Investigation of Improved Labyrinth Seal Stability Accounting for Radial Deformation

Abstract

This study examines the labyrinth seal disc of an aero-engine, specifically analysing the radial deformation caused by centrifugal force and heat stress during operation. This distortion may lead to discrepancies in the performance attributes of the labyrinth seal and could potentially result in contact between the labyrinth seal tip and neighbouring components. A numerical analytical model incorporating the rotor and stator cavities, along with the labyrinth seal disc structure, has been established. The sealing integrity of a standard labyrinth seal disc’s flow channel is evaluated and studied at different clearances utilising the fluid–solid-thermal coupling method. The findings demonstrate that, after considering radial deformation, a cold gap of 0.5 mm in the conventional labyrinth structure leads to stabilisation of the final hot gap and flow rate, with no occurrence of tooth tip rubbing; however, both the gap value and flow rate show considerable variation relative to the cold state. When the cold gap is 0.3 mm, the labyrinth plate makes contact with the stator wall. To resolve the problem of tooth tip abrasion in the conventional design with a 0.3 mm cold gap, two improved configurations are proposed, and a stability study for each configuration is performed independently. The leakage and temperature rise attributes of the two upgraded configurations are markedly inferior to those of the classic configuration at a cold gap of 0.5 mm. At a cold gap of 0.3 mm, the two improved designs demonstrate no instances of tooth tip rubbing.

1. Introduction

The labyrinth seal is a crucial, non-contact, dynamic sealing mechanism. It utilises the convoluted leakage pathway created by the tooth cavity architecture to efficiently mitigate leakage by multi-stage throttling expansion and kinetic energy dissipation of the fluid. Due to its considerable benefits of negligible wear, exceptional reliability, and prolonged lifespan under conditions of high speed, elevated temperature, and intense pressure, enhancing the efficiency of rotating machinery is crucial for ensuring operational safety and maintaining rotor stability. Consequently, labyrinth seals are extensively employed in the essential components of turbine machinery, including gas turbines, steam turbines, aviation engines, compressors, and pumps. The labyrinth plate of the aero-engine frequently features a bolted connection structure. As the turntable spins rapidly, the labyrinth plate experiences radial elongation due to centrifugal force. Simultaneously, the wind resistance of the bolt and the labyrinth plate generates significant heat, leading to radial elongation of the labyrinth plate, which reduces the working gap and diminishes the sealing flow of the labyrinth. Upon reduction in the flow rate, the temperature increase due to wind resistance becomes more pronounced, further exacerbating the elongation of the labyrinth plate, thereby creating a detrimental cycle. Ultimately, when the turntable’s temperature exceeds acceptable limits, the sealing properties of the labyrinth diverge from the intended specifications, potentially leading to friction within the labyrinth, thereby compromising the engine’s safe and stable operation [1,2].

Researchers, both nationally and internationally, have conducted extensive studies on the flow and heat transfer properties of labyrinth seals. Wang et al. [3,4] investigated the flow properties of labyrinth seals under a rotating condition and determined that low rotational speeds had a negligible impact on the flow coefficient of labyrinth seals. Nonetheless, the flow coefficient diminished with an increase in rotational speed. The examination of experimental results for labyrinth seals at elevated rotational speeds revealed that the primary factor influencing the reduction in the flow coefficient was the diminished sealing clearance resulting from the increased rotational speed. Zhiguo et al. [5] investigated the aerodynamic elements affecting the correlation between rotational speed and the leakage characteristics of labyrinth seals, as well as the effects of shape parameter alterations on leakage characteristics at different rotational speeds. The regulations pertaining to flow and heat transfer in labyrinth seals were examined and synthesised under conditions of fluctuating rotational speed and geometric factors. Suryanarayanan and Morrison [6] identified the key elements affecting the leakage characteristics of the labyrinth, such as tooth width, pitch, tooth height, shaft diameter, Reynolds number, and clearance, and analysed their influence on the entrainment coefficient. Micio et al. [7] have experimentally shown that a high Reynolds number and a narrow gap width significantly increase the Nusselt number of the wall. Li Zhigang and Li Jun [8] investigated the leakage flow and cavity pressure characteristics of the rotary staggered labyrinth seal through experimental measurement and numerical analysis, examining the effects of pressure ratio and rotational speed on the seal. Zhang et al. [9] and Lee et al. [10] examined the influence of pressure ratio and geometric parameters on the leakage loss of labyrinth seals through experimental approaches. Their conclusions are similar: a higher number of teeth is associated with diminished tip clearance, and an increased front angle of the tooth improves sealing efficacy. Wasilczuk et al. [11,12] endeavoured to minimise the leakage loss of the seal by optimising the tooth profile and employing a ‘air curtain’. Chougule et al. [13] performed a numerical analysis on the enhanced flow properties of the conventional, straight, four-tooth labyrinth seal. The honeycomb sealing surface design decreased leakage by 13%, while the solid sealing surface design reduced leakage by 7%. Fei et al. [14] conducted a numerical investigation of the sealing gap and step height in the stepped labyrinth seal. The results demonstrate that the sealing gap has a positive association with the leakage rate and a negative correlation with the flow coefficient.

