Design and numerical simulation analysis of a miniature Mixed-flow compressor

This paper completed the pneumatic design of a mixed-flow compressor, adopted RANS turbulence model, and chose Spalart-Allmaras model to complete the numerical simulation of the flow field of the S1 flow surface, meridian flow surface, and impeller surface, and then analyzed the working performance at 90%, 100%, and 110% design speed to explore whether the mixed-flow compressor technology suitable for micro turbojet engines as well as the internal flow field flow state and loss mechanism. The results indicate that the shock wave at the front edge of the main blade and the low energy gas gathered at the tail edge of the main blade and the shunt blade, which causes serious flow blockage, are the main causes of the instability of the mixed-flow compressor at the design point. At 100% design speed, the compressor has higher reliability, stronger adaptability, and the best working performance for its working margin, isentropic efficiency, and pressure ratio with lower loss. The results may provide some references for the further optimization design of the current compressor.


Introduction
In today's local wars throughout the world, the advantages and importance of modern weapons such as drones and cruise missiles have grown increasingly prominent, and they have evolved into a powerful symbol of the nation.As a result, the micro turbojet engine, a key technology of the core component of aircraft like drones, has become a focus of both academic and industrial research in recent years.As one of the most vital components of the turbojet engine, the design of the compressor has a huge impact on the performance of the turbojet engine.In the civil field, the transformation of micro turbojet engine technology may also promote the advancement of pumps, compressors, and other technologies.Compared with axial flow and centrifugal compressors, mixed-flow compressors have excellent characteristics such as minimal windward area, high efficiency, high single-stage pressure ratio, and stability.When compared to centrifugal compressors, the increased power of mixed-flow compressors will slow down, but it can reduce the flow loss [1] .It is becoming widely used in related industries because of its simple structure [2] .However, the design system of the mixed-flow compressor has not been sufficiently improved, so the development of a mixed-flow compressor is far from perfect.Mixed flow compressors have a wide range of prospect applications and great research value in both military and civilian industries due to a series of advantages.
International research on mixed-flow compressor technology began earlier.In 1942, King and Glodeck studied and designed the world's first mixed-flow compressor.Around 2010, the Middle East Technical University of Turkey [3] [4] established a mixed-flow compressor to power a drone.The design point pressure ratio is 4.34 and the efficiency is 0.72.The shock wave is generated on the front edge of the main blade and the downstream small blade flow channel, resulting in a serious flow blockage.The National Space Laboratory of India constructed a mixed-flow compressor with a pressure ratio of 3.8 and a flow rate of 2.72 kg•s -1 to investigate the influence of the tip gap on the mixed-flow compressor in 2015.At 0.65 times the design speed, the stage efficiency is 0.75 and the rotor efficiency is 0.80, respectively [5] .In 2021, Russian researchers used experimental methods to study the flow field characteristics of the miniature mixed-flow compressor, which served as a reference for the design and optimization of the miniature mixed-flow compressor [6] .
In the 1980s, domestic research on mixed-flow compressors began.Gao et al. [7] investigate the aerodynamic design of a small flow mixed-flow compressor based on a "controllable vortex" and establish an integrated design approach for a mixed-flow compressor flow channel based on a segmented Bezier curve.Ao et al. [8] choose and refine the centrifugal compressor flow loss model, resulting in a flow loss model combination that matches the mixed-flow compressor characteristics.The predicted compressor rotor efficiency was closely matched with the trend of the numerical simulation findings when compared to the adiabatic efficiency obtained by the numerical simulation of the 3D flow field.Moreover, the relative composition of the impeller flow loss will change as the working conditions change.However, the law of change for impeller mixing loss, secondary current loss calculated by the loss model, and the change for related flow characteristics calculated by the numerical simulation cannot be mutually verified.Wang et al. [9] employ the fast pneumatic design method, develop the initial geometry of the mixed-flow compressor, then run and analyze the simulation under various speed conditions with CFD software, yielding satisfactory results.Despite the prominent advantages of the mixed-flow compressor, its application in various fields is quite limited.Further research and exploration are still required to realize a larger use of mixed-flow compressors in several domains.This paper selected a miniature mixed-flow compressor as the research object as well as CFD software to set up and run a numerical simulation.It was based on an elementary velocity triangle, the mathematical model of the Turbulent N-S equation, and the S-A turbulence model.It completed the one-dimensional design and three-dimensional model when combined with the relevant requirements of design indicators.According to the results, the pressure ratio is the highest under design speed, with a large working margin and good flow conditions in the flow channel, which meets the standard of a micro turbojet engine.

