## Abstract

The flow in shrouded stator cavities can be quite complex with axial, radial, and circumferential variations. As the leakage flow recirculates and is re-injected into the main flow path upstream of the stator, it deteriorates the near-hub flow field and, thus, degrades the overall aerodynamic performance of the compressor. In addition, the windage heating in the cavity can raise thermal-mechanical concerns. Fully understanding the details of the shrouded-hub cavity flow in a multistage environment can enable better hub cavity designs. In the first part of the paper, the influence of the hub leakage flow on compressor performance and its interactions with the primary flow were investigated. While the impact of hub leakage flow on the primary passage is readily available in the open literature, details inside the cavity geometry are scarce due to the difficulties in instrumenting that region for an experiment or modeling the full cavity geometry. To shed light on this topic, the flow physics in the stator cavity inlet and outlet wells are investigated in this paper using a coupled computational fluid dynamics model with the inclusion of the stator cavity wells for the Purdue 3-stage (P3S) axial compressor, which is representative of the rear stages of a high-pressure-compressor in core engines. At the inlet cavity, the presence of at least one pair of vortices influences the trajectory of the cavity leakage flow. The amount of leakage flow also determines the size of the vortical structures, with larger clearances creating a smaller vortex and vice versa. After passing through the labyrinth seals, the leakage flow travels along the stator landing first and then transitions to the rotor drum. In general, a flow path closer to the rotor drum achieves higher circumferential velocity but also exhibits significant temperature rise. A rise in circumferential velocity directly corresponds to a rise in temperature. In addition, the windage heating increases with increasing seal clearance. Furthermore, the inlet well contributes the most to overall windage, nearly 50% of the total windage heating, while the labyrinth seals and outlet well account for very little.

## Introduction

In axial compressors, there are two common design choices for stator hub configurations: cantilevered stators and shrouded stators. Despite the differences between the two configurations, both cantilevered and shrouded stators have been successfully employed for gas turbine engines [1]. Indeed, in many applications, the decision to use cantilevered stators or shrouded stators is typically driven by mechanical considerations to avoid the primary vibrational modes of first flex, first torsion, and two-stripe frequencies in the compressor operating range [1]. Typically, shrouded stators provide additional mechanical stability compared to cantilevered stators.

For shrouded stators, the flow in the hub cavity involves lower momentum fluid in the circumferential direction than that of the primary flow. As the leakage recirculates back to the stator inlet, it introduces mixing losses to the primary flow and, thus, degrades the performance of the compressor. Besides, the cross-passage pressure gradient moves the low-momentum leakage flow to the stator suction surface. As the low-momentum flow accumulates at the hub corner, it leads to an increase in hub corner blockage, higher total pressure loss, and possible hub corner stall. Over the years, research has confirmed that shrouded stator cavity flows can have a significant impact on the performance of multistage axial-flow compressors. The first systematic investigation of the effects of the shrouded stator hub leakage flow on the performance of a multistage compressor was conducted by Jefferson and Turner [2].

In the first part of this paper, the influence of the hub leakage flow on compressor performance and its interactions with the primary flow were investigated [3]. However, to optimize the near-hub stator performance and minimize the detrimental effects of interactions between the cavity leakage flow and primary flow, the flow structure within the cavity must be understood. Although becoming increasingly important, the literature on details of the characteristics of cavity leakage flow near the cavity and primary flow interface, or the flow path and vortical structures within the cavity geometry, is scarce because a majority of the investigations are carried out using linear cascades that lack the presence of the inlet and outlet cavity wells.

Demargne and Longley [4] conducted an experimental study using a linear cascade and showed that vortical flow structures occur at the cavity interface at low leakage fractions but disappear at leakage fractions greater than 0.66%. These vortical structures played an important role in the exchange of fluid and flow properties between the cavity and the primary passage at low leakage fractions. Wellborn [5] investigated the hub cavity flow in a low-speed axial-flow compressor and also showed vortical flow structures near both the inlet and outlet cavity interfaces. In addition, the majority of the flow entered or exited the cavity near the downstream face of the inlet and outlet cavity interfaces, respectively. While these studies present the details of the vortical flow structures near the hub-cavity interface, the mechanisms that generate these vortical flow structures are not well understood because the inlet and outlet cavity wells are not included in the investigations. In addition, the influence of changes in cavity leakage flow on the vortical flow structures is largely unaddressed.

