. Given the remarkable improvement over the baseline case, it is safe to say that the working mechanism summarized above is correct, and the redesign is successful. The following section will outline the experimental methods used

(**a**) (**b**) **Figure 26.** Pressure rise and loss characteristics for the 3D cantilevered stator. (**a**) Pressure rise; (**b**)

0.06

0.09

ω

0.12

0.15

0.45 0.50 0.55 0.60

ϕ

*Normalized span*

0.45 0.50 0.55 0.60

ϕ

to validate the conclusions.

loss.

0.25

0.30

0.35

*Cps*

0.40

**Figure 25.** Radial distribution of sweep and dihedral for the 3D cantilevered stator.

Orth. Sweep Dihedral Sweep + Dihedral

(**a**) (**b**)

**Figure 24.** The mechanisms of sweep and dihedral on the cantilevered stator: (**a**) sweep; (**b**) dihedral.

(**a**) (**b**)

0.06

0.45 0.50 0.55 0.60

ϕ

**Figure 26.** Pressure rise and loss characteristics for the 3D cantilevered stator. (**a**) Pressure rise; (**b**)

**Figure 25.** Radial distribution of sweep and dihedral for the 3D cantilevered stator. **Figure 25.** Radial distribution of sweep and dihedral for the 3D cantilevered stator. Figure 26 presents the pressure rise and loss characteristics of the 3D cantilevered

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 16 of 25

Figure 26 presents the pressure rise and loss characteristics of the 3D cantilevered stator; results of the orthogonal blade, the bare sweep scheme (no dihedral), and the bare dihedral scheme (no sweep) are also provided for the convenience of comparison. Figure 26 implies that the combination of the sweep and dihedral will intensify the beneficial effects, as the "sweep + dihedral" scheme has the highest static pressure ratio and the Figure 26 presents the pressure rise and loss characteristics of the 3D cantilevered stator; results of the orthogonal blade, the bare sweep scheme (no dihedral), and the bare dihedral scheme (no sweep) are also provided for the convenience of comparison. Figure 26 implies that the combination of the sweep and dihedral will intensify the beneficial effects, as the "sweep + dihedral" scheme has the highest static pressure ratio and the lowest total pressure loss over the whole operating range. At the near-stall point, the static pressure rise coefficient and the total pressure loss coefficient are increased and decreased by 25.5% and 11.1%, respectively. Given the remarkable improvement over the baseline case, it is safe to say that the working mechanism summarized above is correct, and the redesign is successful. The following section will outline the experimental methods used to validate the conclusions. stator; results of the orthogonal blade, the bare sweep scheme (no dihedral), and the bare dihedral scheme (no sweep) are also provided for the convenience of comparison. Figure 26 implies that the combination of the sweep and dihedral will intensify the beneficial effects, as the "sweep + dihedral" scheme has the highest static pressure ratio and the lowest total pressure loss over the whole operating range. At the near-stall point, the static pressure rise coefficient and the total pressure loss coefficient are increased and decreased by 25.5% and 11.1%, respectively. Given the remarkable improvement over the baseline case, it is safe to say that the working mechanism summarized above is correct, and the redesign is successful. The following section will outline the experimental methods used to validate the conclusions.

0.40 0.15 Orth. Sweep Dihedral Sweep + Dihedral **Figure 26.** Pressure rise and loss characteristics for the 3D cantilevered stator. (**a**) Pressure rise; (**b**) loss. **Figure 26.** Pressure rise and loss characteristics for the 3D cantilevered stator. (**a**) Pressure rise; (**b**) loss.

### **4. Application of 3D Blading in a Cantilevered Stator**

### *4.1. The Redesign of the Cantilevered Stator*

loss.

0.25

0.30

0.35 0.09 0.12 *Cps* ω The design parameters of the datum stator are shown in Table 3, with the diffusion factor varying from 0.58 to 0.35 from hub to tip. Moreover, the stator hub clearance occupies 1% of the blade height, which proved beneficial for the stator aerodynamic performance in a previous study [37]. Since the datum stator was comparable to the orthogonal cantilevered stator in the former study, the 3D blading strategy in Figure 25 is adopted for the redesign scheme.