Kong et al. [15] investigated the impact of rotation on the thermal resistance and swirling properties of labyrinth seals. The augmentation of the drag heating coefficient is nearly linearly proportional to the rotational velocity. Moreover, elevating the rotational speed will augment the eddy current ratio at a designated radial position, while the rotational speed will additionally alter the heat transfer and sealing properties of the labyrinth seal [16,17]. The experiments conducted by Willenborg, Kim, and Wittig [18] took into account the Reynolds number and pressure ratio. Their experiment revealed that heat transfer is mostly influenced by Reynolds number rather than pressure ratio. Bo et al. [19] utilised the orthogonal design approach to develop the study scheme and employed the orthogonal test method to examine the overall effect of the sealing structure on the flow coefficient and heat transfer characteristics across different pressure ratios. The relationship between leakage and heat transfer was established and validated, and the sensitivity coefficients of geometric factors related to leakage efficiency and heat transfer at different ratios were identified. Nayak et al. [20,21] numerically assessed the impact of honeycomb pitch and sealing gap on the thermal resistance characteristics of honeycomb labyrinth seals under wind conditions. When the sealing gap is minimal, large-sized honeycomb bushings will exhibit greater leakage and an increase in wind resistance temperature compared to smooth bushings and small-sized honeycomb bushings. When the sealing gap is substantial, the leakage from large-sized honeycomb bushings is comparatively minimal. Wei et al. [22] formulated a flow and heat transfer model for the straight-through labyrinth seal and performed numerical simulations of the flow and heat transfer phenomena under eccentric and rotational settings with an initial gap of 0.15 mm. The influence of different rotational speeds and eccentricity on the leakage coefficient and heat transfer coefficient is further analysed. Sun et al. [23,24] investigated the impact of pressure ratio, rotational speed, and additional factors on the temperature rise characteristics of labyrinth seals through a combination of theoretical, numerical, and experimental approaches, elucidating the mechanism behind the temperature rise effect in labyrinth seals. Szymanski et al. [25] examined the leaking properties of labyrinth seals across different pressure ratios by experimental and numerical simulations, before optimising the structure based on CFD analysis outcomes. Following optimisation, the leakage was reduced by around 24%. Kulkarni et al. [26] proposed a one-dimensional loss model for straight-through labyrinth seals, achieving comparable CFD accuracy through numerical and experimental methodologies. Utilising the K2K method, they systematically elucidated the medium pressure loss and alterations in flow area encountered by the fluid within the sealing flow channel. Xu et al. [27] investigated the changes in leakage characteristics in relation to inlet pressure, sealing gap, and rotational speed, both prior to and following the wear of the honeycomb labyrinth, using experimental and numerical methods. The leakage of the honeycomb bushing labyrinth seal escalates with rising inlet pressure, sealing gap, and wear depth, while it diminishes marginally with an increase in rotational speed.

The evaluation of the operational stability of the labyrinth seal primarily encompasses aeroelastic instability and thermal instability. Scholarly research, both domestically and internationally, predominantly focusses on the aeroelastic instability of labyrinth seals, with minimal investigation into the thermal instability induced by the heat effect of wind resistance. Thermal instability arises from the delayed dissipation of heat generated by wind resistance during the operation of the labyrinth, leading to an increase in its temperature. Thermal expansion causes a reduction in flow rate, leading to excessive heat accumulation and creating a detrimental cycle of instability that damages the seal. This work develops a numerical model for the stability of labyrinth seals. The sealing stability analysis, utilising the fluid–solid-thermal coupling method, is conducted on the common straight-through labyrinth under varying clearances. To address the issue of tooth tip rubbing resulting from thermal instability in conventional labyrinths with minimal clearance, an enhanced configuration is proposed, and a sealing stability analysis is conducted. This offers insights into and methodologies for mitigating clearance instability due to thermal fluctuations in the design of aero-engine labyrinths.

2. Investigation of Labyrinth Seal Stability

2.1. Technique and Procedure

This paper utilises a bidirectional fluid–solid-thermal coupling sequential solution approach, as depicted in Figure 1, which illustrates the flow chart of the numerical simulation analysis. An analysis of the stability of sealing characteristics is conducted for the labyrinth seal structure. The precise procedure is as follows:

1 The cold clearance, velocity, and pressure ratio of the labyrinth seal are determined. A fluid–solid-thermal coupling model of the labyrinth seal has been developed, and the flow field of the disc cavity, along with the temperature field of the labyrinth disc, has been computed and analysed.