Most literature regarding compressor aerodynamic performance assumes an adiabatic nature of the flow and, thus, does not capture the details concerning temperature variations through the cavity. In real applications, as the leakage flow proceeds through the cavity, shearing work is done on the flow between the rotating and stationary wells of the cavity. This shear work causes a rise in the temperature of the fluid inside the cavity known as windage heating. Therefore, the flow affected by windage must be purged or ventilated by allowing some air, driven by the pressure gradient across the stage, to enter the inlet well and flow through the labyrinth seal into the outlet well to emerge back into the primary flow. Thus, in addition to the aerodynamic performance degradation of the primary flow in the hub region, cavity leakage flow also controls windage heating, which is an important mechanical design consideration for component life and high metal temperatures. As designers continue pushing compressor designs toward higher loading and speed, windage heating has now become important in compressor designs.

However, the considerations for aerodynamic versus thermal-mechanical perspectives can compete. On one hand, aerodynamic considerations of cavity flow encourage minimal leakage flow through the labyrinth seal teeth and cavity geometry [6] since most previous studies showed more severe performance degradation with an increase in leakage flow [6,7]. On the other hand, thermal-mechanical considerations of the cavity suggest that sufficient leakage flow is required to maintain an appropriate metal temperature. For instance, a peak temperature rises to 30 K was reported in Bayley and Childs's [8] investigations at a low leakage flow rate for tight clearance cases. Also, in the numerical study of the leakage flow and windage generation within an axial compressor stator well [9], the authors reported the highest temperatures at the rotor drum surface and suggest that too little flow through the cavity results in high air temperatures which, in turn, produces high metal temperatures and reduces component fatigue life.

However, in-engine experimental or coupled stator-cavity numerical investigations of windage heating for air passing through a labyrinth seal cavity are quite scarce in the open literature. Although few, there are articles that investigate standalone labyrinth seals. For instance, Waschka et al. [10] studied the heat transfer and leakage loss for compressible flows in a standalone straight-through labyrinth seals test section at high rotational speeds. The results showed an increase in circumferential velocity, which lead to a decrease in leakage flow rate and increased heat transfer at higher rotational speeds. In addition, the highest local heat flux was observed at the labyrinth seal teeth. Millward and Edwards [11] conducted an experimental study of various stepped and straight-through labyrinth seal geometries in a standalone test section. The results showed a close connection between windage heating and leakage flow. In addition, windage heating increased exponentially with increases in rotational speed. Recently, Li et al. [12] performed a numerical investigation of a straight-through labyrinth seal geometry with seven teeth. Results showed a strong dependence between the reduction of leakage flow rate and the circumferential-to-axial velocity ratio. There was a greater than 20% reduction in leakage flow as the circumferential-to-axial velocity ratio exceeded 5. Those findings suggest strong interdependency among the seal leakage flow rate, circumferential velocity, and windage heating. Denecke et al. [13] performed a numerical study of an inclined labyrinth seal geometry with a focus on the total temperature rise and circumferential velocity development across the rotating seals. This study also showed an increase in windage heating with increases in rotational speed. Also, the results revealed that windage heating increases with an increasing number of labyrinth seal teeth. Most importantly, the study showed a significant influence of the inlet and outlet circumferential velocity on windage heating. A higher inlet circumferential velocity leads to a higher outlet circumferential velocity with less windage heating.

Nevertheless, all these studies feature standalone labyrinth seals with inlet and outlet ducts used to feed airflow at relevant operating conditions. However, in real applications, both the inlet and outlet rotating cavity wells and the long rotating walls affect the flow characteristics and windage heating within the cavity. In this study, a coupled computational fluid dynamics (CFD) model including the primary flow passage, as well as the labyrinth seal and stator cavities, was used to investigate the flow. This work is unique in the sense that it accounts for the inlet and outlet cavity wells inside a coupled stator-cavity model, which provides more representative results related to the development of flow characteristics and windage heating in the stator hub cavity.

## Numerical Setup

The investigations presented in this work were performed on a parametric CFD model of the Purdue 3-stage (P3S) axial compressor research facility using the PAX100 (West Lafayette, IN) configuration. The compressor represents the geometrically scaled-up design of the rear stages of a Rolls-Royce high-pressure compressor used for a jet engine core, matching Mach numbers and Reynolds numbers. The cross section of the computational domain for the coupled numerical model is shown in Fig. 1(a). Since details of the experiment and numerical setup up are documented in part 1 of the study [3], only a brief description of the numerical setup is included in this section.