Design of the mixed-flow compressor
Several indexes, including high efficiency, practicability, and reliability, need to be considered to ensure the design of the mixed-flow compressor is reasonable.The specific design requirements are shown in Table 1

Elementary velocity triangle analysis
The study of elementary velocity triangles is of great benefit to the design of mixed-flow compressors.
The other parameters of the compressor can be changed by appropriately adjusting the size and direction of the several parameters of the outlet and the inlet velocity triangle due to the strong correlation between the geometric size of the compressor impeller and the pneumatic parameters of the compressor.The fundamental velocity triangle of the inlet and outlet of the mixed-flow compressor impeller can be obtained from the velocity synthesis theorem, which states that the absolute velocity of the object is equal to the vector sum of the relative velocity and the convected velocity.This is illustrated in Figure 1.As seen in Figure 1, the inlet velocity triangle is represented by subscript 1, while the outlet velocity triangle is represented by subscript 2. Among the parameters presented in the figure, c 1 is the absolute velocity, the speed at which the air flows into the impeller, u 1 stands for the convected speed, the peripheral velocity of the impeller at the inlet, and ω 1 denotes the relative velocity of the air to the impeller.According to the velocity synthesis theorem, the absolute velocity c 1 is the vector sum of relative velocity ω 1 and convected speed u 1 .
(1) Under the specified working conditions, the direction of the relative speed ω 1 coincides with the tangential direction of the blade's front edge.After entering the impeller at the speed ω 1 , the air applies for work with the impeller, converting phase, accelerating, and pressurizing in the bending expansion channel.As reflected in Figure 1, the airflow gradually changes direction and its relative velocity decreases.Finally, outflows exit at the relative velocity ω 2 from the impeller along the curved channel, which is displayed in Figure 1.Angle β 1 is the airflow inlet angle, whereas angle β 2 represents the airflow outlet angle and β 2 >β 1 .
The wheel rim work equation of the mixed-flow compressor may be obtained by combining the velocity triangle and the Bernoulli equation with the momentum moment equation.

𝑙 𝑀 𝜔 𝜔 𝑐 𝑟 c 𝑟 c u c 𝑢
(2) r 1 and r 2 are the inlet and outlet radii of the blade in this equation.The subscript τ stands for the tangential component of the velocity, and ω is the angular velocity of the compressor.
As indicated in Equation 2, the wheel rim work grows as the increase of the circular velocity of the impeller but the peripheral velocity will lead to excessive centrifugal force.Therefore, retraining factors like material strength must be taken into account.Moreover, an increase in the peripheral velocity will also result in an increase in the relative velocity of the airflow at a constant absolute velocity c 1 , which will cause the shock wave and the shock wave-boundary layer influence in the supersonic situation.The absolute velocity c 1 is increased while keeping the compressor inlet area constant by increasing the axial velocity of the airflow inlet.This will increase the flow at the inlet and allow the turbojet engine to produce more thrust.However, the axial velocity c 1 should not be too high, since this will lead to a significant flow loss in the blade root part.

Design of geometric parameters
According to the design requirements, the one-dimensional design of the mixed-flow compressor is conducted to identify the geometric design parameters, including the number of main blades/shunt blades, the rotor exit radius, blade thickness, and the leading edge thickness of the mixed-flow compressor.The number of compressor blades should not be too large to avoid inlet blockage.Nevertheless, increasing the number of blades appropriately can improve the working efficiency of the compressor and reduce the blade load [2] .In conjunction with the enthalpy increase H, the impeller linear velocity v 2 , and the import total temperature T t1 , the number of blades can be selected by considering the following empirical equation [10] .
At the same time, the choice of blade thickness has a crucial impact on the performance of the compressor.The blade thickness of a mixed-flow compressor should be no less than 0.3% of the outlet diameter, depending on the process technology and application [11] .Given the acceleration of the inlet airflow at the front edge, the front edge thickness of the blade should be as thin as the material and process technologies allow [12] .The high-efficiency compressor impeller has been widely adopted in the form of long and short blades (shunt blades) impellers at home and abroad to reduce the impact loss of the compressor impeller inlet, the obstruction of the blade thickness to the inlet, and to avoid the large spacing between the two blades around the impeller outlet [13] .This design is also used in this paper.Since the working principles of micro turbojet and full-size turbojet engines are the same, the micro engine has unique characteristics known as the "size effect".It means the micro engine has less power, smaller flow, and smaller size for the same order of magnitude in thermodynamic parameters and aerodynamic velocity.The compressor impeller speed must be raised to increase fluid power and achieve a high-pressure ratio [14] .Therefore, the design speed of the compressor should be correspondingly increased.

