# Nonuniform Clearance Effects on Pressure Distribution and Leakage Flow in the Straight-through Labyrinth Seals

Authors: Yu Shi; Shuiting Ding; Tian Qiu; Peng Liu; Chuankai Liu
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

Straight-through labyrinth seal is a simple and reliable noncontact seal structure widely used in aeroengines. During the actual operation of aeroengines, the labyrinth seal clearance might experience a nonuniform variation in the flow direction due to asymmetric structure and uneven temperature. In order to characterize the degree of nonuniformity, the nonuniformity coefficient is defined in current paper. The effect of nonuniform causes (rotor deformation or stator deformation), nonuniform type (convergent clearance or divergent clearance), and nonuniformity coefficient (nonuniformity degree) is carefully studied by numerical simulation. Comparative analysis has shown that there is no obvious difference in flow coefficient (less than 0.8% in current studies) between two nonuniform causes. As for nonuniform type, the nonuniformity impact of divergent type on the flow coefficient is more significant than that of convergent type, as a result of stronger axial inertia. Pressure distributions of teeth cavities indicate that the total pressure drop of the divergent type is more obvious than that of the convergent type when the pressure ratio is the same. With the same dimensionless minimum tip clearance, clearance nonuniformity would result in the increase of leakage flow of the straight-through labyrinth, particularly for the condition with small pressure ratio, large circumferential Mach number, and small dimensionless minimum tip clearance. When the nonuniformity coefficient is within the range of -0.1 to 0.1, the variation curves of nonuniformity impact factor versus the nonuniformity coefficient almost coincide (the maximum deviation is no more than 1.2%) under different operation conditions (Reynolds number, pressure ratio, circumferential Mach number, and dimensionless minimum tip clearance). Current work proves it that the effect of seal clearance nonuniformity on the leakage flow requires special attention for the refined design of aeroengines.