0.45 0.50 0.55 0.60

ϕ


The design parameters of the datum stator are shown in Table 3, with the diffusion factor varying from 0.58 to 0.35 from hub to tip. Moreover, the stator hub clearance occupies 1% of the blade height, which proved beneficial for the stator aerodynamic performance in a previous study [37]. Since the datum stator was comparable to the orthogonal cantilevered stator in the former study, the 3D blading strategy in Figure 25 is adopted for

**Table 3.** Design parameters for the highly loaded cantilevered stator. **Table 3.** Design parameters for the highly loaded cantilevered stator.  **Hub Midspan Tip** 

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 17 of 25

**4. Application of 3D Blading in a Cantilevered Stator** 

*4.1. The Redesign of the Cantilevered Stator* 

the redesign scheme.

Figure 27 presents the comparison of the aerodynamic performances of the datum and redesigned cantilevered stators. To improve the simulation accuracy, the numerical simulation considers the stator blade row separately and imposes the experimentally measured flow fields at the stator inlet according to the specific operating conditions. In the datum case, the pressure rise capability of the cantilevered stator will drop significantly once the mass flow coefficient is below 0.53, due to the deterioration of flow conditions at the hub corner (Figure 28). On the other hand, the stator with 3D blading can overcome the above problem efficiently, as the *Cps* is increased by 71.8% at the mass flow coefficient of 0.45 (P1). The total pressure loss coefficient for the redesigned case is reduced remarkably as well, implying the effectiveness of the 3D blading scheme. Figure 27 presents the comparison of the aerodynamic performances of the datum and redesigned cantilevered stators. To improve the simulation accuracy, the numerical simulation considers the stator blade row separately and imposes the experimentally measured flow fields at the stator inlet according to the specific operating conditions. In the datum case, the pressure rise capability of the cantilevered stator will drop significantly once the mass flow coefficient is below 0.53, due to the deterioration of flow conditions at the hub corner (Figure 28). On the other hand, the stator with 3D blading can overcome the above problem efficiently, as the *Cps* is increased by 71.8% at the mass flow coefficient of 0.45 (P1). The total pressure loss coefficient for the redesigned case is reduced remarkably as well, implying the effectiveness of the 3D blading scheme.

**Figure 27.** Comparison of pressure rise and loss characteristics of the cantilevered stators. **Figure 27.** Comparison of pressure rise and loss characteristics of the cantilevered stators.

**Figure 28.** The comparison of field distribution between the datum and the redesigned stator. **Figure 28.** The comparison of field distribution between the datum and the redesigned stator.

As shown in Figure 28, the comparison of the flow field under different operating conditions (P1~P4, see Figure 27) indicates that the datum stator suffers from severe corner flow separation at the near-stall condition, leading to the sudden drop of the pressure

the flow field; hence, the accumulation of low-energy fluid at the hub corner is relieved significantly, corresponding to the performance improvement in Figure 27. Note that the flow separation in the upper span areas is intensified in comparison with the datum scheme, which is due to the radial migration of the low-energy fluid from the positive

The radial distribution of the aerodynamic parameters is given in Figure 29 to illustrate the effect of 3D blading quantitively. Results show that the 3D blading can enhance the through flow capacity at the hub region over a wide operating range, as the mass flow distribution is more uniform along the radial direction. The beneficial effect extends from 15% to 50% span as the operating point moves from P1 to P4, which is consistent with the former analysis. Moreover, the total pressure loss demonstrates the advantage of 3D blading in reducing the near-stall loss, yet the effect is less remarkable under larger mass flow coefficients when the flow field is naturally healthy. The following section will outline the

experimental methods used for further validation.

(**a**) (**b**)

tions: (**a**) outlet mass flow coefficient; (**b**) total pressure loss.

ϕ*out*

0.3 0.6 0.3 0.6 0.3 0.6

ϕ*out*

P2 P3 P4

**Figure 29.** The radial distribution of mass flow coefficient and loss under different operating condi-

0.0 0.2 0.4

ω

P1

Datum 3D blading

ω

0.0 0.2 0.4 0.0 0.2 0.4 0.0 0.2 0.4

ω

P2 P3 P4

ω

0.0 0.2 0.4 0.6 0.8 1.0

*Normalized span*

dihedral.