Initial conditions
In the vortex viscosity equation, the RANS turbulence model and the Spalart-Allmaras model are used.When the number of iterations reaches 1, 500 steps or the global residual is less than 10 -6 , the CFL number of 3.0 indicates convergence.The total inlet temperature of 288.15 K, the total inlet pressure of 101, 325 Pa, axial intake, given static pressure value, and flow at the outlet are the boundary conditions.A smooth adiabatic non-slip condition is applied to the wall boundary.

Grid-independent validation
The quantity of grid division is directly related to the accuracy of the numerical simulation.Insufficient quantity will cause the calculation results to be distorted, while excessive quantity will take more computational resources.The grid density has to be changed constantly to evaluate the rationality of the grid density.When the calculation results vary by about 0.1%, it is considered that the influence of the grid density on the calculation results may be ignored [15] .
According to the parameters indicated in Table 2 and using AutoGrid5 to take 5 sets of data ranging from 2 million to 5 million grids, the isentropic efficiency and pressure ratio of the compressor achieved are displayed in Figure 3.It can be noticed that the changing rate of isentropic efficiency and pressure ratio between 2.3 million grids and above is not much different.In this research, 2.5 million grids are selected for CFD numerical simulation calculation.Figure 4 depicts the 3D grid model, which is of good quality and satisfies the requirements.

Characteristic curve
Figure 5 demonstrates the performance point results from simulation and characteristic curves at 90%, 100%, and 110% design speed.With the continuous increase of flow, the pressure ratio and efficiency grow first and then fall overall.By observing the flow-pressure ratio curve, it indicates the load capacity of the compressor blade is kept unchanged when the rotational speed remains unchanged but the flow rate decreases, or when the operating condition is close to the near stall point.Although the working margin is large at 100% design speed, the pressure ratio and efficiency show considerable attenuation after the design point, revealing that the work is not stable enough.Table 3 depicts the design results and design indexes at the design speed, and all of the indexes fulfill the design requirements.At 90% design speed, the pressure ratio is significantly reduced and the flow fluctuation is relatively gentle, which results in high work stability.Although the pressure ratio is increased at a design speed of 110%, the working margin is decreased.Besides, the characteristic curve has a steeper trend, the working state is unstable, and the compressor is easy to reach the unstable state.
Although the pressure ratio varies significantly depending on the working situation at each speed, the speed has little influence on the peak efficiency.The compressor can adapt to a range of working conditions, but the working stability is insufficient.