2 The investigation of stress and strain in the labyrinth disc is performed utilising the structural mechanics approach, considering the influences of rotational velocity and temperature. The temperature distribution of the labyrinth plate, obtained from the fluid–solid coupling solution, is utilised, leading to the radial deformation of the plate after considering the influences of velocity and temperature.

3 Extracting the shape variables of the labyrinth disc, rectifying the labyrinth clearance, constructing a new iteration of the fluid domain model of the hollow, and recalculating the flow and temperature fields.

4 Repeat steps 1 to 3, extracting the labyrinth clearance until the relative change between the gap value calculated in the current iteration and that from the previous iteration is less than 0.5%, at which point the iteration is considered convergent and has reached equilibrium. The present flow field characteristics exhibit those of a convergent flow field from the preceding phase.

2.2. Regulatory Equation

2.2.1. Equation for Fluid Domain Control

This paper analyses the interrelation of multiphysical fields. The governing equations of the fluid domain consist of the continuity equation, the momentum equation, and the energy equation, which are as follows:

$${\displaystyle \frac{\partial {\rho }_f}{\partial t}}+\nabla \cdot ({\rho }_{f}v)=0$$

(1)

$${\displaystyle \frac{\partial {\rho }_{f}v}{\partial t}}+(\nabla \cdot {\rho }_{f}vv-{\tau }_f)=f_f$$

(2)

$$\begin{array}{l}{\displaystyle \frac{\partial (\rho {\mathrm{h}}_{\mathrm{tot}})}{\partial t}}-{\displaystyle \frac{\partial p}{\partial t}}+\nabla \cdot ({\rho }_f{v}_f{h}_{tot})=\\ \nabla \cdot (\lambda \nabla T)+\nabla \cdot (v\cdot \tau )+v\cdot \rho f_f+S_E\end{array}$$

(3)

t represents time, $f_f$ depicts the volume force vector, ${\rho }_f$ means fluid density, $v_f$ indicates the fluid velocity vector, h refers to enthalpy, $\lambda$ represents the heat transfer coefficient, $S_E$ is the energy source term, and ${\tau }_f$ is the shear force tensor, which may be expressed as follows:

$${\tau }_f=(-p+\mu \nabla \cdot v)I+2\mu e$$

(4)

In this context, p represents the fluid pressure, $\mu$ denotes the dynamic viscosity, and $e$ signifies the velocity stress tensor, $e={\displaystyle \frac{1}{2}}(\nabla v+\nabla v^T)$.

2.2.2. Robust Domain Control Equation

Fundamental energy equation:

$${\displaystyle \frac{\partial (\rho {\mathrm{h}}_{\mathrm{tot}})}{\partial t}}+\nabla \cdot ({\rho }_s{v}_s{h}_{tot})=\nabla \cdot (\lambda \nabla T)+S_E$$

(5)

Fundamental force equation:

$${\rho }_s{\ddot{d}}_s=\nabla \cdot {\sigma }_s+f_s$$

(6)

In this context, $\rho s$ represents the solid density; $\sigma s$ denotes the Cauchy stress tensor; $fs$ signifies the volume force vector; and $\ddot{d}s$ indicates the local acceleration vector within the solid domain.

2.2.3. Equation for Fluid–Solid-Thermal Coupling

The fluid–solid-thermal coupling interface of the labyrinth seal must ensure that the stress, displacement, heat flow, and temperature of both the fluid and the solid are equivalent. The equation governing fluid–solid coupling control is as follows:

$$\left\{\begin{array}{l}n\cdot {\tau }_f=n\cdot {\tau }_s\\ d_f=d_s\\ q_f=q_s\\ T_f=T_s\end{array}\right.$$

(7)

In this context, n represents the normal vector of the interface, and the stress distribution across the interface must be equilibrated in the normal direction. $\tau$ denotes stress; q signifies heat flow; T represents temperature; the subsequent table s illustrates the solid domain, while the following table f depicts the fluid domain.

3. Validation of Research Object and Methodology

3.1. Subject of Investigation

This research analyses the stability of the sealing characteristics of the labyrinth seal design within the rotor and stator cavities. The centrifugal deformation caused by rotation and the temperature rise from wind resistance leads to changes in the sealing clearance of the labyrinth seal, affected by thermal deformation, which in turn impacts the leakage flow and temperature fluctuations in both the fluid and solid domains. The research focusses on the labyrinth cavity design of an aero-engine. Figure 2 is a two-dimensional schematic illustration of the labyrinth seal cavity concept. The computational domain chiefly includes the solid domain of the labyrinth seal, the fluid domain of the labyrinth seal, and the fluid domain of the rotating cavity. The labyrinth disc consists of five linear teeth, each exhibiting identical tooth profile characteristics. The fluid passes through the labyrinth seal arrangement at the high-pressure compressor exit and into the disc cavity.

