A Numerical Study on Leakage Flow in the Shroud Cavity of a Centrifugal Impeller

Abstract

A high-pressure centrifugal compressor with a shroud cavity is a key piece of energy equipment found in a compressed air energy storage (CAES) system. The leakage flow of the shroud cavity is an important factor affecting the efficiency of the system. Using systematic numerical studies, the influence of leakage flow on high-pressure centrifugal compressors under different leakage rates has been comprehensively elucidated. The study reveals that the leakage flow significantly increases the circumferential swirl angle of the flow at the impeller inlet, thereby exerting a significant influence on the flow characteristics within the impeller. As the leakage rate increases, the stable operating range of the compressor is significantly reduced, with the total pressure ratio decreasing by up to 2% and the isentropic efficiency dropping by up to 4 percentage points.

1. Introduction

With the depletion of fossil fuels, energy storage technology is playing an increasingly important role [1]. Compressed air energy storage (CAES) technology is a crucial supporting technology for distributed energy systems, smart grids, and microgrid advancements. It addresses the challenges of intermittency and grid instability posed by renewable energy sources, such as photovoltaic electricity and wind power [2,3]. CAES provides essential technical support for grid peak shaving and valley filling, significantly enhancing the integration of intermittent renewable energy sources and improving waste heat utilization efficiency. As a result, CAES is widely recognized as one of the most promising large-scale physical energy storage solutions [4,5,6,7].

The centrifugal compressor is the core component of the compressed air energy storage system, efficiently converting electrical energy into the internal and potential energy of air. The impellers of centrifugal compressors are usually of two types: semi-open impellers and shrouded (closed) impellers. Semi-open impellers have the advantages of low cost and ease of manufacturing, while shrouded impellers offer higher flow efficiency and better aerodynamic performance. High-pressure centrifugal compressors tend to have smaller sizes and lower manufacturing costs, making them suitable for adopting shrouded impellers [8]. The leakage within the impeller shroud cavity is a significant factor in determining compressor efficiency. Mitigating air leakage and minimizing the effects of leakage flow are crucial for enhancing compressor efficiency, improving operational reliability and safety and reducing energy consumption and costs [9].

Extensive research has been conducted on shroud cavity leakage in compressors. Babin et al. believe that the influence of leakage flow cannot be neglected in closed impeller designs [10]. Meng et al. [11] utilized numerical simulations to demonstrate the detrimental effects of leakage flow on compressor efficiency and pressure ratio. Prasad et al. [12] investigated the leakage characteristics of straight-through labyrinth seals under varying pressure ratios and radial clearances. Basol et al. [13] and Wang et al. [14] concentrated on optimizing seal structures to curtail leakage through the cavity. Mischo et al. [15] explored the impact of leakage flow injection direction within the impeller’s internal flow passage, revealing its influence on inlet flow turbulence viscosity. The interaction between gap leakage and the mainstream is complex [16]. Huang et al. believe that the leakage flow enhances the effect of the secondary flow within the impeller [17]. However, as the seal teeth wear during the operation of the compressor, the seal gap will gradually increase, leading to a significant increase in leakage [18]. Therefore, research on leakage conditions for different seal gaps is necessary, but there is currently a lack of research in this field.

2. Numerical Methods and Computational Models

2.1. Research Object

The research object of this paper’s research is a centrifugal compressor utilized in a CAES system, with its main flow passage structure and cavity structure illustrated in Figure 1. The main flow passage consists of a shrouded impeller with 13 blades and a diffuser with 10 blades. The inlet radius of the impeller is 51.6 mm, the flow passage width at the inlet is 34.1 mm, the outlet radius is 85.5 mm, and the flow passage width at the outlet is 10.1 mm.

2.2. Method and Verification

2.2.1. Numerical Method

In this paper, the Computational Fluid Dynamics (CFD) software CFX 2020 R1 is used to conduct steady numerical simulation analysis. The meshes of the impeller and diffuser are generated using CFX TurboGrid 2020R1 software. The meshes of the cavity and sealing areas are generated using ICEM CFD 2020 R1 software. All the meshes are structured hexahedrally. In the regions close to blade surfaces and wall, the first boundary layer is set to 10⁻⁶ m. In order to meet the requirements of the SST (Shear Stress Transport) turbulence model, the maximum y+ value of the first boundary layer is below 5.

