Theoretical Model and Numerical Analysis of the Tip Leakage Vortex Variations of a Centrifugal Compressor

Abstract

A centrifugal compressor of a micro turbine generator system is investigated by the theoretical model and numerical analysis to explore the characteristics of the tip leakage vortex as the centrifugal compressor approaches stall. The numerical simulation results show the cross-sectional shape of the tip leakage vortex is elliptical, and its long and short axes are gradually stretched as the compressor approaches stall. Moreover, the vortex trajectory is inclined to the pressure side of the adjacent blade. In addition, the Kirchhoff elliptical vortex model is introduced to analyze the flow passage constriction effect, the passage vortex squeezing effect, and the leakage flow translation effect. Results show that there is no upper limit for the flow passage constriction effect on the tip leakage vortex. Furthermore, relative to the original vortex, the minimum constriction effect depends on the axis ratio of the elliptical tip leakage vortex. The passage vortex has an expansion effect on the tip leakage vortex rather than a squeezing effect, which is limited and also depends on the axis ratio of the ellipse. However, the effect magnitude of the leakage flow depends on the scales both of the long and short axes, which also have no upper limit.

1. Introduction

Centrifugal compressors are widely used in various occasions as components or as an assembly, such as small aircraft propulsion systems, distributed power generation systems, automotive power units (turbochargers, fuel cell compression systems), etc. In order to meet the requirements of compact size and light weight, the design of the current centrifugal compressor must achieve a higher stage pressure ratio. In addition, the relevant design should also take into account high efficiency over the entire operating range. To meet these requirements, the current design trend is towards a highly loaded blade with a view to substantially increasing the stage pressure ratio. It is generally known that the design of high pressure ratio is prone to induce instability, such as stall, surge, etc., which may bring catastrophic damage to the entire equipment [1,2,3]. The design of a high-load centrifugal compressor with high efficiency and a wide stability margin is a major design challenge. Therefore, it is of great significance to study the stable operation of centrifugal compressors in the design stage.

Research about axial compressors has shown that many vortex structures in axial compressors, such as passage vortex [4], tip leakage vortex (TLV) [5], separation flow/vortex (blade corner [6], blade suction surface [7]), etc., have great influences on compressors’ performances. Different from an axial compressor, the blade tip flow structures of a centrifugal compressor often play a more important role in the performance of the compressor due to the influences of the curvature of the radial flow passage, centrifugal force, and load distribution characteristics, etc. Earlier studies, such as Ishida [8], by measuring the pressure at the shroud, suggested that the losses due to the tip clearance are related to the clearance ratio and the clogging caused by the relative speed reduction. Senoo [9], based on Ishida’s research, found that the presence of a tip clearance can cause a local pressure loss, which was verified experimentally with two centrifugal compressors. Harada [10] compared and analyzed the centrifugal impeller with and without tip clearance through experiments, and found that the existence of tip clearance reduces the stable working range of the centrifugal impeller. Recent research has also found these similar phenomena of centrifugal compressors. Chen [11] found that the leakage flow and the tip leakage vortex have a great influence on the performance of the compressor through the research of a centrifugal compressor for a fuel cell vehicle, and the compressor’s efficiency has been improved through optimization. Cao [12] found that the tip leakage vortex has a strong unsteady effect, which will cause the K–H instability of the shear layer and also cause the sharp fluctuation of the tip pressure, thus affecting the performance of the compressor. Tomita [13] conducted a study on a turbocharger compressor and found that the tip leakage vortex has strong fluctuations, and the performance of the compressor can be improved using flow control technology by inducing the tip leakage vortex breakdown. In addition to a TLV, Semlitsch [14] also found a back flow that induces a swirl on the incoming flow being entrained into the impeller, which influences the incidence angles and efficiencies of the blades significantly.

In fact, some measures have already been taken to improve the tip flow structures to improve the compressor performance, such as jet [15], ported shroud [16], self-recirculation casing [17], periodic excitation hole [18], etc. Although these successes have been achieved, the underlying mechanism is not clear enough, so people will have some questions when applying these measures. For example, Park [19] found that the control position is very sensitive to the results, which poses a great challenge to people when adopting such methods from the perspective of universality; For another example, Semlitsch [20] found that the tip flow field was improved with the ported shroud treatment, but the compressor efficiency was not increased obviously. In addition, many conclusions from these studies are also different [15,16,17,18,19,20]. The essence of these reasons is that the understanding of the tip flow structures is not clear enough at present. Therefore, a deeper understanding and mastery of the blade tip flow structures, especially the dynamic behavior of the TLV of the centrifugal compressor is of great significance to design schemes that can suppress TLV and improve the compressor performance.

To understand the inherencies of the TLV with experimental methods, many experimental procedures may need to be designed with massive data needing to be processed, which may bring a high cost. There are also many uncertainties, and it is difficult to be persuasive from the perspective of regularity recognition. While using the method of numerical simulation to summarize the regularities, there are many uncertain factors such as the experimental method, which will be difficult to get a deeper understanding as the researchers [15,16,17,18,19,20] adopted above. Additionally, the flow field structures in the centrifugal compressor are complex, and even if DMD or POD technology [14,21] is used, many dominant structures are still difficult to extract effectively. Therefore, an effective, concise, and regular mathematical and physical understanding would be an appropriate way for the study of TLV.