Fig. 1
Fig. 1
Close modal

The numerical model consists of the primary passage as well as the cavities coupled to the primary passage. A section view of the coupled model with cavities is displayed in Fig. 1(b). The primary flow's computational domain includes the front struts followed by a single passage of each blade from inlet guide vane to stator 3. The modeling of hub cavities includes three parts: the rotating rotor drum wall, the stationary stator landing wall, and the labyrinth seals, as labeled in Fig. 1(b).

ansysturbo-grid and ansys meshing were used to generate the meshes for the primary flow passage and hub cavities, respectively. The same type of mesh (hexahedral) was used for mesh generation, and a similar size factor and boundary layer refinement was applied. Also, all the cavities have the same passage size as the corresponding inlet guide vane or stator domains to ensure consistent information transfer at the cavity inlet and outlet interfaces without having to take pitch angle changes into account. This results in a y+ of the wall-adjacent nodes less than five within both domains. Though not presented in this study, a grid-independent study was conducted to assure grid resolution in axial, radial, and circumferential directions, with details documented in Ref. [14].

ansyscfx was the computational solver used to obtain a steady-state solution of a single passage per blade row. An ideal gas model is selected for the working fluid. The k−ω based shear stress transport model was selected because of the model's ability to account for the transport of turbulent shear stress, which yields highly accurate predictions of the onset and the amount of flow separation under adverse pressure gradients. The model uses an automatic near-wall treatment developed by ansyscfx, which allows for a smooth transition between a low turbulence Reynolds number form to a wall function formulation for accommodating y+ values. Scalable wall functions are used for larger y+ values where low-Reynolds number flows do not exist. Details about the shear stress transport model in cfx can be found in Ref. [15]. Stage mixing planes were employed to convey information between rotor and stator domains, and Rotational Periodicity interfaces were applied to the circumferential walls of the primary flow passage, Fig. 2(a). The hub cavities are attached to the stators, and a frozen rotor model is applied to the interfaces between the stator hub and cavity, as shown in Fig. 2(b).

Fig. 2
Fig. 2
Close modal

Total pressure and temperature profiles measured at the aerodynamic interface plane (Station 0) were used to set the inlet boundary condition and exit corrected mass flow was chosen as the exit boundary condition. The hub and blades are modeled as no-slip, smooth, adiabatic walls, while the shroud surfaces are treated as isothermal walls using the casing surface temperature measurements as boundary conditions. Finally, the exit corrected mass flow rate calculated from the experiment was specified at the exit of stator 3 cavity domains as the exit boundary condition of hub cavity flow.

## Detailed Flow Structures in Stator Wells

To study the influence of hub leakage flow, the stator 1 seal clearance was adjusted, shown in Fig. 1(a), while holding the compressor operating condition constant at the design condition. A total of seven cases were simulated, with seal clearances ranging from 0.25% to 2.5% of the blade height at increments of 0.375%, and details of flow characteristics in the stator 1 hub cavities are presented.

Figure 3 shows the meridional streamlines in the inlet cavity well. Results show the presence of at least one pair of vortices in the stator well. At the interface surface between the primary flow passage and hub cavity, the vortex ❶ lands on the upstream surface of the stator well and drives the flow to enter near the downstream face of the inlet well. In addition, there is a larger vortical structure ❷ at the corner between the rotor landing and downstream face of the stator well. The vortical structures create a pathway for the cavity leakage flow, indicated by the arrowed line. Furthermore, the inlet stator well vortical structures and leakage flow path varies with changes in hub seal clearance. At tight clearances with less hub leakage flow, the corner vortex ❷ takes up more space at the inlet stator well and forces a more arduous path for the leakage flow to navigate. As the hub seal clearance increases, the corner vortex ❷ withdraws, and the near-interface vortex ❶ stretches radially inward. This adjustment of vortex structures ❶ and ❷ alters the leakage flow path. For instance, at hub seal clearances greater than 1.75%, the flow enters the inlet cavity well radially and lands directly on the rotor drum.