To simplify the calculations, the computational models in this paper all utilize a single-passage model. The working fluid is real gas. The compressor rotates at a speed of 28,903 rpm. The inlet of the impeller is configured so that the intake direction is parallel to the rotational axis. The total pressure at the inlet is 5.5 MPa, the total temperature at the inlet is 303 K, and the turbulence intensity is set as 5%. The outlet boundary condition is set as a mass flow rate. The rotor–stator interface adopts the Stage (Mixing-Plane) method, and the wall boundary condition is specified as a no-slip condition. By varying the outlet mass flow rate, calculations are performed for different operating points on the characteristic curve. In steady state calculations, when the mass flow rate error between the impeller inlet and the diffuser outlet for a certain calculation step is less than 0.5 percent, and all residual values are less than 10⁻⁵, the calculation can be considered as converged. The near-stall point is identified using the bisection method [19].

2.2.2. Numerical Method Verification

The reliability of the numerical simulation method is validated by selecting the NASA LSCC impeller [20] as a benchmark case, and a comparative analysis was conducted between the numerical simulations and the experimental data published by NASA [21]. The design flow rate of the NASA LSCC impeller is 30 kg/s. The total pressure at the inlet is 1 bar, and the total temperature at the inlet is 288.15 K. When comparing the results obtained using the aforementioned numerical method for the NASA LSCC with the experimental results published by NASA, good agreement was observed between the two. As shown in Figure 2, under design conditions, the relative error between the experimental and simulated isentropic efficiencies is 0.71%, and the relative error in the total pressure ratio is 0.86%. Both are within the allowable range of less than 2%. These results indicate that the numerical method used in this paper is accurate and reliable, capable of effectively predicting the performance of the model, and thus suitable for subsequent research and analysis.

2.2.3. Mesh Independence Verification

To balance computational efficiency and accuracy, this study conducted a mesh independence verification on the computational model to determine an optimal mesh count that ensures both high accuracy and improved computational efficiency.

Using the isentropic efficiency as an evaluation indicator, we conducted mesh independence verification for both the impeller and the diffuser. Similarly, using the leakage rate as an evaluation indicator, we conducted a mesh independence verification for the cavity. Karim et al. presented a method for verifying mesh independence using the Grid Convergence Index (GCI) parameter [22]. The GCI parameters corresponding to different mesh counts are shown in the figure below. When the GCI is less than 3% [23], it can be considered that this mesh count can ensure the accuracy of the calculation results. The GCI of mesh series are shown in Figure 3. Therefore, the impeller and diffuser structure with 1.79 million mesh elements and the cavity with 1.77 million mesh elements is ultimately selected as the model for numerical computation.

2.3. Equations

The governing equations of the SST turbulence model are as follows [24]:

Continuity equation:

$${\displaystyle \frac{\partial \rho }{\partial t}}+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho u_i\right)=0$$

(1)

Momentum equation:

$${\displaystyle \frac{\partial }{\partial t}}\left(\rho u_i\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho u_i{u}_j\right)=-{\displaystyle \frac{\partial p}{\partial x_i}}+{\displaystyle \frac{\partial }{\partial x_j}}\left(\rho \nu {\displaystyle \frac{\partial u_i}{\partial x_j}}\right)+{\displaystyle \frac{\partial }{\partial x_j}}\left(-\rho \overline{u_i^{\prime }{u}_j^{\prime }}\right)$$

(2)

The Reynolds stress applies the Boussinesq hypothesis, which states that the Reynolds stress is proportional to the average velocity gradient:

$$-\rho \overline{u_i^{\prime }{u}_j^{\prime }}={\mu }_t\left({\displaystyle \frac{\partial u_i}{\partial x_j}}+{\displaystyle \frac{\partial u_j}{\partial x_i}}\right)-{\displaystyle \frac{2}{3}}\left(\rho k+\mu {\displaystyle \frac{\partial u_i}{\partial x_i}}\right){\delta }_{ij}$$