Datum 3D blading

ϕ*out*

0.3 0.6

ϕ*out*

P1

0.0 0.2 0.4 0.6 0.8 1.0

*Normalized span*

Datum

3D blading

> As shown in Figure 28, the comparison of the flow field under different operating conditions (P1~P4, see Figure 27) indicates that the datum stator suffers from severe corner flow separation at the near-stall condition, leading to the sudden drop of the pressure rise capacity. Meanwhile, the utilization of compound sweep and dihedral reorganizes the flow field; hence, the accumulation of low-energy fluid at the hub corner is relieved significantly, corresponding to the performance improvement in Figure 27. Note that the flow separation in the upper span areas is intensified in comparison with the datum scheme, which is due to the radial migration of the low-energy fluid from the positive dihedral. the flow field; hence, the accumulation of low-energy fluid at the hub corner is relieved significantly, corresponding to the performance improvement in Figure 27. Note that the flow separation in the upper span areas is intensified in comparison with the datum scheme, which is due to the radial migration of the low-energy fluid from the positive dihedral. The radial distribution of the aerodynamic parameters is given in Figure 29 to illustrate the effect of 3D blading quantitively. Results show that the 3D blading can enhance

**Figure 28.** The comparison of field distribution between the datum and the redesigned stator.

As shown in Figure 28, the comparison of the flow field under different operating conditions (P1~P4, see Figure 27) indicates that the datum stator suffers from severe corner flow separation at the near-stall condition, leading to the sudden drop of the pressure rise capacity. Meanwhile, the utilization of compound sweep and dihedral reorganizes

The radial distribution of the aerodynamic parameters is given in Figure 29 to illustrate the effect of 3D blading quantitively. Results show that the 3D blading can enhance the through flow capacity at the hub region over a wide operating range, as the mass flow distribution is more uniform along the radial direction. The beneficial effect extends from 15% to 50% span as the operating point moves from P1 to P4, which is consistent with the former analysis. Moreover, the total pressure loss demonstrates the advantage of 3D blading in reducing the near-stall loss, yet the effect is less remarkable under larger mass flow coefficients when the flow field is naturally healthy. The following section will outline the experimental methods used for further validation. the through flow capacity at the hub region over a wide operating range, as the mass flow distribution is more uniform along the radial direction. The beneficial effect extends from 15% to 50% span as the operating point moves from P1 to P4, which is consistent with the former analysis. Moreover, the total pressure loss demonstrates the advantage of 3D blading in reducing the near-stall loss, yet the effect is less remarkable under larger mass flow coefficients when the flow field is naturally healthy. The following section will outline the experimental methods used for further validation.

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 18 of 25

P1 P2 P3 P4

**Figure 29.** The radial distribution of mass flow coefficient and loss under different operating conditions: (**a**) outlet mass flow coefficient; (**b**) total pressure loss. **Figure 29.** The radial distribution of mass flow coefficient and loss under different operating conditions: (**a**) outlet mass flow coefficient; (**b**) total pressure loss.

### *4.2. Discussion of the Experimental Results*

### 4.2.1. Effect of 3D Blading on the Aerodynamic Performance

To start, the overall aerodynamic performance of the compressor stage is compared, as shown in Figure 30. Results highlight that the 3D blading on the cantilevered stator could significantly improve the performance of the compressor stage at small mass flow conditions; at the near stall condition (P4), the total pressure rise coefficient and the efficiency are increased by 3% and 2%, respectively. However, as the operating point moves to the right, the 3D blading will start losing its advantage, and the stage efficiency at *ϕ* > 0.51 will even drop by 0.7%. As only stator blades have been changed, Figure 31 shows the loss characteristics of the cantilevered stators, in which the CFD results are also plotted to present the deviation between the experiment result and the CFD result. Figure 31 indicates the CFD results have high reference value, so the above numerical analysis results are credible.

To start, the overall aerodynamic performance of the compressor stage is compared, as shown in Figure 30. Results highlight that the 3D blading on the cantilevered stator could significantly improve the performance of the compressor stage at small mass flow conditions; at the near stall condition (P4), the total pressure rise coefficient and the efficiency are increased by 3% and 2%, respectively. However, as the operating point moves to the right, the 3D blading will start losing its advantage, and the stage efficiency at *φ* > 0.51 will even drop by 0.7%. As only stator blades have been changed, Figure 31 shows the loss characteristics of the cantilevered stators, in which the CFD results are also plotted to present the deviation between the experiment result and the CFD result. Figure 31 indicates the CFD results have high reference value, so the above numerical analysis results