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## Body

## 1. Introduction

The secondary air system (SAS) plays an important support role to ensure the safe and efficient operation of aeroengines. [1] Increasing demands for high performance, energy-saving, and safety levels for aeroengines are attracting more interest in the accurate analysis and refined design of the secondary air system. Labyrinth, as an important noncontact rotor-stator sealing device, has been widely used in the secondary air system of aeroengines due to its simple structure and high reliability. Labyrinth seal performs as a metering device to control internal airflow, thereby affecting some safety-related functions of SAS (prevention of hot gas ingestion into the turbine disk cavities, turbine blade cooling, and axial bearing loading control). Therefore, the labyrinth sealing flow should be analyzed meticulously for the sake of safety.A large number of scholars have made detailed studies on the factors affecting the sealing flow of labyrinths, including rotational speed [2–4], Reynolds number [5], pressure ratio [6], and geometric parameters [5–9]. The structure of labyrinth is complex, and there are many geometric parameters that affect the sealing flow, such as radius, sealing clearance, tooth tip width, tooth height, tooth pitch, stage number, and inclination angle. Zhao and Wang [10] carried out shape optimization of a straight labyrinth seal to reduce gas leakage and pointed out that the sealing clearance was one of the important optimized parameters. Other researches also revealed that the influence of labyrinth sealing clearance on sealing flow is particularly significant [3, 5, 6, 11–14]. Nayak [3] claimed that the effects of rotation on leakage characteristics of straight-through labyrinths were not exactly the same under different sealing clearance. Li et al. [4] investigated the effects of pressure ratio and rotational speed on leakage flow and cavity pressure in the staggered labyrinth seal. Hu et al. [5] illustrated that flow resistant characteristics of straight-through labyrinths were related to the ratio of equivalent diameter (double sealing clearance) to seal length. Wu and Andrés [6] found cavity pressure reduced, but seal leakage rose with the increase of sealing clearance. Wu and Andrés [11] also investigated the effect of sealing clearance on wall friction factors, relating to sealing flow in labyrinth seals. Yan et al. [12] studied the effect of sealing clearance change caused by teeth bending and mushrooming damages on leakage performance of straight-through labyrinth seals. Andrés et al. [13] compared the sealing leakage and tooth cavity pressure distribution with or without swirl brakes under three different sealing clearances. Lee et al. [14] carried out flow visualization measurement within the labyrinth seal by Schlieren method, investigating the influence of dimensionless sealing clearance on the leakage flow. They also pointed out that the leakage characteristics of stepped and straight-through seals showed different trends. As summarized, sealing clearance is recognized as one of the most important factors affecting sealing flow in labyrinth seals. However, much of the research up to now has been carried out under a fixed clearance.In the actual operations of aeroengines, the sealing clearance of labyrinth would be variable due to the deformation of components caused by mechanical loads and thermal loads. Considering the sealing clearance difference, Li et al. [15] measured leakage coefficients of labyrinth seal in the cold and hot states and compared them with floating ring seal. Kong et al. [16, 17] also paid attention to the variation of labyrinth clearance due to centrifugal deformation when investigating the sealing performance of rotating labyrinth. The labyrinth clearance was measured to include the effects of labyrinth clearance variation on leakage flow. Eastwood et al. [18] built an advanced multiconfiguration stator well cooling test facility and carried out displacement measurements for sealing clearance. The hot running seal clearance grew up by approximately 30% of cold build clearance. Ganine et al. [19] used solid finite element model coupled with one-dimensional flow network to predict radial clearances evolution of the 14 labyrinth seals in a square cycle. From their simulation results, the widest range of radial clearance variation was about from 40% to 180% of cold built clearance.However, the sealing clearance profile of the deformed labyrinth may not always keep uniform during the engine operation process. Recent evidence has shown that thermal expansion may also make sealing clearance nonuniform along the flow direction due to the uneven temperature distribution [20]. Sun et al. [21] found that convergent and divergent clearance profiles have different effects on leakage flow when the nonuniformity degree is the same. Shi et al. [22] discussed the effects of nonuniform clearance on windage heating and swirl development under different Reynolds numbers, pressure ratios, circumferential Mach numbers, and dimensionless minimum clearances. Up to now, the nonuniform clearance effects on pressure distribution and leakage flow in the straight-through labyrinth seal is still lack of systematic and in-depth research. The leakage flow characteristics of the labyrinth with a nonuniform clearance are not clear, leading to an inaccurate flow prediction of SAS in the operation of aeroengines. In current studies, the straight-through labyrinths are taken as the investigation objective, and the nonuniformity coefficient is proposed to characterize the degree of the sealing clearance nonuniformity. The influences of the sealing clearance nonuniformity (nonuniform causes, nonuniform type, and nonuniformity coefficient) on the pressure distribution and the leakage flow are discussed in detail. This investigation reveals new understanding into the labyrinth sealing performance, contributing to the performance assessment and safety design of the aerothermomechanical system, including aeroengines.


## 2. Methodology

### 2.1. Description of Nonuniform Sealing Clearance

#### 2.1.1. Geometric Configuration

A straight-through labyrinth is regarded as the research objective of this investigation, which consists of four stages of labyrinth teeth, as shown in Figure1. The reasons of nonuniform deformation of sealing clearance usually include asymmetric structure and uneven temperature, resulting in the monotonic profile of the labyrinth tip clearance. The labyrinth clearance becomes larger or smaller gradually along the flow direction as shown in Figure 2. Generally, the labyrinth seal has a pretty tiny axial dimension, structured no more than tens of millimeters. In such a limited space, the tip clearances of different stages caused by the nonuniform deformation can be simplified to be linear distribution according to the previous investigation [21–23]. Therefore, in this work, the assumption of the linear clearance profile is also adopted, and the nonuniformity coefficient u is employed to depict the nonuniformity of the labyrinth seal clearance as defined in the authors’ previous investigation [22]. The absolute value of the nonuniformity coefficient u can symbolize the nonuniformity degree of the labyrinth seal clearance, which is defined as the ratio of the maximum tip clearance deviation to twice of the average sealing clearance.
(1)u=cmax−cmin2cav.Figure 1
Geometric model.Figure 2
Nonuniform seal clearance.cmax is the maximum tip clearance, cmin is the minimum tip clearance, and cav is the average tip clearance.As for a labyrinth with four stages of teeth, Equations (2) and (3) provide the calculation method of the average labyrinth tip clearance and the nonuniformity coefficient.
(2)cav=14∑i=14ci,(3)u=c1−c42cav.ci is the tip clearance of the ith stage. u=0 implies a uniform sealing clearance, and u=1 implies a zero minimum labyrinth tip clearance, meaning a situation of rub-impact.The convergent type of sealing clearance is characterized byu>0, while the divergent type of sealing clearance is characterized by u<0. The critical safety-limit parameter for the seal device is the minimum tip clearance. Consequently, the minimum clearance, replacing the mean clearance, together with the nonuniformity coefficient is regarded as the key geometric parameter for clearance nonuniformity description. In this investigation, the minimum tip clearance range is 0.2mm≤cmin≤0.6mm, and the nonuniformity coefficient range is −0.5≤u≤0.5. Table 1 includes other detailed geometric dimensions. The meaning of each parameter has been explained in Figure 1.Table 1
Geometric parameters.
ParameterValueα10°B6 mmH4 mmLi6 mmLo6 mmr254 mmt0.5 mm