Figure 3 illustrates the schematic diagram of the structural properties of the labyrinth teeth examined in this study. The primary structural characteristics include the following: seal clearance c, number of labyrinth teeth N, tooth spacing B, height of labyrinth teeth H, tooth top width w, forward inclination angle α₁, backward inclination angle α₂, among others. The precise parameters are presented in

Table 1. The structural parameters of the labyrinth seal research object.

Structural Parameters	Value
Number of Teeth (N)	5.00
Tooth Pitch (B)/mm	4.50
Tooth Height (H)/mm	4.80
Clearance (c)/mm	0.50, 0.30
Tooth Width (w)/mm	0.25
Front inclination angle (α1)/°	10.00
Rear inclination angle (α2)/°	10.00
Outer radius of labyrinth/mm	247.00
Inner radius of labyrinth/mm	66.00
Bolt diameter/mm	10.00
Bolt length/mm	35.00
Distance of bolt and tooth tip/mm	39.50

, and the numerical simulation model is illustrated in Figure 4.

3.2. Grid Partitioning and Autonomy Validation

This paper discusses ANSYS 19.2. Meshing is employed to create a mesh for the revolving disc chamber of a labyrinth seal. Unstructured grids are utilised in their entirety. Five sets of grids of varying dimensions are established for the model, with the grids encrypted at the tooth apex of the labyrinth. The mesh size of the tooth tip is established at 0.06 mm. Simultaneously, boundary layer grids are created on the wall surface to guarantee that the Y + value of the simulation results complies with the turbulence model requirements.

Table 2. Grid parameters and calculation results.

Num.	Grid Size (mm)	Grid Number (×104)	Mass Flow (g/s)
1	4	146.65	6.967
2	3	205.57	7.897
3	2	312.27	8.03
4	1	580.87	7.978
5	0.5	1515.98	7.967

and Figure 5 present the primary mesh tooth size, mesh number, and flow calculation outcomes for the five grid sets. The graphic demonstrates that a mesh size of 0.5 mm yields a relative error of 0.14% compared to the flow calculation result of 1 mm, which meets the specified standards. Figure 6 and Figure 7 depict local grids with grid scales of 1 mm and 0.5 mm, respectively.

3.3. Establishment of Boundary Conditions

The boundary for the fluid–solid coupling computation is defined as illustrated in Figure 8 and

Table 3. Boundary conditions parameter.

Boundary	Value/Condition
Inlet Pressure (P*in)/atm	1.6, 2.0, 2.4
Outlet Pressure (P*out)/atm	1
Inlet Temperature (T)/K	298.15
Rotational velocity/rev min−1	15,000
Fluid media	Perfect gas
Turbulence model	SST k-Omega
Wall condition	Anti-slip wall

. The ideal gas serves as the fluid medium, with the static pressure boundary set at the outlet and the total temperature and total pressure boundaries specified at the inlet. The SST k-Omega turbulence model has been chosen as the turbulence model. The periodic boundary is established on both sides of the fluid and solid domains of the labyrinth, the coupling heat transfer boundary is defined at the interface between the fluid and solid domains, and the wall condition is designated as no slip.

A fixed constraint is imposed on the mounting bolt of the labyrinth plate to restrict its radial displacement; the velocity boundary is applied to the labyrinth plate; the temperature field boundary, computed by CFD 19.2 software, is applied to the entire solid domain of the labyrinth plate to assess the deformation resulting from the combined effects of centrifugal force and thermal resistance due to wind. The specific solid domain boundary conditions are shown in Figure 9.

3.4. Verification of Solution Accuracy

3.4.1. Validation of the Temperature Rise Characteristics Due to Wind Resistance

This paper replicates the model from the literature [28] to validate the correctness of the numerical method, as illustrated in Figure 10. The wind resistance temperature rise characteristics of the labyrinth seal are analysed using the identical meshing technique and boundary condition settings outlined in Section 2.2, along with the same operational conditions referenced in the literature. The temperature differential between the inlet and outlet sections of the labyrinth seal is extracted as the wind resistance temperature increase.

Table 4. Comparison of solution results.

	ΔTtot	Relative Error
Reference [28]	4.08	-
Results of CFD	3.92	4.06%

presents the comparison between the calculated values and the literature [28]. The relative inaccuracy of the wind resistance temperature increase in the labyrinth seal section is 4.06%, satisfying the accuracy criteria of the solution.