(3)

Equations (4)–(6) define the dimensionless flow rate, total pressure ratio, and isentropic efficiency, respectively.

$$m_{nor}={\displaystyle \frac{m}{m_{design}}}$$

(4)

$${\pi }_t={\displaystyle \frac{P_{t,02}}{P_{t,01}}}$$

(5)

$${\eta }_{isen}={\displaystyle \frac{h_{\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{n}}}{h_{02}-h_{01}}}$$

(6)

Equation (7) defines the stall margin (SM).

$$SM=\left({\displaystyle \frac{{\pi }_{t,stall}{m}_{design}}{{\pi }_{t,design}{m}_{stall}}}-1\right)100\mathrm{\%}$$

(7)

The nondimensional circumferential velocity at a point and the normal velocity of a plane inside the compressor are defined as follows:

$$V_c^{*}=V_c/\left(2\pi \omega r\right)$$

(8)

$$V_n^{*}=(V_a\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{\pi }{4}}+V_r\mathrm{s}\mathrm{i}\mathrm{n}{\displaystyle \frac{\pi }{4}})/(2\pi \omega r)$$

(9)

Equation (10) defines the swirling angle (SA_c).

$${SA}_c=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}{\displaystyle \frac{{\mathrm{V}}_c}{{\mathrm{V}}_a}}$$

(10)

Khabibova [25] et al. defined the depreciation of energy as dissipation in an isolated system based on the second law of thermodynamics, and provided a detailed derivation process of the dissipation function. In a compressor, when secondary factors such as heat transfer and radiation are neglected, flow loss becomes the primary mechanism causing an entropy increase. The tensor form of the dissipation function that reflects flow losses is defined by Equation (11).

$$\begin{array}{c}\varphi ={\displaystyle \frac{2\mu }{\rho }}\left[{\left({\displaystyle \frac{\partial u_1}{\partial x_1}}\right)}^2+{\left({\displaystyle \frac{\partial u_2}{\partial x_2}}\right)}^2+{\left({\displaystyle \frac{\partial u_3}{\partial x_3}}\right)}^2\right]-{\displaystyle \frac{2\mu }{3\rho }}{\left({\displaystyle \frac{\partial u_1}{\partial x_1}}+{\displaystyle \frac{\partial u_2}{\partial x_2}}+{\displaystyle \frac{\partial u_3}{\partial x_3}}\right)}^2\\ +{\displaystyle \frac{\mu }{\rho }}\left[{\left({\displaystyle \frac{\partial u_1}{\partial x_2}}+{\displaystyle \frac{\partial u_2}{\partial x_1}}\right)}^2+{\left({\displaystyle \frac{\partial u_1}{\partial x_3}}+{\displaystyle \frac{\partial u_3}{\partial x_1}}\right)}^2+{\left({\displaystyle \frac{\partial u_2}{\partial x_3}}+{\displaystyle \frac{\partial u_3}{\partial x_2}}\right)}^2\right]\end{array}$$

(11)

Based on the conservation relationship of energy transfer according to the First Law of Thermodynamics, an energy loss coefficient, denoted as ${\zeta }_{EL}$, is defined, as shown in Equation (12). This coefficient is used to quantitatively evaluate the losses caused by different gap leakages.

$${\zeta }_{EL}={\displaystyle \frac{{\left(P_3/P_{3,t}\right)}^{\frac{\gamma -1}{\gamma }}-{\left(P_3/P_{3,t,in}\right)}^{\frac{\gamma -1}{\gamma }}}{1-{\left(P_3/P_{3,t,in}\right)}^{\frac{\gamma -1}{\gamma }}}}$$

(12)

Liu et al. [26] and Shao et al. [27] presented the differential equations for the conservation of mechanical energy in steady flow fields.

$$w·{\displaystyle \frac{\partial p}{\partial x}}=-\sigma ·{\displaystyle \frac{{\partial }^{2}p}{\partial x_i^2}}$$

(13)

Using the linear velocity at the outlet of the impeller and the chord length C of the blade as the unit quantities, the aforementioned equation, after being nondimensionalized and simplified, is reduced to the following:

$$w^{*}·\nabla p^{*}=-\left({\displaystyle \frac{\sigma }{UC}}\right){\nabla }^2{p}^{*}$$

(14)

Define $∆p^{*}$ as the nondimensional pressure difference. $Eu=∆p^{*}={\displaystyle \frac{∆p}{\rho U^2}}$.