Fig. 3
Fig. 3
Close modal

Figure 4 shows the meridional streamlines in the outlet cavity well. There is a wide vortical structure ❻ along the stator root surface, which yields a significant detour of the leakage flow. For instance, at tight clearances with less hub leakage flow, the vortical structure ❻ forces the leakage flow exiting from labyrinth seals to travel along the stator landing and follow the rotor drum geometry curvature until the leakage flow exits to the primary passage. In addition, this vortical structure ❻ also forces the flow closer to the rotor drum, which aids in the development of circumferential momentum. As the hub seal clearance increases, the vortex ❻ contracts. Therefore, the majority of the leakage flow will travel closely along the stator landing but for a shorter distance.

Fig. 4
Fig. 4
Close modal

In summary, the simulations show complex vortical structures in both the inlet and outlet stator cavities. Those vortices ❶–❽ form the leakage flow path in the hub cavity. In addition, the vortical structures change as the hub seal clearance (or hub leakage flow) changes resulting in different flow topologies in the inlet stator well. For instance, at tighter clearance cases (hub seal clearances no greater than 1.375%), the vortical structures ❶–❻ yield a “geometrically dependent” leakage flow path. In other words, the leakage flow follows the geometric contour of the stator well and the rotor drum. On the other hand, at larger clearance cases with hub seal clearance greater than 1.75%, deviation of the leakage flow path from stator well geometry occurs. Overall, tighter clearance cases show a more abrupt switch from the stator landing to the rotor drum, while that transition is gradual for larger clearance cases.

## Radial Variations of Flow Profiles in Stator Wells

The interactions between the hub leakage flow and the rotor drum determine the development of the circumferential velocity of the cavity leakage flow, which further influences the near-hub performance and windage heating. In this section, the quantitative details of the flow parameters in the cavity at different seal clearances are investigated. As indicated in Fig. 5, the analysis was conducted at four different radial locations from the cavity top interface surface to the bottom rotor drum surface.

Fig. 5
Fig. 5
Close modal

The radial direction is normalized with hub radius and the axial direction is normalized with the total cavity length Along the radial direction, a notation of 0 indicates the rotor drum surface and 1 represents the top interface surface between the primary and hub leakage flow. Similarly, the axial locations in the hub cavity are also represented using numerical values between 0 and 1. The inlet stator well holds a range from 0.75 to 1.0 while the outlet stator well covers a range from 0 to 0.175. The hub clearance gap ranges from 0.175 to 0.15, respectively. All the profiles presented in this section are using the circumferentially area-averaged results.

### Radial Variations in Inlet Stators Wells.

In this section, the results of inlet cavity flow were presented in the order of leakage flow streamlines. As denoted in Fig. 3, the leakage flow enters the cavity inlet well through the cavity inlet face $(r/ro=0.99)$ and travels radially inwards before arriving at the rotor drum $(r/ro=0.01)$.

Figure 6(a) shows the flow parameters at the cavity inlet interface ($r/ro=0.99$). The radial velocity profiles show where the fluid enters the inlet cavity well with a negative value, indicating flow entering the cavity. At all clearances, the majority of the leakage flow enters the hub cavity downstream of the hub cavity ($x/L>0.8$). This trend gets more extreme with tighter clearances. For instance, more than 80% of flow enters the inlet stator well through the rear half of the interface surface ($x/L>0.8)$ at 0.25% hub seal clearance. This is caused by the vortex ❶ on the upstream surface of the stator well, shown in Fig. 3. In addition, the magnitude of the radial velocity increases with increasing seal clearances, indicating more leakage flow at larger seal clearances. On the other hand, the circumferential velocity of hub leakage flow decreases significantly with increasing seal clearance. There is more than a 70% drop in hub leakage flow circumferential velocity as the hub seal clearance increases from 0.25% to 2.5%. Lastly, the trend of the total temperature profiles is governed by a compound effect of seal clearance and hot-leakage flow re-ingestion. This results in the lowest total temperature at 1.0% span clearance instead of 2.5% span clearance. Though not presented here, a detailed analysis of flow near the cavity interface indicates that the 0.25% span clearance case shows the possibility of re-ingestion, but the 1.00% span clearance case avoids the re-ingestion zone altogether. Further discussion on this topic can be found in part 1 of the paper [3].