3.4.2. Validation of Leakage Characteristics Accuracy

An experimental apparatus to assess the leaking characteristics of labyrinth seals was conceived and constructed [29]. The test portion consists of a detachable and replaceable labyrinth test component and a sealing element. The assembly configuration is illustrated in Figure 11. The bottom end of the test specimen is secured and calibrated using a dual-direction precision vice to modify the sealing gap between the labyrinth test specimen and the sealing component. The upper portion of the test specimen is secured with bolts to an adjustable inlet segment. Figure 12 illustrates the labyrinth seal test specimen, whereas Figure 13 and

Table 5. Labyrinth seal experimental structure parameters.

Structural Parameters	Value
Tooth Pitch (B)/mm	3.00
Number of Teeth (N)	5
Tooth Height (H)/mm	3.20
Clearance (c)/mm	0.57, 0.67, 0.77
Tooth Width (w)/mm	0.30
Front inclination angle (α1)/(°)	10.00
Rear inclination angle (α2)/(°)	10.00
Piece width/mm	200.00

provide the precise dimensional specifications of the labyrinth seal test specimen. The sealing gap was adjusted with the stopper ruler during the experiment. Following the modification and securing of the gap, its dimensions were remeasured using the stopper ruler to ensure precision. The exit pressure is atmospheric, and the leakage characteristics at different pressure ratios are ascertained by adjusting the inlet pressure.

The conversion flow serves as the evaluative metric for the leaking properties of labyrinth seals, with the calculation procedure outlined as follows:

$$\psi ={\displaystyle \frac{\dot{m}\sqrt{T_{\mathrm{in}}^{\ast }}}{p_{\mathrm{in}}^{\ast }}}$$

(8)

where m represents the mass flow rate in kg/s; $T_{\mathrm{in}}^{\ast }$ denotes the total inlet temperature in K; $p_{\mathrm{in}}^{\ast }$ indicates the total inlet pressure in Pa. Figure 14 illustrates the trend of the actual and simulated values of the conversion flow of the labyrinth test specimen in relation to the pressure ratio. The chart indicates that the overall pattern of the experimental results aligns well, with the experimental data generally exceeding the simulation data, particularly exhibiting higher errors at lower pressure ratios. The greatest inaccuracy of a 0.57 mm gap is 7.72% at a pressure ratio of 1.1. The largest mistake of 0.67 mm occurs at a pressure ratio of 1.2, with a maximum error percentage of 8.56%. The highest mistake of 0.77 mm occurs at a pressure ratio of 1.2, with a maximum error percentage of 8.14%. The reliability of the numerical method utilised in this study is validated.

4. Stability Assessment of Traditional Configuration

4.1. Establishment of Model Grid and Boundary Conditions

The traditional geometric model aligns with the model presented in Section 2.1, as illustrated in Figure 4. The grid division utilises the validated 1 mm grid scale. The temperature monitoring sites are established in both the fluid and solid computational domains, as illustrated in Figure 15.

4.2. Analysis of Sealing Stability in Conventional Configurations

4.2.1. Analysis Under a Cold Gap Condition of 0.5 mm

Figure 16 depicts the meridian streamline diagram for the conventional configuration at a pressure ratio of 2.0 and a rotational speed of 15,000 revolutions per minute. The streamline diagram illustrates that as airflow traverses the gap between the labyrinth and the stator wall, a throttling effect occurs due to the abrupt decrease in flow area, resulting in diminished static pressure and augmented flow velocity. A jet is generated after traversing the sealing gap. A portion of the airflow enters the tooth cavity, creating a vortex, while the remaining airflow traverses along the stator wall to the subsequent stage of the labyrinth gap.

Figure 17 and Figure 18 illustrate the correlation between the radial deformation of the labyrinth disc and the sealing gap with respect to the number of iterations. Figure 19 and Figure 20 illustrate the temperature variation within the computational domain and the leakage flow via the labyrinth sealing gap, respectively. Following centrifugal analysis, the traditional arrangement undergoes radial deformation, leading to a reduction in the sealing gap and a diminished flow rate via the labyrinth seal gap, which therefore results in a temperature increase. The application of the temperature field to the labyrinth disc results in an augmented radial deformation due to the synergistic effects of centrifugal force and thermal expansion from the rotating disc. As a result, the flow through the sealing gap and labyrinth seal decreases, leading to increased temperatures in both the fluid and solid domains. With an increase in the number of repetitions, the radial deformation, clearance, leakage flow, and temperatures of both the fluid and solid domains tend to reach a state of equilibrium.