#### 2.1.2. Dimensionless Parameters

In the labyrinth seal, after the airflow is accelerated by the throttling of the former tip clearance, the kinetic energy of the fluid cannot be completely dissipated into thermal energy in the following cavity. Because of the inertia, part of the kinetic energy is directly transported from the upstream throttle to the downstream throttle. The kinetic energy carry-over effect will weaken the sealing ability. The axial Mach numberMA is an important parameter to describe the axial inertia in the labyrinth seal, which is defined as
(4)MA=VAa.VA is the axial velocity of airflow, and a is the local sound speed.In this study, the pressure coefficientψ is adopted to characterize the dimensionless total pressure [4, 24]. The definition is shown in
(5)ψ=pt−ps,outpt,in−ps,out.pt is total pressure in a tooth cavity, pt,in is total pressure of seal inlet, and ps,out is static pressure of seal outlet.Total pressure drop is an important characterization of the labyrinth seal. For the whole labyrinth seal, the pressure load of labyrinth seal∆ψseal is defined as the pressure coefficient difference between the inlet and the outlet, as shown in Equation (6). For a single stage of the labyrinth tooth, the pressure load of labyrinth tooth ∆ψi is defined as the pressure coefficient difference between the front and rear tooth cavity, regarded as the contribution to the total pressure drop, shown in Equation (7).
(6)∆ψseal=ψinlet−ψoutlet,(7)∆ψi=ψi−ψi+1.ψinlet and ψoutlet are the pressure coefficient of labyrinth seal inlet and outlet, respectively. ψi is the pressure coefficient in the ith teeth cavity.As one of most important characteristic quantity, the flow coefficient [4, 24, 25] is introduced to characterize the dimensionless mass flow rate. The definition is shown in
(8)φ=ṁRTt,in1/2Apt,in.ṁ is leakage mass flow rate, Tt,in is inlet total temperature of seal inlet, R is gas constant, and A is the cross-section area of the labyrinth tip clearance. For labyrinths with nonuniform tip clearances, cross-section area of the labyrinth minimum tip clearance Amin is used in Equation (8) instead of A.Denecke et al. [26] made a dimensional analysis on the leakage flow of rotating labyrinth seal and simplified the parameters affecting flow characteristics according to the results of numerical simulation. Denecke et al. thought that flow characteristics were in connection with axial Reynolds number, effective pressure ratio, circumferential Mach number, geometric parameters, inlet preswirl ratio, specific heat ratio, and Prandtl number. In this study, the inlet preswirl is not the focus of the nonuniform clearance effect, so the air enters the sealing structure along the axial direction in the subsequent numerical simulation, in that situation where the ratio of inlet total pressure to outlet static pressure π=pt,in/ps,out is equivalent to the effective pressure ratio [26]. Specific heat ratio and Prandtl number are considered as temperature-independent since the air would get little temperature increase within the sealing device. As for the geometric parameters, only the seal clearance, including the minimum tip clearance and the nonuniformity coefficient, is investigated to get insight into the influence of the nonuniform clearance. To improve the universality of research results, the minimum tip clearance is normalized by the labyrinth pitch. Nonuniformity coefficient u and dimensionless minimum tip clearance C=cmin/B are employed jointly to depict the sealing clearance. Consequently, flow coefficient and pressure coefficient of the labyrinth seal can be expressed as
(9)φ=fπ,Re,MU,C,u.Definitions of Reynolds number Re and circumferential Mach numberMU are shown in Equations (10) and (11), respectively.
(10)Re=2ρVcminμ=ṁμΠr,(11)MU=Ωra.ρ, μ, and V are the density, the dynamic viscosity, and the velocity of the labyrinth seal inlet, respectively. r is the radius of the labyrinth teeth tip, Π is circumference ratio, Ω is angular velocity of labyrinth rotor, and a is local sound speed.In this investigation, the effect of clearance nonuniformity on the leakage flow is focused on, so the nonuniformity impact factorE is defined to describe the influence of nonuniformity degree on the flow coefficient, shown as
(12)E=φuφ0.φu is the flow coefficient of the nonuniform clearance labyrinth (u≠0), while φ0 is the flow coefficient of the uniform clearance labyrinth (u=0) with the same minimum tip clearance.