Translate the above equation into its integral form using cylindrical coordinates.

$$Eu=\mathrm{R}\mathrm{e}·\mathrm{L}\mathrm{i}{\int }_0^{r^{*}}{\int }_0^{\theta }{\int }_0^{z^{*}}\left[w^{*}·\nabla \left(-\nabla p^{*}\right)\right]d{r}^{*}d\theta d{z}^{*}$$

(15)

$Eu$ represents the Euler number, which reflects the relative relationship between the pressure drop across the fluid and the fluid’s dynamic pressure. $\mathrm{R}\mathrm{e}$ is the Reynolds number, and $\mathrm{L}\mathrm{i}$ reflects the relative relationship between the momentum diffusion through the fluid and the power consumption within the fluid.

$$w^{*}·\nabla (-p^{*})=\left|w^{*}\right|·\left|\right(-\nabla p^{*}\left)\right|·\mathrm{c}\mathrm{o}\mathrm{s}\beta$$

(16)

The value of $\mathrm{c}\mathrm{o}\mathrm{s}\beta$ reflects the flow loss within the fluid.

3. Research and Analysis

3.1. Overall Performance

We conduct a numerical simulation study on shrouded impeller centrifugal compressors with different seal structures. The characteristics of these models and the leakage amount at the design operating point are presented in

Table 1. Characteristics of various calculation models.

Name	Characteristics	Leakage Rate
None	No Cavity	0
Case 1	Cavity and Seal Teeth 1 (5 Teeth 0.1 mm)	0.60%
Case 2	Cavity and Seal Teeth 2 (5 Teeth 0.2 mm)	1.28%
Case 3	Cavity and Seal Teeth 3 (5 Teeth 0.3 mm)	1.86%
Case 4	Cavity and Seal Teeth 4 (5 Teeth 0.4 mm)	2.41%

, and the leakage rates of all conditions are shown in Figure 4.

The aerodynamic performance, including total pressure ratio and isentropic efficiency, of the model None and the four different seal teeth configuration models (Case 1, Case 2, Case 3, and Case 4) of all flow conditions is illustrated in Figure 5.

After adding the cavity structure, the isentropic efficiency and total pressure ratio of the models from Case 1 to Case 4 under all operating conditions exhibited varying degrees of decline compared to the baseline model without cavity leakage. Furthermore, as the leakage increases, the isentropic efficiency and the total pressure ratio of these models show a downward trend, and the stable operating range of the compressor also tends to narrow. Under design conditions, compared to None, the isentropic efficiency of Case 1, Case 2, Case 3, and Case 4 decreases by 1.71%, 2.59%, 3.61%, and 4.31%, respectively. Similarly, the total pressure ratios decrease by 0.56%, 1.05%, 1.73%, and 2.11%.

To visually analyze the impact of different leakage flows on the performance of the compressor, Figure 6 presents a comparison of the stall margin (SM, defined by Equation (7)) and peak isentropic efficiency (PIE) for models with varying seal teeth clearances.

To further analyze the impact of the interface between the shroud cavity and the impeller on the performance of the compressor, four models are constructed based on Case 3 for analysis. The specific settings for these computational models are detailed in

Table 2. Connection status of the cavity inlet and outlet of four models.

Name	Cavity Inlet	Cavity Outlet
Case 3a	Closed	Connected to the impeller inlet
Case 3b	Connected to the impeller outlet	Closed
Case 3c	Connected to the impeller outlet	Pressure outlet
Case 3d	Pressure flow inlet	Connected to the impeller inlet

.