P1

Prototype 3D blading

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 19 of 25

4.2.1. Effect of 3D Blading on the Aerodynamic Performance

To start, the overall aerodynamic performance of the compressor stage is compared,

as shown in Figure 30. Results highlight that the 3D blading on the cantilevered stator

could significantly improve the performance of the compressor stage at small mass flow

conditions; at the near stall condition (P4), the total pressure rise coefficient and the effi-

ciency are increased by 3% and 2%, respectively. However, as the operating point moves

to the right, the 3D blading will start losing its advantage, and the stage efficiency at *φ* >

0.51 will even drop by 0.7%. As only stator blades have been changed, Figure 31 shows

the loss characteristics of the cantilevered stators, in which the CFD results are also plotted

to present the deviation between the experiment result and the CFD result. Figure 31 in-

dicates the CFD results have high reference value, so the above numerical analysis results

0.84

η

ϕ

0.88

0.92

*4.2. Discussion of the Experimental Results* 

are credible.

0.70.8

0.9

are credible.

*Cpt*

4.2.1. Effect of 3D Blading on the Aerodynamic Performance

P4 P3 P2

*4.2. Discussion of the Experimental Results* 

**Figure 30.** Comparison of performance characteristics of the compressor stage: (**a**) total pressure rise coefficient; (**b**) efficiency. **Figure 30.** Comparison of performance characteristics of the compressor stage: (**a**) total pressure rise coefficient; (**b**) efficiency. coefficient; (**b**) efficiency.

the absolute reduction reaches 0.035 at the near-stall point (P4), or 20.5%. However, the loss of the cantilevered stator will increase slightly at higher mass flow conditions; the **Figure 31.** Comparison of loss characteristics of the cantilevered stators between experiment and **Figure 31.** Comparison of loss characteristics of the cantilevered stators between experiment and CFD.

maximum increment is 0.008 at *φ* = 0.53, which is acceptable in consideration of its advantage under other conditions. Additionally, the variation of the stator loss coincides with the trend on the stage level, thus implying that the variation of compressor stage performance in Figure 30 is attributed mainly to the 3D bladed stator. In order to further CFD. Turning to the stator blade row, Figure 32a presents the variation of the stator total pressure loss coefficient with the stage mass flow coefficient. In comparison with the datum scheme, the total pressure loss of the 3D cantilevered stator is reduced significantly; the absolute reduction reaches 0.035 at the near-stall point (P4), or 20.5%. However, the loss of the cantilevered stator will increase slightly at higher mass flow conditions; the maximum increment is 0.008 at *φ* = 0.53, which is acceptable in consideration of its advantage under other conditions. Additionally, the variation of the stator loss coincides with the trend on the stage level, thus implying that the variation of compressor stage performance in Figure 30 is attributed mainly to the 3D bladed stator. In order to further Turning to the stator blade row, Figure 32a presents the variation of the stator total pressure loss coefficient with the stage mass flow coefficient. In comparison with the datum scheme, the total pressure loss of the 3D cantilevered stator is reduced significantly; the absolute reduction reaches 0.035 at the near-stall point (P4), or 20.5%. However, the loss of the cantilevered stator will increase slightly at higher mass flow conditions; the maximum increment is 0.008 at *ϕ* = 0.53, which is acceptable in consideration of its advantage under other conditions. Additionally, the variation of the stator loss coincides with the trend on the stage level, thus implying that the variation of compressor stage performance in Figure 30 is attributed mainly to the 3D bladed stator. In order to further quantify the impact of 3D blading on different regions of the cantilevered stator, the total loss is decomposed along the blade span, as shown in Figure 32b–d. The blade is classified into three regions according to the mass flow, i.e., the hub region (0~25% total mass flow), the middle region (25~75% total mass flow), and the tip region (75~100% total mass flow). Results show that the loss reduction of the 3D stator stems mainly from the hub and midspan regions, which is consistent with the weakening of corner flow separation in these areas. Moreover, the utilization of the forward sweep turns out to improve the tip region over the whole operating range, which signifies the necessity of balancing radial flow in the design process.

3D blading

0.1

0.2

ω

0.3

0.3

0.4

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 20 of 25

the design process.

 Prototype 3D blading

the design process.