This paper conducts the research based on a centrifugal compressor of a micro turbine generator system. The main content of this paper can be divided into the following three parts: Firstly, the numerical simulation method of the compressor is introduced, and the results of which are discussed; Then, according to the features shown by the CFD results, a two-dimensional vortex model of the TLV is introduced; Finally, the built model is used to analyze the vortex’s behaviors under the influences of three typical external factors.

2. The Investigated Centrifugal Compressor and the Numerical Simulation Method

2.1. The Investigated Centrifugal Compressor

The investigated object is a transonic micro centrifugal compressor used in a micro turbo-generator of 35 kW power. The impeller is semi-open with backward bending that consists of 12 blades with an inlet tip diameter of 58.0 mm and an outer diameter of 108.0 mm. The blade height at the inlet and the outlet are 22.75 mm and 4.65 mm, respectively. The tip clearance is 0.3 mm. At the design point, the total pressure ratio is 3.2 with a mass flow rate of 0.3 kg/s. The design rotational speed is 80,000 RPM. The relative flow angles at the hub and the shroud of the leading edge are 68.0° and 25.7°, respectively, and both angles at the trailing edge are 35.1°. The impeller and the tip clearance gap were meshed using 153 and 17 elements in the span-wise direction. The computational grid for a single blade passage consists of 81 nodes in the blade-to-blade direction and 217 nodes in the stream-wise direction. Figure 1 shows the computational domain of about 3,800,000 grids in total. The Reynolds number is about 3 × 10⁵ and the non-dimensional wall distance y+ varies from 1−4.

2.2. The Numerical Simulation Methods

Three-dimensional steady-state simulation is carried out based on Reynolds-averaged Navier–Stokes equations by ANSYS/CFX. For simplification, calculations are performed only in the single rotor passage. The governing equation is discretized by the finite volume method with a two-order central difference scheme. Multigrid strategy and implicit residual smoothing are used to accelerate convergence. The total pressure and total temperature are given at inlet by 101325 Pa and 293 K. Normal flow direction is set to the inlet with a medium turbulence intensity. Ideal air model is selected. The solid wall is set to adiabatic no-slip condition. Counter rotating wall is employed for the shroud. An average static pressure over the whole outlet is given for the outlet boundary condition. The k–ε model is used as the turbulence model and the total-energy model is used as the heat transfer model. The calculation condition is gradually approached to the stall point by continuously increasing the back pressure, and the previous convergence condition when the calculation diverges is regarded as the near-stall condition. The compressor performance is shown in Figure 2. We believe that the reliability of this numerical simulation method is based on the following two points: 1. In previous studies by us and our former research group, CFX or Numeca/Fine are used to study the centrifugal compressor, and the results are in good agreement with the experiment [21,22,23]; 2. Other scholars used similar methods and settings to conduct numerical analysis of the compressor, which also showed good reliability [24,25]. Furthermore, our subsequent research focuses on the dominant structure of the TLV, rather than the synthesis of the various complex flows. When we began to plan to use an appropriate model to describe the state of the TLV, we realized that this is not an easy job, which is also a difficult problem faced and encountered by many researchers. Therefore, our intention is not to accurately describe the details of the TLV. In addition to the difficulty of a description, it is even impossible to distinguish the TLV from the surrounding flow field strictly, because the boundary between them is relatively fuzzy. In this case, our idea is to describe the main characteristics of the dominant TLV and analyze its performance under external influences. Therefore, what we really concern is a relative relationship, that is, the relationship between the dominant TLV and the external influences. To sum up, we believe that our calculation method is suitable for our research and has a certain degree of reliability.

3. Numerical Result Discussion

We selected four working conditions for comparative analysis. These four working conditions are the near-stall point, the quasi-stall point, the design point, and the large margin point, as shown in Figure 2, marked as S4, S3, S2, and S1. In this comparative analysis, we focus on the variations of the TLV. One of the reasons for this concern is that our and previous studies have shown that the TLV often plays a key role in the centrifugal compressor stall, and the second is that our subsequent modeling is also based on the TLV.

3.1. Cross-Sectional Characteristics

The cross-sectional structure and the influence scope of the TLV can be roughly visualized by the eddy viscosity and streamlines as shown in Figure 3. Looking further at these figures, we found, as in previous studies [26,27,28], that the flow structure at the blade tip of a centrifugal compressor is more complex than the root. Among these working conditions, the special one is the working condition S1 (the large margin point). Due to the influence of the negative attack angle, there are tip leakage vortices on both the pressure side and the suction side of the blade. By comparing all the TLV structures on the suction side, we can draw the following two main conclusions: one is that the structure of the TLV is roughly elliptical in cross-section, and the other is that the influence area of the TLV gradually expands as the compressor moves toward stall. The expansion range is reflected in both the long and short axes of the ellipse. In order to better explain the TLV expansion phenomenon, we make a rough quantitative comparison of the sizes of the vortices based on the cross-sectional structures presented by the eddy viscosity. Figure 4 shows the structures of the TLVs under these working conditions at the cross-section of about 15% chord length (S4 working condition is not shown in the figure because S4 and S3 are relatively close, and we also consider to avoid error interference). In this figure, we use the eddy viscosity threshold of 0.002 Pa·s to calculate the TLV area. The ratio of the TLV area to the inlet area under these three working conditions is about 1.46% (even including the TLV caused by the negative attack angle), 2.37%, and 12.33%, respectively.