Design point flow field analysis
4.2.1.Relative Mach number of the S1 plane.Figure 6 illustrates the distribution of the Mach number of the S1 plane at 10%, 50%, and 90% leaf height under the design conditions to investigate the reason for the flow instability of the compressor at 100% design speed.At 10% blade height, there is low-energy gas accumulation near the front edge of the shunt blade pressure surface, and supersonic airflow forms a very thin shock wave at the front edge of the main blade suction surface.The airflow will slow down and raise pressure after accelerating on the suction surface of the main blade [16] , which is favorable for improving the pressure ratio.In addition, the flow in other areas is in the subsonic flow state.There is low-energy gas distribution towards the tail edge of the main blade and the shunt blade at 50% blade height.The sudden entropy increase and flow separation caused by the interference of the shock wave and boundary layer at the front edge of the main blade will reduce the efficiency of the compressor.At 90% blade height, the relative Mach number of the airflow at the inlet increase further, reaching 1.65.The shock wave range at the main blade suction surface front edge is further expanded, increasing the block of airflow.Furthermore, low-energy gas almost covers the main blade and shunt blade tail edge area.These changes significantly improve the pressure from the entrance to the outlet area, but the flow in the flow channel is also more obstacles, resulting in large flow loss and work instability of the compressor.4.2.2.Flow field analysis of the meridian flow surface.Figure 7 outlines the cloud diagram of the total temperature distribution of the meridian flow surface of the compressor at the design point.There is a temperature increase along the side of the flow channel at the entrance of the diffuser and it extends to the tail edge of the diffuser while observing the total temperature distribution of the fluid from the inlet of the compressor to the flow channel of the tail edge.The temperature between the center and the other side is lower.Therefore, the temperature differential on both sides will create a rotating airflow in the flow channel, hindering the normal flow of the fluid in the flow channel of the compressor.Figure 8 displays a cloud diagram of the velocity distribution of the meridian flow surface of the compressor.The airflow reaches its maximum velocity at the entrance of the impeller and there is a small part of reflux at the edge.The high-speed reflux will collide with the impeller blade, producing a lot of pneumatic noise.Since the power generated by the reflux is invalid power and the reflux gas will occupy a certain cross-sectional area of the flow channel, the cross-sectional area occupied along the flow channel by the normally flowing gas will decrease.It is not conducive to the normal flow of fluid in the flow channel and will reduce the efficiency of the compressor.Hence, the design can properly increase the number of blades to reduce the distance between the blade pressure surface and the suction surface to limit the generation of backflow in the flow channel.

Conclusion 1.
A mixed-flow compressor is designed and its performance parameters are obtained based on the fundamental velocity triangle, the Turbulent N-S equation mathematical model, and the S-A turbulence model.The flow is 1.5 kg•s -1 , the pressure ratio is 4.309, and the isentropic efficiency is 81.83%, which complies with the requirements of the given technical indexes.
2. In the mixed-flow compressor with three different speeds of the characteristic curve, the compressor with the speed near stall point efficiency is greater than the compressor with the speed near blockage point efficiency.At 100% design speed, the working margin has a higher equal entropy efficiency and pressure ratio, while the loss is lower.The compressor has higher reliability, stronger adaptability, and the best working performance under these circumstances.
3. According to the analysis of Mach numbers in various blades under design conditions, it is found that the instability of the mixed-flow compressor at the design point is primarily affected by the shock wave of the main blade at the leading edge as well as a large amount of low-energy gas gathered at the tail edge of the main blade and the shunt blade, which causes serious flow blockage.
4. Rotating air flow and reflux were discovered in the flow channel after analyzing the total temperature and fluid velocity of the radial flow surface of the compressor, which harms the efficiency of the compressor.
5. Analysis of the pressure study of different leaves under design conditions demonstrates that the loading of the inclined flow compressor gradually rises with the increase of leaf height, and its position is mostly concentrated in the leaf and tail areas.
Figure 2 displays the 3D model of the mixed-flow compressor created from one-dimensional design parameters.

Figure 5 .
Figure 5. Characteristic curves of the compressor.

Figure 6 .
Figure 6.Distribution of the relative Mach number of the S1 plane.

Figure 7 .
Figure 7. Absolute total temperature.Figure 8. Speed vector diagram of the meridian flow surface.

Figure 8 .Figure 9 .Figure 10 .
Figure 7. Absolute total temperature.Figure 8. Speed vector diagram of the meridian flow surface.4.2.3.Profile surface pressure.The blade static pressure cloud diagram may very clearly reflect the pressure loading situation of the pressure surface and the suction surface of the mixed-flow compressor blade.Figures 9 (a) and 10 (a) depict the pressure loading of the pressure and suction surfaces of the main blade, respectively.Figures 9 (b) and 10 (b) illustrate the pressure loading of the pressure surface and suction surface of the shunt blade, respectively.As seen from the figures, the loading mode is considerably different from that of the axial flow compressor.However, it is similar to the centrifugal compressor.The static pressure on the pressure surface and the suction surface of the blade is progressively growing.Moreover, the reverse pressure gradient is gradually increasing.When observing the static pressure distribution along the front edges of the main blade and the shunt blades, there is an obvious pressure drop in the front edge of the shunt blade.The main and shunt blade loads are both good.20%~60% and 60%~90% of the area are larger in the axial chord length direction,

Table 1 .
. Design requirements for the mixed-flow compressor.

Table 2 .
Settings of the mesh division parameters.

Table 3 .
Comparison of compressor design results and design indexes.