The aerodynamic performance of the models (None, Case 3, Case 3a, Case 3b, Case 3c, and Case 3d) of full flow conditions is illustrated in Figure 7.

The cavity structure of the Case 3a model is only connected to the impeller inlet of the compressor while remaining disconnected to the impeller outlet. The results indicate that the compressor’s total pressure ratio and isentropic efficiency do not exhibit significant changes compared to the model named None without a shroud cavity. The area where mass exchange occurs with the main flow passage is confined to a limited cavity space behind the sealing teeth. This area is not only small in volume but also contains relatively simple vortex structures, resulting in minimal impact on the mainstream flow field. Therefore, the performance of this model shows no significant performance differences from the model without a cavity structure.

The cavity structure of the Case 3b model is only connected to the impeller outlet of the compressor, while remaining disconnected to the impeller inlet. The results indicate that the compressor’s total pressure ratio does not exhibit noticeable changes compared to the model named None without a shroud cavity. However, there is a noticeable decrease in the isentropic efficiency. In this case, there is a significant mass exchange between the fluid in the shroud cavity structure and the main fluid from the outlet of the impeller. Due to the higher entropy of the fluid within the shroud cavity compared to fluid within the impeller, mixing losses occur when the two mix. This mixing results in a reduction in isentropic efficiency.

The cavity structure of the Case 3c model is only connected to the impeller outlet of the compressor, with the static pressure at the cavity outlet set to match it at the impeller inlet. This is attributed to fluid leakage within the compressor shroud cavity, which creates a low-pressure area and causes the fluid at the impeller outlet to expand, resulting in a decrease in compressor pressure. However, the fluid leakage improves the flow conditions near the shroud surface at the impeller outlet. In conjunction with the fluid mixing losses analyzed in Case 3b, these effects collectively contribute to the compressor’s isentropic efficiency remaining virtually unchanged.

The cavity structure of the Case 3d model is only connected to the impeller inlet of the compressor, with the total temperature and total pressure at the cavity inlet set to match those at the impeller outlet. The calculation results indicate noticeable reductions in both the total pressure ratio and the isentropic efficiency of the centrifugal compressor, along with a considerable decrease in the stable operating range. The subsequent text will provide a more detailed analysis.

From the simulations of these non-actual cases, it can be observed that the two major compressor shroud cavity losses are lost due to mass exchange between the fluid within the shroud cavity and the fluid at the impeller outlet, as well as the loss when the leakage flow is injected into the main flow passage.

3.2. Flow Analysis Within the Shroud Cavity

In Figure 8, under design conditions, within the shroud cavity, the main flow path of the fluid is as follows: the fluid flows along the stationary wall from the cavity inlet towards the sealing teeth. A portion of the fluid flows along the moving wall, ultimately flowing back to the cavity inlet, forming a circulatory loop. However, there are also several counter-rotating vortex structures alongside the main flow path of the shroud cavity. These vortices exacerbate the turbulence in the flow and significantly increase the flow losses, placing the fluid within the cavity in a state of high entropy production with significant energy dissipation.

For an in-depth analysis, the dissipation function is adopted for specific research. The dissipation function is defined by Equation (11). This function is used to analyze the underlying causes of energy loss and to determine their locations. Based on this, we have plotted the contour plot of energy dissipation in the cavity, as shown in Figure 9.

In Figure 10, the flow within the cavity differs significantly with varying leakage rates. The horizontal axis represents the positional relationship along the red line in Figure 10a. As the leakage rate increases, the velocity gradient in the vicinity of the static wall exhibits a decreasing trend. Conversely, the velocity gradient at the moving wall increases. The impact of changes in velocity gradient can be intuitively reflected in the contour plot of energy dissipation shown in Figure 9. As the leakage amount increases, the velocity gradient at the moving wall decreases, leading to a reduction in shear force and hence a decrease in energy dissipation. Conversely, at the stationary wall, the velocity gradient increases, causing an increase in shear force and consequently an increase in energy dissipation.