Comparing the sealing qualities of the typical arrangement under several pressure ratios reveals that the radial distortion of the labyrinth disc diminishes as the pressure ratio increases. The increase in pressure ratio leads to heightened leakage flow through the labyrinth seal’s rotating disc cavity structure, causing a reduction in the perceived wind resistance temperature. This, in turn, diminishes the thermal expansion of the labyrinth disc due to wind resistance heat, resulting in a decrease in the labyrinth seal clearance trend.

4.2.2. Analysis Under a Cold Gap Condition of 0.3 mm

Figure 21 and Figure 22 illustrate the radial deformation and the variation in sealing gap of the standard labyrinth seal, featuring a cold gap of 0.3 mm. Figure 23 and Figure 24 illustrate the temperature variation within the computational domain and the leakage flow via the labyrinth seal gap, respectively. Figure 21 and Figure 22 illustrate that as the number of repetitions grows, radial deformation progressively rises while the sealing gap diminishes. The reduction in seal clearance diminishes the flow rate through the labyrinth structure, significantly enhancing the wind resistance temperature rise effect. This results in a notable increase in the temperatures of both the fluid and solid domains, ultimately resulting in increased radial deformation of the labyrinth plate due to thermal expansion, which further reduces the seal clearance. The reduced sealing gap diminishes the flow rate throughout the disc cavity, leading to an elevated temperature due to wind resistance, which subsequently raises the temperature of both the airflow and the labyrinth disc. This creates a detrimental cycle that ultimately results in the rubbing phenomenon between the labyrinth tooth tip and the stator wall.

Using a pressure ratio of 2.0 as an example, after three iterations, the radial deformation is 0.33 mm, while the sealing gap is −0.03 mm. The collision between the labyrinth disc and the stator wall is deemed to occur at this moment.

5. Enhanced Configuration Design and Stability Assessment

5.1. Enhanced Configuration Model and Study of Mechanical Properties

In the conventional configuration, with a cold gap of 0.3 mm and a rotation speed of 15,000 rev/min, the radial deformation of the labyrinth plate, resulting from the combined effects of wind resistance heating and centrifugal force, is excessive, leading to contact between the labyrinth’s tooth tip and the stator wall. A revised arrangement is suggested to mitigate this problem.

To enable further studies, the enhanced grate tooth structure is designated, and the locations of the grate tooth (inner ring, outer ring, and side) are identified. The identical position is designated in accordance with the airflow direction, as seen in Figure 25.

Figure 26 illustrates the simulation results of centrifugal deformation for several enhanced designs. The calculation results indicate that the largest deformation of configuration A occurs at the cantilever end, with a maximum deformation of 0.289 mm at that location. The deformation of the outer ring labyrinth measures 0.097 mm and 0.105 mm, while the deformation of the inner ring labyrinth measures 0.162 mm, 0.210 mm, and 0.258 mm, respectively. The maximum distortion of configuration B occurs at the terminal end of the cantilever of the labyrinth plate. The maximum deformation of the end labyrinth is 0.106 mm, the outer ring labyrinth is 0.104 mm, and the inner ring labyrinth is 0.099 mm.

5.2. Analysis of Sealing Stability for Enhanced Configuration

Figure 27 illustrates the geometric model of the enhanced configuration calculation domain. Temperature monitoring stations are established in both the fluid and solid computational domains, with their locations depicted in Figure 28.

5.2.1. Analysis of Sealing Stability for Configuration A

Figure 29 illustrates the meridional streamline diagram of the enhanced configuration A under conditions of a cold gap of 0.3 mm, a pressure ratio of 2.0, and a spinning speed of 15,000 revolutions per minute. The graphic illustrates that as airflow traverses the gap between the outer ring labyrinth and the stator wall, a throttling effect occurs due to the abrupt decrease in flow area, resulting in diminished static pressure and heightened flow velocity. A jet is generated after traversing the sealing gap. A portion of the airflow enters the tooth cavity, creating a vortex, while the remaining airflow travels down the stator wall to the subsequent stage of the labyrinth gap. Upon traversing the outer ring labyrinth, the airflow proceeds into the inner ring labyrinth via the diversion cavity, then entering the disc cavity after undergoing throttling through the inner ring labyrinth.