0.3

0.4

**Figure 32.** Comparison of loss characteristics of the cantilevered stator: (**a**) total loss, (**b**) hub loss, (**c**) midspan loss, and (**d**) tip loss. **Figure 32.** Comparison of loss characteristics of the cantilevered stator: (**a**) total loss, (**b**) hub loss, (**c**) midspan loss, and (**d**) tip loss. The radial distribution of the mass flow coefficient for the cantilevered stator is illustrated in Figure 33. Results show that the 3D blading barely influences the mass flow dis-

quantify the impact of 3D blading on different regions of the cantilevered stator, the total loss is decomposed along the blade span, as shown in Figure 32b–d. The blade is classified into three regions according to the mass flow, i.e., the hub region (0~25% total mass flow), the middle region (25~75% total mass flow), and the tip region (75~100% total mass flow). Results show that the loss reduction of the 3D stator stems mainly from the hub and midspan regions, which is consistent with the weakening of corner flow separation in these areas. Moreover, the utilization of the forward sweep turns out to improve the tip region over the whole operating range, which signifies the necessity of balancing radial flow in

> Prototype 3D blading

0.3

0.4

 Prototype 3D blading

quantify the impact of 3D blading on different regions of the cantilevered stator, the total loss is decomposed along the blade span, as shown in Figure 32b–d. The blade is classified into three regions according to the mass flow, i.e., the hub region (0~25% total mass flow), the middle region (25~75% total mass flow), and the tip region (75~100% total mass flow). Results show that the loss reduction of the 3D stator stems mainly from the hub and midspan regions, which is consistent with the weakening of corner flow separation in these areas. Moreover, the utilization of the forward sweep turns out to improve the tip region over the whole operating range, which signifies the necessity of balancing radial flow in

The radial distribution of the mass flow coefficient for the cantilevered stator is illustrated in Figure 33. Results show that the 3D blading barely influences the mass flow distribution at the stator inlet, whereas it could improve the throughflow capacity at the hub region over a wide operating range. The distribution of mass flow coefficient at the stator outlet exhibits a similar pattern to that in Figure 29, which proves the accuracy of the numerical simulation. On the other hand, the mass flow coefficient in the lower span areas of the stator outlet sees a remarkable increment at P2~P4 conditions, implying the flow capacity near the endwall is enhanced via the combination of the forward sweep and dihedral. It should be noted that at the P1 condition, the mass flow coefficient of a 10~40% span is reduced for the 3D blading scheme, which indicates the flow capacity is weakened in these regions. The mass flow at the midspan areas is first increased and then decreased as the operating point moves from P1 to P4, owing to the redistribution of radial flow. The effect of 3D blading is more pronounced at small mass flow coefficients, thus confirming the former investigation. The radial distribution of the mass flow coefficient for the cantilevered stator is illustrated in Figure 33. Results show that the 3D blading barely influences the mass flow distribution at the stator inlet, whereas it could improve the throughflow capacity at the hub region over a wide operating range. The distribution of mass flow coefficient at the stator outlet exhibits a similar pattern to that in Figure 29, which proves the accuracy of the numerical simulation. On the other hand, the mass flow coefficient in the lower span areas of the stator outlet sees a remarkable increment at P2~P4 conditions, implying the flow capacity near the endwall is enhanced via the combination of the forward sweep and dihedral. It should be noted that at the P1 condition, the mass flow coefficient of a 10~40% span is reduced for the 3D blading scheme, which indicates the flow capacity is weakened in these regions. The mass flow at the midspan areas is first increased and then decreased as the operating point moves from P1 to P4, owing to the redistribution of radial flow. The effect of 3D blading is more pronounced at small mass flow coefficients, thus confirming the former investigation. tribution at the stator inlet, whereas it could improve the throughflow capacity at the hub region over a wide operating range. The distribution of mass flow coefficient at the stator outlet exhibits a similar pattern to that in Figure 29, which proves the accuracy of the numerical simulation. On the other hand, the mass flow coefficient in the lower span areas of the stator outlet sees a remarkable increment at P2~P4 conditions, implying the flow capacity near the endwall is enhanced via the combination of the forward sweep and dihedral. It should be noted that at the P1 condition, the mass flow coefficient of a 10~40% span is reduced for the 3D blading scheme, which indicates the flow capacity is weakened in these regions. The mass flow at the midspan areas is first increased and then decreased as the operating point moves from P1 to P4, owing to the redistribution of radial flow. The effect of 3D blading is more pronounced at small mass flow coefficients, thus confirming the former investigation.

(**a**) inlet mass flow coefficient; (**b**) outlet mass flow coefficient. **Figure 33.** The radial distribution of mass flow coefficients under different operating coefficients: (**a**) inlet mass flow coefficient; (**b**) outlet mass flow coefficient. **Figure 33.** The radial distribution of mass flow coefficients under different operating coefficients: (**a**) inlet mass flow coefficient; (**b**) outlet mass flow coefficient.