Fig. 6
Fig. 6
Close modal

Figure 6(b) shows the flow parameters at the radial location directly below the rotor ledge ($r/ro=0.70$). Since most leakage flow is localized further downstream of the hub cavity ($x/L>0.8$) at the interface surface, this imbalance in leakage flow distribution gets significantly improved with a more uniform flow distribution in the inlet stator well. For the circumferential velocity profile, the same trend applies with changes in seal clearance. A tighter hub seal clearance results in a higher circumferential velocity. In addition, compared to the magnitude of the circumferential velocity near the interface surface ($r/ro=0.99)$, there is a significant increase in leakage flow circumferential velocity at surface $r/ro=0.70$, suggesting that the rotor ledge does a significant amount of work on the incoming cavity leakage fluid. While re-ingestion continues to dominate the total temperature profiles, there is a decline in total temperature near the stationary surface where the vortical structure ❶ is present. This decline in temperature is consistent across all seal clearance cases indicating that the vortical structure exists at a lower temperature than the cavity leakage fluid, which is moving radially inwards near the rotating surface. Similarly, the circumferential velocity profiles show that the vortical structures ❶ and ❷ move at a lower circumferential velocity than the cavity leakage flow.

Figure 6(c) shows the flow parameters at the radial location in between the labyrinth seal teeth and the stator landing $(r/ro=0.21).$ In addition to the inlet stator well, the development of circumferential velocity as the flow progresses from the inlet well toward the labyrinth seals is also evident. The same trend for the circumferential velocity with changes in hub seal clearance is observed. Tighter clearances result in higher circumferential velocity for the cavity leakage flow. This observation also holds for the size of the vortical structure ❷ with changes in seal clearances. However, in contrast to the lower circumferential velocity associated with the vortical structure ❶ observed in Fig. 6(b), Fig. 6(c) shows higher circumferential velocity for the vortical structure ❷ because it is closer to the rotor drum. This suggests that the vortical structure ❷ may not negatively impact, or decrease, the circumferential momentum. Rather, when the cavity leakage flow path is in direct contact with the stationary surface, it hinders the transfer of circumferential momentum. In addition, at all seal clearances, there is a sharp drop in the circumferential velocity near the region of flow exiting the inlet well ($x/L=0.75)$. This sharp drop off in the circumferential velocity is because the plane $(r/ro=0.21)$ is so close to the stationary surface. At tighter clearances, the majority of the flow lands on the rotor drum as it proceeds axially, resulting in an increase in axial velocity but also a decrease in radial and circumferential velocities. Furthermore, the leakage flow quickly regains circumferential momentum as it progresses toward the labyrinth seals ($0.5.

Finally, Fig. 6(d) shows the flow parameters at the radial location just above the rotor drum $(r/ro=0.01).$ First, there is barely any momentum in the radial direction as the leakage flow approaches the rotor drum, and this is the case for all hub seal clearances. For the circumferential velocity profiles, there is a significant increase in circumferential velocity as the flow progresses toward the labyrinth seals ($0.5). This suggests that the transfer of viscous work increases drastically as the flow approaches the rotor drum. This increase in viscous work is also supported by the temperature profiles. Comparing the magnitude of total temperature profiles across Figs. 6(c) and 6(d) show a steady rise in temperature due to frictional heating between the stator landing and the cavity leakage fluid. In this study, the highest temperature at the inlet hub cavity is observed close to the rotor drum surface, indicating that the rotor drum surface is a critical component for mechanical considerations. Also, as the flow approaches the labyrinth seals, the temperature rises quickly subjecting the labyrinth seal teeth to heating and thermal stresses. For all seal cases, the rise in total temperature near the labyrinth seal is as much as 10 °F hotter than the fluid in the cavity wells.

### Radial Variations in Outlet Stators Wells.

In this section, the results presented to investigate the outlet cavity follow the same logic as the inlet cavity. As denoted by the streamlines in Fig. 4, the flow enters the cavity outlet well near the stator landing $(r/ro=0.21)$, then proceeds to the rotor drum $(r/ro=0.01)$ before traveling radially outward toward the cavity outlet interface $(r/ro=0.99)$. Therefore, the results are presented in that order.

Figure 7(a) shows the flow parameters of the cavity leakage fluid exiting the labyrinth seals at a radial location right below the stator landing $(r/ro=0.21).$ The radial velocity profiles show that the flow continues to travel along the stator landing until $x/L=0.25$, after which the flow transitions radially inwards (toward the rotor drum), and finally turns outwards as the flow approaches the further upstream of outlet cavity surface ($x/L=0)$. This radially inward momentum is caused by the vortex ❻ shown in Fig. 4. The peaks in the radial velocity profiles in the axial range from 0 to 0.20 tracks where the flow moves radially outwards after traveling along with the rotor drum. In addition, as the leakage fluid travels along the stator landing, there is a quick drop in the circumferential velocity and a significant increase in the total temperature at tighter clearances. While at larger clearances, these trends are less drastic—there are only minor increases in the circumferential velocity and total temperature.