Another important point is that in the quasi-stall condition S3 and the near-stall condition S4, a separation flow on the blade back is gradually aggravated, and eventually it will be mixed with the flow structure at the blade tip. Therefore, we can find that the elliptical areas affected by the TLVs under these two working conditions are somewhat damaged. However, in the other two working conditions, S1 and S2, the influences of the separation flows on the blade back are not so great, and the areas affected by the TLVs in these two working conditions present a very obvious regular narrow and long elliptical structure.

In fact, all the changes of the tip leakage vortex mentioned above are the result of the combined actions of various complex factors, and it is difficult for us to say whether it is due to the increased pressure or the cause of the early generation of the tip leakage vortex. Therefore, in subsequent analyses, we sought to extract the effects of some single factors on the tip leakage vortex in order to support the future adoption of some suitable compressor flow control techniques.

3.2. Trajectory Variation

In the previous section, we have explained the structure of the TLV, and here we intend to explain the movement of the TLV. Q-criteria and streamlines were applied to identify the TLVs, as shown in Figure 5. It can be seen from the figure that the trajectories of the TLVs in the S3 and S4 operating conditions are relatively similar. Unlike the other operating conditions, the trajectories of the TLVs in S3 and S4 are more close to the pressure side of the adjacent blades. The approximate quantitative TLV angles of three working conditions are shown in Figure 6 (S4 is not shown for the same reason as mentioned above). We define the TLV angle that is between the center line of the vortex shown by the Q-criterion and the chord line of the guide vane. The value of Q is 0.01 in Figure 6. The TLV angle under these three working conditions is about 15°, 18.2°, and 20.4°, respectively. This phenomenon is similar to the criterion derived by Vo et al. [29] from observations of axial compressor stalls. Since the TLV is generated at the induce vane of the centrifugal compressor, which is similar to the axial compressor in terms of the flow passage and the airfoil. Therefore, it is not surprising that the trajectory characteristics of centrifugal compressors are similar to those of axial compressors. It is also for this reason that the subsequent theoretical analysis in this paper is based on centrifugal compressors, but the conclusions can also be applicable to axial compressors. For the TLV trajectory movement, it is also the result of the combined actions of many factors, such as leakage flow, back pressure, wall viscous effect, attack angle, etc. However, from the final result of the cross-sectional shape of the TLV, it is equivalent that only the position of the TLV has changed. Thus, we approximate this change in position as translation motion, which we will use in the subsequent modeling analysis.

4. A Two-Dimensional Vortex Model for TLV

In different stages of the compressor development, various methods are used for stability analysis, such as mathematical model analysis, numerical simulation analysis, and integrated-machine experimental test. The first two methods are mainly used in the early design process. Since the 1950s, some researchers have successively established different mathematical models to analyze the unstable behaviors of the compressors, such as the concept of diffusion factor proposed by Lieblein [30], the approximate criterion of rotating stall proposed by Dunham [31], the Emmoons model [32], the Stenning model [33], the Greitzer model [34,35], etc. Although these models have achieved some success, there are still three main problems: 1. Many models are built on the cascade channel, which is difficult to be applied for centrifugal compressors; 2. Some models combine the compression system with the spring damping system. By analogy, the influences of the flows are ignored; 3. Part of them only consider one-dimensional effects, which has many limitations. In our previous studies, we also conducted a one-dimensional model to explain the cascade separation flow and its flow control mechanism [36]. We found that the model also encountered difficulties in analyzing the TLV of the centrifugal compressor.

Another method is to solve the Reynolds-averaged N–S equations, which is more widely used, especially in checking the characteristics of the compressor’s flow field. However, during the calculation of off-design conditions, especially the stall boundary, we usually judge whether the flow is stable or not by calculating the convergence. The flow is considered unstable if the calculation no longer converges by changing different outlet boundary conditions. However, this method lacks a solid theoretical basis. If this judgment method is strictly followed, it may even mislead the designers’ judgments in some cases. Therefore, to better understand the TLV, we introduce a mathematical model for analysis by referring to its behavior characteristics displayed by CFD results. Furthermore, we intend to combine the theoretical model and numerical analysis for further research.

4.1. Kirchhoff Elliptical Vortex

The region where the vorticity is concentrated in a two-dimensional inviscid flow field is called a vortex patch. For example, the vortex core of a Rankine vortex is the simplest circular vortex patch, which is often used by researchers to analyze some vortex structures. Generally speaking, the boundary shape of a vortex patch of any shape should change continuously when the vortex patch moves. There is a special case: an elliptical vortex patch with uniform vorticity will rotate around itself at a constant angular velocity and remain unchanged; this kind of vortex is called Kirchhoff elliptical vortex. The constant angular velocity Ω is expressed by the following formula:

$$\mathsf{\Omega }=\frac{\mathrm{a}\mathrm{b}}{{\left(\mathrm{a}+\mathrm{b}\right)}^2},$$

(1)

where a and b are the long and short axes of the ellipse, respectively. From the previous CFD results, the core region of the centrifugal compressor’s TLV is very close to the Kirchhoff elliptical vortex. Therefore, this paper does not use the process developed by many researchers based on circular vortex patch, but starts on the basis of the elliptical vortex.