When the cold gap measures 0.3 mm, the alterations in radial deformation and sealing gap for the enhanced configuration A are illustrated in Figure 30 and Figure 31, while the temperature variations within the calculation domain and the leakage flow through the labyrinth seal gap are depicted in Figure 32 and Figure 33. Figure 30 and Figure 31 illustrate that when the number of iterations grows, the radial distortion of both the inner and outer ring labyrinths progressively escalates. The outer ring labyrinth gradually nears the stator wall, causing a reduction in the sealing gap; in contrast, the inner ring labyrinth slowly withdraws from the stator wall, resulting in an expansion in the sealing gap. The sealing effectiveness of the outer ring labyrinth has improved, whereas that of the inner ring labyrinth has decreased. Analysis of the leakage flow rate depicted in Figure 33, in conjunction with the temperature variations at the monitoring point illustrated in Figure 32, reveals that the sealing performance diminishes due to radial deformation caused by the denser arrangement of labyrinth teeth in the inner ring, while the sealing efficacy of the less densely distributed labyrinth teeth in the outer ring is enhanced. The alteration in the sealing efficacy due to the radial deformation of the entire labyrinth disc is minimal, resulting in little overall reduction in the leakage flow rate, as well as a marginal increase in the temperature of the calculation domain, which ultimately stabilises. The phenomenon of friction between the labyrinth tooth tips and the stator wall, induced by the adverse cycle resulting from wind resistance, is mitigated.

Figure 30, Figure 31, Figure 32 and Figure 33 illustrate that as the pressure ratio increases during iteration to a stable state, the radial deformation of the labyrinth disc progressively diminishes, the sealing gap of the outer ring labyrinth gradually decreases, and the sealing gap of the inner ring labyrinth progressively increases. The leakage flow rate escalates with an increased pressure ratio, but the temperatures of both the fluid and solid domains diminish as the pressure ratio ascends. The increased pressure ratio enhances the flow rate in the cavity, resulting in improved heat dissipation, hence lowering the temperatures of both the solid and fluid domains. This consequently causes radial deformation as a result of wind resistance and centrifugal forces.

5.2.2. Analysis of Sealing Stability for Configuration B

Figure 34 illustrates the streamline diagram of the enhanced configuration B under the conditions of a cold gap of 0.3 mm, a pressure ratio of 2.0, and a speed of 15,000 revolutions per minute. The streamline diagram illustrates that as airflow traverses the gap between the outer ring labyrinth and the stator wall, a throttling effect occurs due to the abrupt decrease in flow area, resulting in diminished static pressure and heightened flow velocity. A jet is generated after passing through the sealing gap. A portion of the airflow enters the tooth cavity, creating a vortex, while the remaining airflow travels down the stator wall to the subsequent stage of the labyrinth gap. Upon traversing the outer ring labyrinth, the airflow enters the inner ring labyrinth via the diversion cavity and subsequently flows into the disc cavity after undergoing throttling through the inner ring labyrinth.

When the cold gap measures 0.3 mm, the alterations in radial deformation and sealing gap of the enhanced configuration B labyrinth disc are illustrated in Figure 35 and Figure 36. The temperature variation within the computational domain and the leakage flow via the labyrinth seal gap are illustrated in Figure 37 and Figure 38. Figure 35 and Figure 36 illustrate that when the number of iterations grows, the radial deformation of the inner ring labyrinth, outer ring labyrinth, and end labyrinth progressively escalates. The outer ring labyrinth progressively nears the stator wall, resulting in a diminishing sealing gap. The inner ring labyrinth progressively distances itself from the stator wall, resulting in an increasing sealing gap. The radial displacement of the terminal labyrinth minimally influences its sealing gap. The sealing efficacy of the outer ring labyrinth has been enhanced, the sealing efficacy of the inner ring labyrinth has diminished, while the sealing efficacy of the end labyrinth remains mostly unaltered. The variation in leakage flow rate depicted in Figure 38, in relation to the number of iterations and the temperature fluctuations at the monitoring point shown in Figure 37, indicates that a greater concentration of labyrinth teeth is present at the terminus of the labyrinth, with only two labyrinth teeth allocated in both the inner and outer rings. The sealing effectiveness of the complete labyrinth disc is limited due to radial distortion; consequently, the leakage flow rate decreases somewhat, the temperature within the calculation domain increases slightly, and ultimately reaches stability. The friction between the labyrinth teeth tips and the stator wall, caused by adverse wind resistance, is reduced.

Nonetheless, the enhanced structure exhibits increased sensitivity to axial movement because to the greater distribution of labyrinth teeth at the terminus. The impact of axial movement on the sealing clearance of the enhanced configuration end labyrinth teeth is not the focus of this study.

Figure 35, Figure 36, Figure 37 and Figure 38 illustrate that as the pressure ratio escalates during iteration towards a stable state, the radial deformation of the labyrinth disc progressively diminishes, the sealing gap of the outer ring labyrinth decreases, and the sealing gap of the inner ring labyrinth increases, while the end labyrinth gap remains unchanged. The leakage flow rate escalates with an increase in pressure ratio, whereas the temperatures of both the fluid and solid domains diminish as the pressure ratio rises. The rationale is akin to the enhanced configuration A, which is not reiterated.