Fig. 7
Fig. 7
Close modal

Nevertheless, the overall trends are the same with higher circumferential velocity and total temperature at tighter clearances because there is less leakage flow rate and a shorter distance traveled along the stator landing before transitioning to the rotor drum.

Next, Fig. 7(b) shows the flow parameters of the leakage fluid at the surface close to the rotor drum $(r/ro=0.01).$ As for the inlet cavity, at all seal clearance cases, the radial velocity of the leakage flow is negligible near the rotor drum due to the zero flow through the rotor drum boundary condition. As the flow exits the labyrinth seal, the circumferential velocity of the hub leakage flow first decreases rapidly in the range of $0.25, then starts to recover in the cavity well ($0). There is a higher axial velocity at tighter seal clearances. Lastly, profiles near the rotor drum report the highest temperature. While discussion regarding temperature development in the cavity wells is largely nonexistent in the literature, the study by Bayley et al. [8] and Ozturk et al. [9] showed similar temperature developments. In addition, the total temperature profiles near the rotor drum seem less sensitive to changes in seal clearance except for the tightest clearance case, where the total temperature is more than 10 deg higher.

Figure 7(c) shows the flow parameters of the leakage fluid as the flow travels radially outwards toward the outlet interface at the radial location directly below the rotor ledge ($r/ro=0.7$). Like the trend at the inlet cavity, there is a larger radial velocity at the larger hub seal clearances, indicating higher leakage flow rates. In addition, there are larger radial velocities near the further upstream cavity well ($x/L=0$). This is caused by the large vortex structure ❻ near the stator root surface, shown in Fig. 4. There is still a significant portion of circumferential momentum with the higher total temperature at this radial location because of the viscous work input as the leakage flow travels along with the rotor drum. Like the trend at the inlet cavity, tighter seal clearances result in higher circumferential velocity and total temperature. Finally, Fig. 7(d) shows the flow parameters as the leakage fluid exits the outlet interface. The radial velocity profiles show that most of the leakage flow exits near the downstream surface. Like the trend at other radial locations, a tighter clearance results in higher circumferential velocity and hotter temperatures of the exhausting leakage flow.

Overall, these profiles show how the cavity leakage flow interacts with the vortical structures ❶–❽ and the cavity geometry features. More importantly, these profiles show how the flow characteristics change with labyrinth seal clearance. In particular, the radial and circumferential profiles allow for tracking the cavity leakage flow path and the viscous work being imparted on the flow, respectively. The total temperature profiles are also a good indicator of areas subject to heating and thermal stresses, which, in this case, are the rotor drum and the labyrinth seals as they indicate hotter overall temperatures than the rest of the geometry. This allows designers to not only predict the flow path and investigate the development of circumferential velocity and temperature in the cavity leakage fluid but also employ changes to the cavity geometry for optimization.

## Overall Flow Characteristics

In this section, the overall flow characteristics at the inlet and outlet cavity interfaces with changes in hub seal clearance are presented to understand the interdependence of the variables including circumferential and radial velocities, leakage flow rate, total temperature rise, and windage heating. The flow properties at the inlet and outlet cavity interfaces are area-averaged to represent the overall flow parameter.

### Radial and Circumferential Velocities at Inlet and Outlet Stators Wells.

Figure 8(a) shows the changes in radial velocity at the hub-cavity interfaces with changes in seal clearance. A negative value represents flow entering the cavity with a radially inward direction, while a positive value represents flow exiting the cavity. As expected, there is an increase in radial velocity with increasing seal clearance allowing more leakage flow. In addition, at each seal clearance, the magnitude of area-averaged radial velocity at the cavity entrance and exit are almost identical, indicating a negligible change in leakage flow density across the cavity. Also, the trend in radial velocity is opposite to the trend in circumferential velocity, as shown in Fig. 8(b). The radial velocity decreases with increasing circumferential velocity and vice versa.