Based on this ellipse vortex model, we can divide the flow field of the centrifugal compressor into the following two parts: inside the ellipse, the vorticity is known, and its velocity field is determined; the outside of the ellipse is equivalent to an elliptical column affected by the compressor flow field. Finally, the two parts are matched together to form the entire velocity field inside the centrifugal compressor.

4.2. Flow Function Outside the Elliptical Vortex

The governing equation of the flow function is:

$${\nabla }^2\mathsf{\psi }=0.$$

(2)

The ellipse boundary in Cartesian coordinates (x, y) is represented as:

$$\mathrm{F}\left(\mathrm{x},\mathrm{y}\right)=\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,$$

(3)

The unit outer normal vector of which can be written by:

$$n=\frac{\partial \mathrm{y}}{\partial \mathrm{s}}{e}_{\mathrm{x}}-\frac{\partial \mathrm{x}}{\partial \mathrm{s}}{e}_{\mathrm{y}},$$

(4)

where s is the surface arc length measured counterclockwise. When the elliptical cylinder rotates counterclockwise, the velocity of a point on the surface is $V_{\mathrm{b}}=\mathsf{\Omega }{e}_{\mathrm{z}}\times r$. Therefore, we can generate the normal velocity:

$$V_{\mathrm{b}}\cdot n=-\mathsf{\Omega }\mathrm{y}\frac{\partial \mathrm{y}}{\partial \mathrm{s}}-\mathsf{\Omega }\mathrm{x}\frac{\partial \mathrm{x}}{\partial \mathrm{s}}=-\mathsf{\Omega }\mathrm{r}\frac{\partial \mathrm{r}}{\partial \mathrm{s}},$$

(5)

where ${\mathrm{r}}^2={\mathrm{x}}^2+{\mathrm{y}}^2$. On the other hand, if we express the normal velocity with a flow function. Then, the normal velocity has the form:

$$V_{\mathrm{b}}\cdot n=\frac{\partial \mathsf{\psi }}{\partial \mathrm{s}}{n}_{\mathrm{x}}-\frac{\partial \mathsf{\psi }}{\partial \mathrm{s}}{n}_{\mathrm{y}}=\frac{\partial \mathsf{\psi }}{\partial \mathrm{y}}\frac{\partial \mathrm{y}}{\partial \mathrm{s}}+\frac{\partial \mathsf{\psi }}{\partial \mathrm{x}}\frac{\partial \mathrm{x}}{\partial \mathrm{s}}=\frac{\partial \mathsf{\psi }}{\partial \mathrm{s}}.$$

(6)

Comparing Equations (5) and (6), we can obtain:

$$\frac{\partial \mathsf{\psi }}{\partial \mathrm{s}}=-\mathsf{\Omega }\mathrm{r}\frac{\partial \mathrm{r}}{\partial \mathrm{s}}.$$

(7)

After integrating the above equation, the surface flow function expression can be obtained as:

$${\mathsf{\psi }}_{\mathrm{b}}=-\frac{1}{2}\mathsf{\Omega }{\mathrm{r}}^2+\mathrm{const},$$

(8)

Which has a more concise form if we use elliptic coordinates defined by:

$$\{\begin{array}{l}\mathrm{x}=\mathrm{c}\cosh \mathsf{\xi }\cos \mathsf{\eta }\\ \mathrm{y}=\mathrm{c}\sinh \mathsf{\xi }\sin \mathsf{\eta }\end{array},$$

(9)

where the elliptic focus c² = a² − b² and on the ellipse border ξ = ξ₀. According to the CFD results, it can be obtained that the velocity on the boundary of the elliptical vortex is very close to the blade tip velocity due to the influence of the viscous force and the relative movement of the shroud. Therefore, we assume:

$$\begin{array}{l}{{\mathsf{\psi }}^{\prime }}_{\mathsf{\xi }}|{}_{\mathsf{\xi }={\mathsf{\xi }}_0}=-\frac{1}{2}\mathsf{\Omega }{\left({\mathrm{c}}^2{\cosh }^2\mathsf{\xi }{\cos }^2\mathsf{\eta }+{\mathrm{c}}^2{\sinh }^2\mathsf{\xi }{\sin }^2\mathsf{\eta }\right)}_{\mathsf{\xi }}{}^{\prime }|{}_{\mathsf{\xi }={\mathsf{\xi }}_0}\\ =-\mathsf{\Omega }{\mathrm{c}}^2\cosh \mathsf{\xi }\sinh \mathsf{\xi }|{}_{\mathsf{\xi }={\mathsf{\xi }}_0}=\mathrm{U}\end{array}$$

(10)