3.3. Influence on Impeller

Figure 11 displays, from left to right, contour plots of the static entropy increase on the meridional surface for None (no cavity), Case 3b (cavity without leakage), and Case 3 (cavity with leakage). The mass exchange between the impeller outlet and the cavity inlet has a relatively minor direct impact on the performance on the impeller, but it does have a negative influence on the diffuser, increasing the entropy of the fluid near the shroud surface of the diffuser. The injection of leakage flow into the compressor impeller inlet results in significant mixing losses, which involve the mass exchange of high-entropy fluid with the mainstream and the relative mixing of fluids flowing in different directions.

After adding the cavity leakage structure, the leakage flow alters the flow near the shroud surface at the impeller inlet, increasing the circumferential flow velocity of the fluid and disrupting the flow at the shroud surface. Figure 12 illustrates the circumferential swirling angle of the main flow passage at 99% of the blade height under different leakage flow rates, where the horizontal axis represents the position within the single channel.

After adding a shroud cavity structure, as the leakage increases, the circumferential swirling angle (SA, defined by Equation (10)) also increases, resulting in an increased impact angle between the flow and the blade.

Figure 13 compares the static entropy distributions at three blade height surfaces (10%, 50%, and 90% blade height) with various sealing structures, at a mass flow rate of 28.1 kg/s. It can be observed that at the 10% blade height, there are no significant differences among the contour plots at different leakage rates. However, at 90% blade height, the static entropy near the suction surface of the blade increases significantly. As the blade height increases, the influence of the shroud cavity becomes more pronounced. The effect of increased leakage on the flow passage becomes more evident with higher blade heights, especially on the suction surface of the blade, where the leakage flow significantly increases flow losses.

The angle (β, defined by Equation (16)) between the velocity vector and the direction of the inverse pressure gradient directly indicates the state of the fluid flow. Inside the impeller of a compressor, the fluid should flow along the positive pressure gradient, meaning that the angle between the velocity vector and the direction of the inverse pressure gradient directly is greater than 90°. When the fluid flows along the positive pressure gradient, backflow occurs within the impeller.

As shown in Figure 14, at a mass flow rate of 28.1 kg/s, as the leakage increases, inverse pressure gradient flow appears on the suction surface of the blade. And, as the leakage volume increases, this area will expand, accompanied by a more significant inverse pressure gradient. In this area, secondary flows such as backflow, vortices, and reverse bubbles occur. By observing the streamlines generated along the leading edge of the blade, it can be seen that flow separation occurs near the suction surface, corresponding to the inverse pressure gradient flow inside the compressor, resulting in compressor stall and reducing operating range, increasing losses, and decreasing isentropic efficiency.

3.4. Influence on Diffuser

Based on the analysis presented earlier, the fluid within the shroud cavity exchanges mass with the fluid within the main flow passage. This exchanged mass has an impact on the diffuser. In Figure 15, when leakage flow enters the cavity, its axial velocity distribution is uneven. The fluid at the impeller outlet mainly flows into the cavity near the pressure side of the blade and flows back into the main flow passage near the suction side of the blade. Therefore, there is significant mass exchange at the impeller outlet. When the fluid in the main flow passage enters the cavity and then flows back into the main flow passage due to the influence of secondary flow within the cavity, it carries a relatively high entropy. This has adverse impacts on the flow within the diffuser, thereby reducing the efficiency of the centrifugal compressor. Additionally, as the leakage increases, the amount of fluid flowing into the cavity increases, while the amount of fluid flowing back into the main flow passage decreases.

For the design conditions of Case 1, Case 2, Case 3, and Case 4, the entropy increases of the outflow from the inflow fluids at the cavity inlet are 224.1 J/(kg·K), 109.1 J/(kg·K), 91.4 J/(kg·K), and 81.6 J/(kg·K), respectively. This indicates that when the fluid within the cavity mixes with the fluid in the main flow passage, the difference in their aerodynamic parameters decreases with increasing leakage volume, leading to a reduction in flow loss at the cavity inlet.