**Figure 33.** The radial distribution of mass flow coefficients under different operating coefficients:

Figure 34 presents the distributions of the blockage coefficient and the total pressure loss coefficient. The variation of the blockage denotes that the 3D blading could generally relieve the flow blockage at the hub corner, thereby creating a healthier flow field. However, the blockage coefficient of a 10~40% span is increased at the P1 condition, which corresponds to the reduction of mass flow ratio in Figure 33b. As for loss characteristics, large amounts of loss reduction are brought by the 3D blading at small mass flow conditions (P3 and P4), whereas the beneficial effect is less distinctive at large mass flow ratios (P1 and P2).

(P1 and P2).

**Figure 34.** The radial distribution of blockage and loss under different operating coefficients: (**a**) blockage; (**b**) total pressure loss. **Figure 34.** The radial distribution of blockage and loss under different operating coefficients: (**a**) blockage; (**b**) total pressure loss.

Figure 34 presents the distributions of the blockage coefficient and the total pressure loss coefficient. The variation of the blockage denotes that the 3D blading could generally relieve the flow blockage at the hub corner, thereby creating a healthier flow field. However, the blockage coefficient of a 10~40% span is increased at the P1 condition, which corresponds to the reduction of mass flow ratio in Figure 33b. As for loss characteristics, large amounts of loss reduction are brought by the 3D blading at small mass flow conditions (P3 and P4), whereas the beneficial effect is less distinctive at large mass flow ratios

### 4.2.2. Effect of 3D Blading on the Flow Field Distribution

4.2.2. Effect of 3D Blading on the Flow Field Distribution To obtain an overview of the flow structures in the cantilevered stator, Figure 35 presents the oil-flow results of the datum stator. The separation line (SL) is denoted by the red solid line, the attachment line (AL) is denoted by the red dotted line, the spiral node is denoted by the letter F, and the saddle point is denoted by the letter S. It can be seen that at the P1 condition, under the blowing effect of the endwall leakage flow, the corner separation at the stator hub starts from the saddle point S1 and separates into S1-F1 and S1-F3 along the radial direction; the tip region also suffers corner flow separation. The upper and lower separation areas bounded by S1 are approximately symmetrical and located close to the blade trailing edge, whereas the separation region at the blade tip is independent of the hub corner separation. With the decrease of the mass flow coefficient, the corner flow separation first enlarges its radial scale at P2 and then changes the topology at P3: the separation line S1-F1 heads upstream, pushing F1 to the endwall and incurring the corner stall. According to Figure 35, the stator hub is severely blocked at the P3 condition, represented by the large-scale low-speed zone at the outlet. Further observation of the tip flow shows that at the near-stall conditions (P3, P4), the separation region at the stator hub and the tip will gradually merge at the trailing edge, which is also demon-To obtain an overview of the flow structures in the cantilevered stator, Figure 35 presents the oil-flow results of the datum stator. The separation line (SL) is denoted by the red solid line, the attachment line (AL) is denoted by the red dotted line, the spiral node is denoted by the letter F, and the saddle point is denoted by the letter S. It can be seen that at the P1 condition, under the blowing effect of the endwall leakage flow, the corner separation at the stator hub starts from the saddle point S1 and separates into S1-F1 and S1-F3 along the radial direction; the tip region also suffers corner flow separation. The upper and lower separation areas bounded by S1 are approximately symmetrical and located close to the blade trailing edge, whereas the separation region at the blade tip is independent of the hub corner separation. With the decrease of the mass flow coefficient, the corner flow separation first enlarges its radial scale at P2 and then changes the topology at P3: the separation line S1-F1 heads upstream, pushing F1 to the endwall and incurring the corner stall. According to Figure 35, the stator hub is severely blocked at the P3 condition, represented by the large-scale low-speed zone at the outlet. Further observation of the tip flow shows that at the near-stall conditions (P3, P4), the separation region at the stator hub and the tip will gradually merge at the trailing edge, which is also demonstrated in Figure 35. Finally, under the cantilevered geometry, the occurrence of the corner stall will only induce local separation flow at SL2, which is distinctively different from the conventional shrouded stator (represented by the large-scale endwall separation and a rapid expansion of the separation zone) [37–39].