Then, we substitute:

$$\{\begin{array}{l}\mathrm{a}=\mathrm{c}\cosh {\mathsf{\xi }}_0\\ \mathrm{b}=\mathrm{c}\sinh {\mathsf{\xi }}_0\end{array},$$

(11)

Into the above formula to get:

$$-\mathsf{\Omega }\mathrm{a}\mathrm{b}=-\mathrm{U},$$

(12)

where U is approximately equal to the rotational velocity of the blade tip. Since the rotational direction of the vortex is opposite to that of the blade, a minus sign should be added in front of U. Combining Equations (8) and (12) we can obtain the flow function expression on border:

$${\mathsf{\psi }}_{\mathrm{b}}=-\frac{1}{2}\mathsf{\Omega }{\mathrm{c}}^2\cos 2\mathsf{\eta }+\mathrm{const}=-\frac{1}{2}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2\cos 2\mathsf{\eta }+\mathrm{const}.$$

(13)

Furthermore, the external flow function should simultaneously satisfy the governing Equation (2), the surface boundary condition (13), and the condition that the velocity at infinity is 0, so its solution should be of the form:

$$\mathsf{\psi }=\mathrm{A}{\mathrm{e}}^{-2\mathsf{\xi }}\cos 2\mathsf{\eta }.$$

(14)

Comparing Equations (13) with (14), we can obtain the constant:

$$\mathrm{A}=-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2{\mathrm{e}}^{2{\mathsf{\xi }}_0},$$

(15)

Then, the external flow function becomes:

$$\mathsf{\psi }=-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2{\mathrm{e}}^{2{\mathsf{\xi }}_0}{\mathrm{e}}^{-2\mathsf{\xi }}\cos 2\mathsf{\eta }.$$

(16)

Considering that the elliptical vortex has a circulation Γ = πabω, that is, the vortex flux passing through the cross-sectional area of the ellipse, the external flow function can finally be expressed as:

$${\mathsf{\psi }}^{\left(\mathrm{o}\right)}=-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2{\mathrm{e}}^{2{\mathsf{\xi }}_0}{\mathrm{e}}^{-2\mathsf{\xi }}\cos 2\mathsf{\eta }-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }\left(\mathsf{\xi }-{\mathsf{\xi }}_0\right),$$

(17)

where superscript o in the formula means external and ω is vorticity.

4.3. Flow Function Inside the Elliptical Vortex

The governing equation of the flow function is:

$${\nabla }^2\mathsf{\psi }=-\mathsf{\omega }.$$

(18)

We choose a flow function of the form:

$${\mathsf{\psi }}^{\left(\mathrm{i}\right)}=\frac{1}{2}\mathsf{\omega }\left(\mathrm{A}{\mathrm{x}}^2+\mathrm{B}{\mathrm{y}}^2\right),$$

(19)

which is the solution of Equation (18). By plugging it into Equation (18), we get:

$$\mathrm{A}+\mathrm{B}=-1.$$

(20)

There are three undetermined constants in this problem. In addition to the above equation, the other two related equations can be determined by the continuity of the normal and tangential velocity components on the boundary of the inner and outer flow. The normal velocity component is:

$$V\cdot n=\frac{\partial \mathsf{\psi }}{\partial \mathrm{y}}\frac{\partial \mathrm{F}}{\partial \mathrm{x}}-\frac{\partial \mathsf{\psi }}{\partial \mathrm{x}}\frac{\partial \mathrm{F}}{\partial \mathrm{y}}.$$

(21)

Thus, the second related equation can be determined by the continuity according Equations (3), (8), and (19), that is:

$$\mathrm{A}{\mathrm{a}}^2-\mathrm{B}{\mathrm{b}}^2=\frac{\mathsf{\Omega }}{\mathsf{\omega }}\left({\mathrm{b}}^2-{\mathrm{a}}^2\right).$$

(22)

From the continuity of tangential velocity:

$${{\psi }^{\prime }}_{\xi }^{\left(o\right)}|{}_{\xi ={\xi }_0}={{\psi }^{\prime }}_{\xi }^{\left(i\right)}|{}_{\xi ={\xi }_0},$$

(23)

which holds true for any value of η, we can get the last relation:

$$\mathrm{A}-\mathrm{B}=\frac{\mathsf{\Omega }}{\mathsf{\omega }}\frac{\left({\mathrm{a}}^2-{\mathrm{b}}^2\right)}{\mathrm{a}\mathrm{b}}.$$

(24)

Combining Equations (20), (22), and (24), we get the result of the three undetermined constants:

$$\mathrm{A}=-\frac{\mathrm{b}}{\mathrm{a}+\mathrm{b}},\mathrm{B}=-\frac{\mathrm{a}}{\mathrm{a}+\mathrm{b}},\mathsf{\Omega }=\frac{\mathrm{a}\mathrm{b}}{{\left(\mathrm{a}+\mathrm{b}\right)}^2}\mathsf{\omega }.$$

(25)