Figure 16 presents a contour plot of the meridional surface energy loss coefficient defined by Equation (12) within the diffuser. In this figure, “None” represents the structure without a shroud cavity, while “Case 3b” represents the structure with a shroud cavity added but without injection into the impeller inlet. Cases 1, 2, 3, and 4 represent four structures with increasing leakage rate.

It can be observed that the performance of the diffuser with a shroud cavity regardless of the type of structure is lower than that without shroud cavity. As the leakage rate increases, the energy loss coefficient first decreases and then increases near the shroud surface of the diffuser. This trend does not completely align with the entropy increase observed for the outflow and inflow fluids at the cavity inlet. During the gradual increase in leakage rates in Cases 2, 3, and 4, the performance of the diffuser deteriorates, which is contrary to the previously mentioned trend of entropy increase.

To construct four computational models for numerical simulation analysis, we designate them as Case 1*, Case 2*, Case 3*, and Case 4*, corresponding to the original Case 1, Case 2, Case 3, and Case 4 models, respectively. In these revised models, leakage flow is only present when exiting from the shroud cavity outlet, with no leakage flow being injected into the impeller inlet.

Based on the numerical simulation results, in Figure 17, it can be observed that the energy loss coefficients of the fluid within the diffusers of Case 1, Case 2, Case 3, and Case 4 decrease sequentially. Therefore, it can be inferred that the performance within the diffuser is influenced by the flow at both the inlet and outlet of the shroud cavity, and the effects of leakage at these two locations are opposite. The leakage flow can improve the flow field in partial areas of the diffuser. Conversely, the leakage flow can have a negative impact on the interior of the impeller. These adverse impacts also propagate downstream along the flow direction. In cases of small leakage, the inlet of the shroud cavity is the primary influencing factor, whereas in cases of large leakage, the outlet of the shroud cavity becomes the primary factor.

4. Conclusions

This paper presents an extensive numerical simulation study on the flow characteristics of both the main flow passage and the shroud cavity of a shrouded impeller centrifugal compressor under various leakage conditions. The conclusions are drawn as follows:

• The shroud cavity and leakage flow within the shrouded impeller centrifugal compressor significantly impair its overall performance, including the reduction in isentropic efficiency, pressure ratio, and stable operating range. This detrimental impact is observed to escalate with increasing leakage amounts due to wear, further exacerbating the compressor’s performance decline.
• When leakage flow is injected into the main flow passage at the impeller inlet from the shroud cavity, the circumferential mixing exhibits a uniform distribution. In contrast, when injected into the shroud cavity, the leakage flow of the main flow fluid from the impeller outlet exhibits a nonuniform characteristic. Particularly, fluid near the pressure side of the blades, upon injection into the shroud cavity, forms complex circulatory structures and ultimately flows back into the main flow passage near the suction surface at the impeller outlet.
• The leakage flow from the shroud cavity increases the swirling angle of the gas near the shroud surface at the impeller inlet. The heightened leakage flow rate directly results in a more significant increase in the swirling angle, consequently intensifying the impact loss inflicted by the airflow on the blades and detrimentally impacting the compressor’s stable operation.
• The primary cause of the performance degradation in shrouded impeller centrifugal compressors is the mixing loss that occurs between the leakage flow and the main flow. Under low flow conditions, this mixing loss becomes especially pronounced, resulting in the generation of separation vortices in the vicinity of the shroud surface, thereby further impairing the compressor’s performance.
• The flow conditions at the inlet and outlet of the shroud cavity exert a significant influence on the energy loss within the diffuser. Under small flow rate conditions, the flow conditions at the shroud cavity’s inlet are the primary determinant. Conversely, under large flow rate conditions, the flow at the shroud cavity’s outlet becomes the primary factor. Furthermore, studies have revealed the existence of an optimal value of leakage rate that minimizes energy loss within the diffuser.

In summary, this paper provides a detailed analysis of the impact of different gap leakage flows on the overall performance and internal flow of the compressor, offering references for the design of compressor seal teeth. Further research is still needed on the unsteady state study of compressor seal leakage. Understanding how to reduce leakage and mitigate the adverse effects of leakage flows on compressors would have more significant practical implications.