Fig. 8
Fig. 8
Close modal

Figure 9(a) shows the changes in circumferential velocity at the hub-cavity interfaces with changes in seal clearance. First, the flow exiting the cavity has a higher circumferential velocity than the entering flow due to viscous work input from the rotor drum. Also, the circumferential velocity of both the flow entering and leaving the cavity decreases with increasing hub seal clearance. Second, the rise in circumferential velocity across the cavity decreases with increasing seal clearance, as shown in Fig. 9(b). Similar observations were reported by Wellborn [6] and Heidegger [7], where increasing the leakage generally reduced the rise in circumferential velocity across the cavity.

Fig. 9
Fig. 9
Close modal

As the seal clearance increases, allowing more leakage flow rate, the frictional momentum is transferred over more fluid which, in turn, results in a smaller circumferential velocity rise. Also, though the compressor speed remains constant in this study, the rise in the circumferential velocity of the leakage flow can be affected by compressor speed. Since the leakage fluid travels along the path close to the rotor drum, its velocity is of similar magnitude as the wheel speed. Therefore, an increase in wheel speed will lead to a higher circumferential velocity rise across the leakage cavity. As a result, circumferential velocity rise across the cavity can be affected by both wheel speed and leakage flow rate. This is an important takeaway for compressor designers since both the cavity leakage flow rate and the circumferential velocity change affect the near-hub performance in the primary flow passage.

### Overall Total Temperature Rise and Windage Heating in Stator Wells.

Figure 10 shows the total temperature at the inlet and outlet cavity interfaces as well as the rise in total temperature across the cavity. First, the tightest clearance case, 0.25% span, results in the highest total temperature at both cavity inlet and exit interfaces. The high temperatures correspond to the re-ingestion observed for tighter clearances. The seal clearance case of 1.0% span renders the lowest temperature as it avoids re-ingestion. As the seal clearance further increases from 1.0% to 2.5% span, there is only a slight increase in the total temperature because more leakage flow at larger clearances makes the momentum transfer less effective. Once again, the highest temperature at the tightest seal clearance confirms the competing mechanisms for design considerations from aerodynamic versus thermal-mechanical perspectives. While a tight hub seal clearance is favored from the aerodynamic perspective, it can raise thermal-mechanical concerns due to the significant temperature rise that occurs in the cavity. Furthermore, the interdependence of circumferential velocity and total temperature rise between the cavity inlet and exit interfaces is shown in Fig. 10(b). A tighter seal clearance results in both a higher increase in circumferential velocity and a higher rise in total temperature.

Fig. 10
Fig. 10
Close modal
Finally, the influences of hub seal clearance on windage heating are analyzed. Windage heating, described by
$W=m˙cpΔTo$
(1)

Depends on both the leakage mass flow rate and the total temperature rise across the hub cavity. Figure 11(a) shows the changes in the overall windage heating as well as the contributions of the cavity wells and labyrinth seals with changes in tip clearance. The windage heating increases with increasing seal clearance. This trend is nearly opposite of the trend for total temperature rise across the cavity. This shows that leakage mass flow rate dominates this correlation instead of the total temperature rise. For a tighter seal clearance, the leakage flow through the cavity is relatively low, which results in less shear work done on the fluid as opposed to larger seal clearance cases. Figure 11(a) also shows the contributions of individual stator wells and labyrinth seals to windage heating. For all seal clearance cases, the inlet well contributes the most, accounting for more than 50% of the overall windage heating. The labyrinth seals generate approximately 30% of the windage work, while the outlet cavity well accounts for the remaining 20%. At all seal clearance cases except for the 1.0% and 1.375% span clearance cases, the inlet cavity well tends to show increasing contributions to windage heating. The drop in inlet cavity well windage heating for 1.0% and 1.375% span is considered to be associated with the drop in total temperature of the incoming cavity flow, as shown in Fig. 10. This is additional evidence that there is no cavity flow re-ingestion for these cases.

Fig. 11
Fig. 11
Close modal

Besides, the windage heating can also be gauged by the shear work from the rotor drum in terms of the inlet circumferential velocity, as shown in Fig. 11(b). A tighter clearance results in a higher circumferential velocity at the inlet cavity interface, which further leads to a smaller velocity gradient across the rotor drum boundary layer, and therefore, less shear work and windage heating. On the other hand, a large clearance results in a lower circumferential velocity at the inlet cavity interface and a deeper velocity gradient across the rotor drum boundary layer, leading to more shear work and windage heating.

In summary, there are tends to a higher total temperature rise across the cavity at a tighter seal clearance, which can raise thermal–mechanical concerns. However, despite the higher total temperature rise, there is less overall windage heating at tighter seal clearances primarily due to the lower leakage flow rate. Finally, both the total temperature rise and windage heating show a strong dependence on inlet tangential momentum.