The internal flow function finally is:

$$\begin{array}{l}{\mathsf{\psi }}^{\left(\mathrm{i}\right)}=-\frac{1}{2}\mathsf{\omega }\frac{{\mathrm{c}}^2}{\mathrm{a}+\mathrm{b}}\left(\mathrm{b}{\cosh }^2\mathsf{\xi }{\cos }^2\mathsf{\eta }+\mathrm{a}{\sinh }^2\mathsf{\xi }{\sin }^2\mathsf{\eta }\right)\\ =-\frac{1}{2}\frac{\mathrm{U}\left(\mathrm{a}+\mathrm{b}\right){\mathrm{c}}^2}{{\mathrm{a}}^2{\mathrm{b}}^2}\left(\mathrm{b}{\cosh }^2\mathsf{\xi }{\cos }^2\mathsf{\eta }+\mathrm{a}{\sinh }^2\mathsf{\xi }{\sin }^2\mathsf{\eta }\right)\end{array}$$

(26)

5. Vortex Behaviors under Imposed Strain

The content of this section is to use the model introduced in the previous section to analyze TLV characteristics under typical strains. The flow field inside a centrifugal compressor is intricate. Therefore, generally speaking, it is difficult to describe the strain field with an accurate model. However, we can consider some basic strain fields. On this basis, we can try to analyze the combination of different strain fields, and use these combinations to help analyze the influences of the complex flow field of the compressor on the TLV, and finally look into the dynamic behavior of the vortex. In this section, we analyze three basic effects as follows. More other basic effects can also be analyzed in the following way.

5.1. Flow Passage Constriction Effect

When the airflow in the centrifugal compressor moves from the guide vane to the outlet, the flow passage gradually shrinks in the span direction. We assume that this effect is equivalent to being acted upon by a virtual force, denoted as F^image shown in Figure 7. On the meridian plane, this virtual force can be decomposed along the radial and axial directions, and the decomposed virtual forces are denoted as ${\mathrm{F}}_{{\mathrm{x}}^{\prime }}^{\mathrm{image}}$ and ${\mathrm{F}}_{{\mathrm{y}}^{\prime }}^{\mathrm{image}}$, respectively. As shown in the figure, the effect of the radial component on the elliptical vortex can be equivalent to being compressed on the short axis of the elliptical vortex. Meanwhile, the long axis is subject to a stretching effect due to the continuity of the fluid. In fact, this effect should include a radial translation effect. However, here we only analyze the constriction effect. That is because the translation effect is also affected by other factors, such as meridian curvature. Another reason, as mentioned earlier, is that complex behaviors can be analyzed by superimposing simple behaviors.

The effect of the above virtual force can be transformed by introducing the following matrix:

$$V=\left[\begin{array}{cc}\mathrm{e}& 0\\ 0& -\mathrm{e}\end{array}\right],$$

(27)

whose corresponding flow function is:

$$\begin{array}{l}\mathsf{\psi }=\mathrm{e}\mathrm{x}\mathrm{y}=\mathrm{e}{\mathrm{c}}^2\left(\cosh \mathsf{\xi }\cos \mathsf{\eta }+\sinh \mathsf{\xi }\sin \mathsf{\eta }\right)\\ =\frac{1}{4}\mathrm{e}{\mathrm{c}}^2\sinh 2\mathsf{\xi }\sin 2\mathsf{\eta }\end{array}$$

(28)

where e is a strain rate strength that is determined here by the geometry of the flow passage. According to the superposition principle of the harmonic functions, after the strain effect is applied to the original external flow function, the new external flow function becomes:

$${\mathsf{\psi }}_{\mathrm{c}\mathrm{e}}^{\left(\mathrm{o}\right)}=\frac{1}{4}\mathrm{e}{\mathrm{c}}^2\sinh 2\mathsf{\xi }\sin 2\mathsf{\eta }-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2{\mathrm{e}}^{2{\mathsf{\xi }}_0}{\mathrm{e}}^{-2\mathsf{\xi }}\cos 2\mathsf{\eta }-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }\left(\mathsf{\xi }-{\mathsf{\xi }}_0\right),$$

(29)

where the subscript ce means constriction effect. Continuity conditions for the flow function and tangential velocity on the ellipse boundary are satisfied that provides:

$$\begin{array}{l}\frac{1}{4}\mathrm{e}{\mathrm{c}}^2\sinh 2{\mathsf{\xi }}_0\sin 2\mathsf{\eta }-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2\cos 2\mathsf{\eta }\\ =-\frac{1}{2}\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{a}{\cos }^2\mathsf{\eta }+\mathrm{b}{\sin }^2\mathsf{\eta }\right)\end{array}$$

(30)

$$\begin{array}{l}\frac{1}{2}\mathrm{e}{\mathrm{c}}^2\cosh 2{\mathsf{\xi }}_0\sin 2\mathsf{\eta }+\frac{1}{2}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2\cos 2\mathsf{\eta }-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }\\ =-\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{b}{\cos }^2\mathsf{\eta }+\mathrm{a}{\sin }^2\mathsf{\eta }\right)\end{array}$$

(31)

We can multiply Equation (30) by 2 and add it to Equation (31) to eliminate the second term to get:

$$\frac{1}{2}\mathrm{e}{\mathrm{c}}^2\sin 2\mathsf{\eta }\left(\sinh 2{\mathsf{\xi }}_0+\cosh 2{\mathsf{\xi }}_0\right)-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }=-\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{a}+\mathrm{b}\right)\left({\cos }^2\mathsf{\eta }+{\sin }^2\mathsf{\eta }\right),$$