5.3. Comparison of Sealing Stability Between Traditional Setup and Enhanced Configuration

The stability of the sealing properties of the conventional and enhanced configurations is assessed by utilising the cold gap, temperature, and leakage flow rate during iterations until final stability is achieved as assessment criteria.

The leakage and temperature increase of various constructions with a 0.5 mm cold gap are illustrated in Figure 39 and Figure 40. Figure 39 illustrates that, with a 0.5 mm cold gap, the conventional hot leakage flow rate decreases by 45.2% relative to the cold leakage flow rate. The hot flow rate of improved structure A is 4.61% compared to the cold flow rate, while improved structure B exhibits a hot flow rate that is 1.44% relative to the cold flow rate. Figure 40 illustrates that, at a cold state gap of 0.5 mm, the temperature of the typical fluid domain exceeds the cold state by 9.38%, whereas the temperature of the solid domain surpasses the cold state by 5.39%. Relative to the cold state, the temperature of the fluid domain in the enhanced structure A has risen by 0.27%, while the temperature of the solid domain has increased by 0.28%. The temperature of the fluid domain in the enhanced structure B is 0.17% elevated compared to the cold state, whereas the temperature of the solid domain is 0.05% elevated relative to the cold state.

The leakage and specific temperature increase of various constructions with a cold gap of 0.3 mm are illustrated in Figure 41 and Figure 42. The standard design is undermined by the gap. The leakage and temperature values from the previous iteration stage before rubbing are compared and evaluated. Figure 41 indicates that, with a 0.3 mm cold gap, the conventional hot leakage flow rate diminishes by 97.31% relative to the cold leakage flow rate. The hot flow rate of improved structure A varies by 44.79% compared to the cold flow rate, while the hot flow rate of improved structure B varies by 18.11% compared to the cold flow rate. Figure 42 illustrates that the temperature of the traditional hot fluid domain and the solid domain increased by 81.13% and 58.79%, respectively, compared to the cold state with a 0.3 mm cold gap. In comparison to the cold state, the temperature of the hot fluid domain and the solid domain of the enhanced structure A rose by 15.93% and 9.81%, respectively. In comparison to the cold state, the temperature of the hot fluid domain and the solid domain of the enhanced structure B rose by 9.55% and 4.09%, respectively.

In the context of a cold gap of 0.5 mm, the temperature variations at the leakage and monitoring points in both cold and hot states are more pronounced, whereas the temperature changes in the improved design are less significant compared to the conventional configuration. In the context of a cold gap of 0.3 mm, the conventional type experienced significant rubbing, resulting in a marked reduction in flow rate and a substantial increase in temperature. Conversely, the improved configuration exhibited reduced leakage and temperature to varying extents, while the rubbing phenomenon was absent, thereby mitigating the instability of the labyrinth clearance under conditions of a small cold gap.

6. Conclusions

The primary conclusions derived from the aforementioned research are as follows:

1. A numerical model was created to assess the stability of the sealing characteristics of the labyrinth seal. The stability of the sealing qualities of the conventional labyrinth seal configuration was attained by the sequential solution method of fluid–solid-thermal coupling. The conventional labyrinth seal configuration ultimately exhibited stability at a cold gap of 0.5 mm. Under a cold gap of 0.3 mm, thermal instability ultimately resulted in rubbing failure, markedly decreasing leakage flow and drastically elevating temperature.

2. In response to the rubbing failure phenomenon of the traditional labyrinth in a cold gap of 0.3 mm, two enhanced configurations are provided, and a stability analysis of the sealing features of the improved configurations is conducted. The two enhanced configurations mitigate the rubbing failure phenomenon of the traditional configuration caused by thermal instability at a cold gap of 0.3 mm, resulting in increased stability of the gap.

3. Upon comparing the stability of the sealing characteristics between the conventional configuration and the two enhanced configurations, it is evident that the leakage and temperature rise characteristics of the two enhanced configurations are significantly lower than those of the conventional configuration at a cold gap of 0.5 mm. At a cold gap of 0.3 mm, thermal instability due to the wind resistance heat effect in the standard arrangement results in rubbing failure, whereas the two enhanced configurations exhibit stability. The sealing characteristic stability of the enhanced configuration surpasses that of the traditional arrangement.

4. Under varying clearances, the leakage and temperature fluctuations in the hot state of the enhanced configuration B are less pronounced than those in the cold state, compared to the enhanced configuration A. Furthermore, the sealing characteristics of the enhanced configuration B exhibit superior stability relative to those of the enhanced configuration A. However, the enhanced configuration B incorporates an end labyrinth structure, which is influenced by axial movement factors in practical applications, necessitating further research.