## Discussion and Conclusions

In this study, the flow physics in the stator cavity inlet and outlet wells were investigated to understand the flow path of the leakage fluid and windage heating within the cavity. The investigations presented in this work consist of a series of numerical simulations using a coupled CFD model with the inclusion of the stator cavity wells which replicates the P3S axial compressor representing the rear stages of a high-pressure-compressor in core engines.

At the inlet cavity, simulation results show the presence of at least one pair of vortices, the structures of which vary with changes in hub seal clearance (or hub leakage flow). These vortical structures determine the path of the leakage flow across the cavity. For large clearance, cavity leakage flow proceeds straight inward radially and lands on the rotor drum, while the vortical structures of small clearance force the leakage path to stay near the stationary surface creating a more arduous path for the cavity leakage flow. After passing through the labyrinth seals, the leakage flow travels along the stator landing first and then transitions to the rotor drum before turning toward the interface for the outlet cavity. The amount of leakage flow also determines the size of the vortical structures, with larger clearances creating smaller vortices and vice versa.

The radial variations in flow profiles showed an increase in circumferential velocity for tighter clearances. The radial variations also showed that the re-ingestion cases achieved much higher temperatures. On the other hand, the cases that avoided re-ingestion altogether registered the lowest temperatures. The rotor ledge also imparted a significant amount of work on the incoming cavity leakage fluid, as indicated by an immediate rise in total temperature and circumferential velocity. More importantly, the vortical structures did not negatively impact the development of tangential momentum as they followed the rotor surface with the same circumferential momentum. Rather, whether the cavity leakage flow path was in contact with the stationary stator landing or rotating rotor drum determined the transfer of circumferential momentum because the viscous work transfers from near the rotor drum to radially outward toward the stator landing. Similarly, vortical structures also exist near the rotor drum with a uniform but higher temperature profile due to frictional heating from the rotor drum to hub leakage flow. In general, the simulations showed that the size of vortical structures upstream and downstream of the hub seal scale such that the distance traveled by the cavity leakage along the stator landing and rotor drum changed. Therefore, the seal clearance, the size of vortical structures, and flow path along the stator landing and rotor drum are closely interdependent and play a major role in determining the development of circumferential velocity as well as the temperature rise of the cavity leakage flow in the cavity wells.

Overall, in addition to the deteriorated near-hub performance caused by the leakage flow discussed in the part 1 paper [3], there is a higher total temperature at the hub-cavity interfaces due to windage heating, raising potential thermal-mechanical concerns. A tighter clearance results in a larger total temperature rise and higher circumferential velocity at both the inlet and exit cavity interfaces. These findings reveal the competing mechanisms for design considerations from aerodynamic versus thermal-mechanical perspectives. While a tight seal clearance is favored from the aerodynamic perspective, it can raise thermal–mechanical concerns due to the significant temperature rise that occurs in the cavity. Despite the higher total temperature rise, there is less overall windage heating at tighter seal clearances. This shows that leakage mass flow rate dominates the correlation with the overall windage heating. For a tighter seal clearance, the leakage flow through the cavity is relatively low, which results in less shear work done on the fluid as opposed to larger seal clearance cases. On the other hand, the overall windage heating decreases with increasing inlet circumferential velocity because, for higher inlet circumferential flow, the difference between the rotor drum and cavity leakage flow velocity is smaller, which results in less viscous work dissipation. Furthermore, it is the inlet stator well that contributes the most to overall windage, nearly 50% of the total windage heating, while the outlet cavity well and labyrinth seals accounted for 30% and 20% of the work, respectively.

This study provides insight to flow through cavity wells to understand the development of circumferential velocity and temperature in the cavity wells. The details inside the cavity wells of a coupled cavity model, which are largely missing in the existing literature due to linear cascades or standalone labyrinth seal investigations, are revealed by this study. As compressor designs evolve to higher wheel speeds, analysis of the cavity wells at higher rotational speeds becomes essential due to the development of higher circumferential velocities. The consideration of heat transfer is necessary to fully assess the thermal loads induced by the different cavity flows. The lessons learned from this study can provide insight into how the added complexities of the cavity wells and long rotating walls on the cavity leakage flow might influence the design considerations for optimizing stator passage aerodynamics as well as minimizing stator cavity heating.

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