(32)

which can be further simplified to:

$$\mathrm{e}/\mathsf{\omega }=\frac{-\mathrm{a}\mathrm{b}}{\sin 2\mathsf{\eta }{\left(\mathrm{a}+\mathrm{b}\right)}^2}=\frac{-1}{\sin 2\mathsf{\eta }\left(\mathsf{\epsilon }+1/\mathsf{\epsilon }+2\right)},$$

(33)

where ε = a/b is the axis ratio of the ellipse. In fact, when we introduced the strain rate matrix, e was already set to be positive. Therefore, it is easy to obtain the minimum value of $\mathrm{e}/\mathsf{\omega }$ from the above formula is $1/\left(\mathsf{\epsilon }+1/\mathsf{\epsilon }+2\right)$. That is to say, inside the centrifugal compressor, there is no upper limit for the compression effect of the flow passage on the TLV, and relative to the original vortex, the minimum constriction effect depends on the axis ratio of the elliptical TLV.

5.2. Passage Vortex Squeeze Effect

In the centrifugal compressor passage, the size of the passage vortex is often larger than other vortices, so it has a greater impact on the TLV. Therefore, here, we take an example on analyzing the effect of the passage vortex. It can be seen from Figure 8 that the generation of the large-scale passage vortex will squeeze the tip leakage vortex’s space. As with the previous processing method, we equivalently apply this squeezing effect as adding a virtual force in the direction of the arrow shown in Figure 8, denoted by ${\mathrm{F}}_{\mathrm{s}\mathrm{e}}^{\mathrm{image}}$.

The effect of this squeezing can be achieved by introducing the following matrix:

$$V=\left[\begin{array}{cc}0& -\mathrm{e}\\ -\mathrm{e}& 0\end{array}\right],$$

(34)

whose corresponding flow function is:

$$\mathsf{\psi }=\frac{1}{2}\mathrm{e}\left({\mathrm{x}}^2-{\mathrm{y}}^2\right)=\frac{1}{2}\mathrm{e}{\mathrm{c}}^2\left({\cosh }^2\mathsf{\xi }{\cos }^2\mathsf{\eta }+{\sinh }^2\mathsf{\xi }{\sin }^2\mathsf{\eta }\right),$$

(35)

where e is a strain rate strength that is determined here by the strength of the passage vortex. Once again, we can get a new flow function by superimposing the above flow function on the external flow function that becomes:

$$\begin{array}{l}{\mathsf{\psi }}_{\mathrm{s}\mathrm{e}}^{\left(\mathrm{o}\right)}=\frac{1}{2}\mathrm{e}{\mathrm{c}}^2\left({\cosh }^2\mathsf{\xi }{\cos }^2\mathsf{\eta }+{\sinh }^2\mathsf{\xi }{\sin }^2\mathsf{\eta }\right)\\ -\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2{\mathrm{e}}^{2{\mathsf{\xi }}_0}{\mathrm{e}}^{-2\mathsf{\xi }}\cos 2\mathsf{\eta }-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }\left(\mathsf{\xi }-{\mathsf{\xi }}_0\right)\end{array}$$

(36)

where the subscript se means squeeze effect. Again, according to the continuity condition, we can get:

$$\frac{1}{2}\mathrm{e}\left({\mathrm{a}}^2{\cos }^2\mathsf{\eta }+{\mathrm{b}}^2{\sin }^2\mathsf{\eta }\right)-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2\cos 2\mathsf{\eta }=-\frac{1}{2}\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{a}{\cos }^2\mathsf{\eta }+\mathrm{b}{\sin }^2\mathsf{\eta }\right),$$

(37)

$$\mathrm{e}\mathrm{a}\mathrm{b}+\frac{1}{2}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2\cos 2\mathsf{\eta }-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }=-\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{b}{\cos }^2\mathsf{\eta }+\mathrm{a}{\sin }^2\mathsf{\eta }\right).$$

(38)

The above two equations can be simplified to:

$$\mathrm{e}\left({\mathrm{a}}^2{\cos }^2\mathsf{\eta }+{\mathrm{b}}^2{\sin }^2\mathsf{\eta }\right)+\mathrm{e}\mathrm{a}\mathrm{b}-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }=-\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{a}+\mathrm{b}\right)\left({\cos }^2\mathsf{\eta }+{\sin }^2\mathsf{\eta }\right).$$

(39)

Therefore, the ratio of strain rate to vorticity is:

$$\mathrm{e}/\mathsf{\omega }=-\frac{1}{2\left(\left(\mathsf{\epsilon }-\frac{1}{\mathsf{\epsilon }}\right){\cos }^2\mathsf{\eta }+1+\mathsf{\epsilon }\right)},$$

(40)

whose value range is between −1/2(2ε − 1/ε + 1) and −1/2(1 + ε). From this value, it can be seen that the squeezing effect of the passage vortex on the TLV is opposite to the direction we set earlier, indicating that the squeezing of the passage vortex has a stretching effect. Therefore, we can infer that the entrainment effect of the passage vortex has a greater effect on the TLV rather than squeezing effect inside the compressor, and the TLV expands along this effect. Meanwhile, we can conclude that the expansion of the TLV or the entrainment effect of the passage vortex has a range from −1/2(2ε − 1/ε + 1) to −1/2(1 + ε) relative to the TLV, and this value also depends on the axis ratio of the ellipse.