strated in Figure 35. Finally, under the cantilevered geometry, the occurrence of the corner stall will only induce local separation flow at SL2, which is distinctively different from the conventional shrouded stator (represented by the large-scale endwall separation and a rapid expansion of the separation zone) [37–39]. Figure 36 presents the flow field distribution at the stator outlet, where the results of both the datum stator and the 3D bladed stator are demonstrated. It can be observed that the 3D blading can effectively push the leakage flow away from the blade suction surface, yet it will incur secondary leakage at the large flow conditions (in P1 and P2, the leakage flow moves into adjacent blade channel). As a result, the flow separation at the blade trailing edge is enhanced slightly. With the decrease of mass flow rate, both the leakage flow and the transverse secondary flow will be enhanced by the increase of the circumferential Figure 36 presents the flow field distribution at the stator outlet, where the results of both the datum stator and the 3D bladed stator are demonstrated. It can be observed that the 3D blading can effectively push the leakage flow away from the blade suction surface, yet it will incur secondary leakage at the large flow conditions (in P1 and P2, the leakage flow moves into adjacent blade channel). As a result, the flow separation at the blade trailing edge is enhanced slightly. With the decrease of mass flow rate, both the leakage flow and the transverse secondary flow will be enhanced by the increase of the circumferential pressure gradient, while the difference in their variation rate makes the leakage flow approach the blade suction surface and finally accumulate toward the hub corner of the blade surface (P3 and P4). In fact, it is only at the near-stall conditions when the 3D blading manifests its advantage: at P3 and P4, the flow separation is weakened in the middle and lower part of the stator, thereby alleviating the blockage in the middle and lower span areas. Moreover, the forward sweep at the stator tip turns out to improve the flow field in the meantime, as the wake is narrowed correspondingly.

pressure gradient, while the difference in their variation rate makes the leakage flow approach the blade suction surface and finally accumulate toward the hub corner of the blade surface (P3 and P4). In fact, it is only at the near-stall conditions when the 3D blading

P1 P2 P3 P4 Suction surface Hub  P1 P2 P3 P4 Suction surface Hub 

the meantime, as the wake is narrowed correspondingly.

the meantime, as the wake is narrowed correspondingly.

*Appl. Sci.* **2023**, *13*, x FOR PEER REVIEW 22 of 25

**Figure 35.** Oil-flow results on the blade suction surface and hub wall for the datum stator. **Figure 35.** Oil-flow results on the blade suction surface and hub wall for the datum stator. **Figure 35.** Oil-flow results on the blade suction surface and hub wall for the datum stator.

manifests its advantage: at P3 and P4, the flow separation is weakened in the middle and lower part of the stator, thereby alleviating the blockage in the middle and lower span areas. Moreover, the forward sweep at the stator tip turns out to improve the flow field in

manifests its advantage: at P3 and P4, the flow separation is weakened in the middle and lower part of the stator, thereby alleviating the blockage in the middle and lower span areas. Moreover, the forward sweep at the stator tip turns out to improve the flow field in

**Figure 36.** The distribution of the normalized axial speed at the stator outlet. **Figure 36.** The distribution of the normalized axial speed at the stator outlet. **Figure 36.** The distribution of the normalized axial speed at the stator outlet.