5.3. Leakage Flow Translation Effect

As mentioned above, the movement of the TLV trajectory is the result of many factors, but from the perspective of lateral displacement, the effect of the leakage flow is more directly compared with other factors. Therefore, what we discuss in this section is the effect of the leakage flow, as shown in Figure 9.

Assuming that the leakage flow intensity is constant in the tip clearance, we can write the flow function corresponding to the translation action by:

$$\mathsf{\psi }=\mathrm{A}\mathrm{x}-\mathrm{B}\mathrm{y}+\mathrm{C},$$

(41)

where A, B, and C are constants determined by the strength of the leakage flow. Since our analysis takes the long axis of the ellipse vortex as the x-axis, a simpler flow function can be written as:

$$\mathsf{\psi }=-\mathrm{B}\mathrm{y}.$$

(42)

We replace B with e, for notation consistency with the previous analysis. The external flow function becomes:

$${\mathsf{\psi }}_{\mathrm{l}\mathrm{f}\mathrm{e}}^{\left(\mathrm{o}\right)}=-\mathrm{e}\mathrm{c}\sinh \mathsf{\xi }\sin \mathsf{\eta }-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2{\mathrm{e}}^{2{\mathsf{\xi }}_0}{\mathrm{e}}^{-2\mathsf{\xi }}\cos 2\mathsf{\eta }-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }\left(\mathsf{\xi }-{\mathsf{\xi }}_0\right),$$

(43)

where the subscript lfe means leakage flow effect. Using the continuity condition, we can get:

$$-\mathrm{e}\mathrm{b}\sin \mathsf{\eta }-\frac{1}{4}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2\cos 2\mathsf{\eta }=-\frac{1}{2}\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{a}{\cos }^2\mathsf{\eta }+\mathrm{b}{\sin }^2\mathsf{\eta }\right),$$

(44)

$$-\mathrm{e}\mathrm{a}\sin \mathsf{\eta }+\frac{1}{2}\frac{\mathrm{U}}{\mathrm{a}\mathrm{b}}{\mathrm{c}}^2\cos 2\mathsf{\eta }-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }=-\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{b}{\cos }^2\mathsf{\eta }+\mathrm{a}{\sin }^2\mathsf{\eta }\right).$$

(45)

The above two equations can be combined as:

$$-\mathrm{e}\sin \mathsf{\eta }\left(2\mathrm{b}+\mathrm{a}\right)-\frac{1}{2}\mathrm{a}\mathrm{b}\mathsf{\omega }=-\mathsf{\omega }\frac{\mathrm{a}\mathrm{b}}{\mathrm{a}+\mathrm{b}}\left(\mathrm{a}+\mathrm{b}\right)\left({\cos }^2\mathsf{\eta }+{\sin }^2\mathsf{\eta }\right).$$

(46)

Finally, we get the ratio of strain rate to vorticity:

$$\mathrm{e}/\mathsf{\omega }=\frac{1}{2\sin \mathsf{\eta }}\mathrm{a}\mathrm{b}/\left(2\mathrm{b}+\mathrm{a}\right),$$

(47)

whose minimum absolute value is ab/2(2b + a). It reflects the influence of the leakage flow on the TLV trajectory which has no upper limit as with the influence of the aforementioned passage compression effect. Additionally, relative to the original vortex, the minimum effect depends on both sizes of the long axis and short axis of the elliptical vortex.

6. Conclusions

In order to explore the characteristics of the tip leakage vortex of a centrifugal compressor as it approaches stall, this paper conducts the research based on a centrifugal compressor of a micro turbine generator system with a theoretical model and numerical analysis. The conclusions are as follows:

(1)

The numerical simulation results show that the flow structure at the blade tip of the centrifugal compressor is more complicated than other regions. The cross-sectional shape of the tip leakage vortex is elliptical, and its long and short axes are gradually stretched as the compressor approaches the stall. Moreover, the vortex trajectory is also gradually inclined to the pressure side of the adjacent blade;

(2)

We introduced Kirchhoff elliptical vortex and combined numerical simulation and the flow characteristics of the centrifugal compressor to analyze the behaviors of the tip leakage vortex. We analyzed three typical cases that have important influences on the tip leakage vortex, namely the flow passage constriction effect, the passage vortex squeeze effect, and the leakage flow translation effect. The results show there is no upper limit for the flow passage constriction effect on the tip leakage vortex, and relative to the original vortex, the minimum constriction effect depends on the axis ratio of the elliptical tip leakage vortex. The passage vortex has an expansion effect on the tip leakage vortex rather than a squeezing effect, which is limited and also depends on the axis ratio of the ellipse. However, the effect magnitude of the leakage flow on the tip leakage vortex depends on the scales of the long and short axes, which also have no upper limit as with the influence of the passage constriction effect. As for other influencing factors and combined effects, further research is needed in the future.