To further reveal the mechanism of 3D blading in the corner flow in the cantilevered stator, Figure 37 presents the comparison of the secondary flow velocity vector at the stator outlet. Traces of the secondary flow (CF) and the leakage flow (LF) are depicted to demonstrate the flow structures with better clarity. Apparently, the leakage flow that travels from the blade pressure surface to the suction pressure surface could hinder the circumferential migration of the secondary flow. In the datum scheme, the CF travels through the bottom of the LF and climbs to the blade suction surface, inducing a counterclockwise vortex on its left side and a clockwise vortex on its right side. Upon the utilization of 3D blading, the strengthening of the leakage flow enhances the inhibition effect of the LF on the CF, leading to secondary leakage at the P1 and P2 conditions. As for P3 and P4 conditions, the flow separation in the corner region is reduced, owing to the weakening of the secondary flow. To further reveal the mechanism of 3D blading in the corner flow in the cantilevered stator, Figure 37 presents the comparison of the secondary flow velocity vector at the stator outlet. Traces of the secondary flow (CF) and the leakage flow (LF) are depicted to demonstrate the flow structures with better clarity. Apparently, the leakage flow that travels from the blade pressure surface to the suction pressure surface could hinder the circumferential migration of the secondary flow. In the datum scheme, the CF travels through the bottom of the LF and climbs to the blade suction surface, inducing a counterclockwise vortex on its left side and a clockwise vortex on its right side. Upon the utilization of 3D blading, the strengthening of the leakage flow enhances the inhibition effect of the LF on the CF, leading to secondary leakage at the P1 and P2 conditions. As for P3 and P4 conditions, the flow separation in the corner region is reduced, owing to the weakening To further reveal the mechanism of 3D blading in the corner flow in the cantilevered stator, Figure 37 presents the comparison of the secondary flow velocity vector at the stator outlet. Traces of the secondary flow (CF) and the leakage flow (LF) are depicted to demonstrate the flow structures with better clarity. Apparently, the leakage flow that travels from the blade pressure surface to the suction pressure surface could hinder the circumferential migration of the secondary flow. In the datum scheme, the CF travels through the bottom of the LF and climbs to the blade suction surface, inducing a counterclockwise vortex on its left side and a clockwise vortex on its right side. Upon the utilization of 3D blading, the strengthening of the leakage flow enhances the inhibition effect of the LF on the CF, leading to secondary leakage at the P1 and P2 conditions. As for P3 and P4 conditions, the flow separation in the corner region is reduced, owing to the weakening of the secondary flow.

of the secondary flow.

**Figure 37.** Distribution of secondary flow velocity vector at the stator outlet. **Figure 37.** Distribution of secondary flow velocity vector at the stator outlet.

To sum up, the interaction between the endwall leakage flow and the transverse secondary flow determines the effect of the 3D blading. For the present cantilevered stator, at large mass flow rates, the experimental results demonstrate that the 3D blading pushes the leakage flow too close to the pressure side of the adjacent blade, thereby inducing secondary leakage. However, at small mass flow rates, the inhibition of the leakage flow to the endwall secondary flow is not strong enough; hence, the corner separation requires further elimination. The above results also imply that the design of 3D blading needs to optimize the evolution of corner flow structures at different operating conditions. To sum up, the interaction between the endwall leakage flow and the transverse secondary flow determines the effect of the 3D blading. For the present cantilevered stator, at large mass flow rates, the experimental results demonstrate that the 3D blading pushes the leakage flow too close to the pressure side of the adjacent blade, thereby inducing secondary leakage. However, at small mass flow rates, the inhibition of the leakage flow to the endwall secondary flow is not strong enough; hence, the corner separation requires further elimination. The above results also imply that the design of 3D blading needs to optimize the evolution of corner flow structures at different operating conditions.

### **5. Conclusions**

**5. Conclusions**  This study focuses on the utilization of 3D blading in the cantilevered stator and seeks to reveal its mechanism in improving the compressor aerodynamic performance. This study focuses on the utilization of 3D blading in the cantilevered stator and seeks to reveal its mechanism in improving the compressor aerodynamic performance. The main conclusions are drawn as follows:


of the leakage flow to the endwall secondary flow is not strong enough; hence, the corner separation needs further elimination. The design of the 3D cantilevered stator needs to optimize the evolution of corner flow structures over the operating range.

The validation of the 3D blading is conducted in the low-speed compressor test facility under the single-stage environment in the present study, and it would be meaningful if experiments could be implemented in high-speed and multi-stage environments in the future to reveal more flow mechanisms while validating the present conclusions.

**Author Contributions:** Conceptualization, G.A. and X.Y.; methodology, X.X., R.W. and G.A.; software, G.A.; validation, Y.Q., G.A. and X.Y.; formal analysis, X.X., R.W. and G.A.; investigation, X.X., Y.Q., R.W., B.L., G.A. and X.Y.; resources, Y.Q. and G.A.; data curation, Y.Q., X.X., R.W. and G.A.; writing—original draft preparation, X.X. and R.W.; writing—review and editing, R.W.; visualization, Y.Q., X.X. and R.W.; supervision, B.L.; project administration, B.L. and X.Y.; funding acquisition, B.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China (Grant Nos. 52276025, 52206038), the National Science and Technology Major Project (J2019-II-0004-0024, J2019-II-0003-0023), the Advanced Jet Propulsion Innovation Center/AEAC (funding number HKCX2022-01- 008), and the Fundamental Research Funds for the Central Universities.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

### **References**


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