### 2.2. Numerical Simulation Method

#### 2.2.1. Numerical Model

The commercial solver ANSYS CFX is adopted for the numerical simulation. Due to the rotational periodical characteristics of geometry and flow structure, a periodic symmetry model of three-dimensional labyrinth seal flow is adopted in the numerical simulation, in which the hexahedral structured grids are shown in Figure3. The numerical setup is depicted in Table 2. Steady state analysis is conducted to simulate the seal flow. The working medium in the fluid domain is compressible ideal air. The heat transfer option is selected as the total energy model in the ANSYS CFX preprocessor. The Shear Stress Transport (SST) turbulent model is used to close the Reynolds stress terms. According to the recommendation of SST turbulent model, the maximum wall y+ is near 1, so the grids around the labyrinth rotor and stator are refined locally to satisfy the computation requirement. The convection terms discretization and the turbulence terms discretization are all carried out by high resolution scheme.Figure 3
Mesh.Table 2
Numerical method and boundary conditions.
Analysis typeSteady stateFluidAir ideal gasHeat transferTotal energyTurbulence modelSSTDiscretization schemeHigh resolutionGridPeriodic hexahedral elementsWallAdiabaticInlet total temperature300 KInlet turbulence intensity5%Pressure ratio1.03≤π≤4Reynolds number500≤Re≤25000Circumferential Mach number0.23≤MU≤0.92Dimensionless minimum tip clearance0.033≤C≤0.100Nonuniformity coefficient−0.5≤u≤0.5Table2 also includes the boundary conditions. The labyrinth rotor and stator are assigned as adiabatic wall. The rotating speed of the labyrinth rotor is within the range of 3000rpm≤Ω≤12000rpm corresponding to the circumferential Mach number scope of 0.23≤MU≤0.92. The total temperature of 300 K and the turbulence intensity of 5% are set for the labyrinth inlet. As for the pressure boundaries, the total pressure is assigned to the inlet, and the back pressure (static pressure) is assigned to the outlet. Moreover, a negative feedback method [22] is adopted in order to accurately control the pressure ratio independently with the Reynolds number. The pressure ratio scope of 1.03≤π≤4 and the Reynolds number scope of 500≤Re≤25000 can cover the most common operating state of aeroengine labyrinth seals. The influence of the sealing clearance is the core of this study. Typical labyrinth clearance profiles in the aeroengine are usually within the dimensionless minimum tip clearance scope of 0.033≤C≤0.100 (corresponding to 0.2mm≤cmin≤0.6mm) and the clearance nonuniformity coefficient scope of −0.5≤u≤0.5.


#### 2.2.2. Validation of Simulation Results

A grid sensitivity analysis has also been performed to define the mesh size. Three sets of grids with different element numbers are established. Element numbers of coarse, medium, and refined meshes are 21.2 thousand, 44.6 thousand, and 91.4 thousand, respectively. The calculations are carried out under the condition of uniform sealing clearance, that isu=0. The grid independence validation is carried out under the uniform sealing clearance situation. The dimensionless tip clearance C is 0.067 (corresponding to a sealing clearance of 0.4 mm). In the verification case, the pressure ratio varies from 1.03 to 4, and the circumferential Mach number is 10000 (corresponding to a rotating speed of 3000 rpm). The computational convergence is accepted when the residuals of continuity equation, momentum equation, and energy equation converge to the order of 2×10−6.The results of flow coefficients are depicted in Figure 4. The results illustrate that the maximum flow coefficient deviation of different grids is no more than 1.5%. Especially, less than 0.2% difference is observed between refine grid and medium grid. The flow coefficient results of medium grids can be convinced as independent from the mesh size. Considering the compromise between simulation accuracy and calculation cost, the medium grids are selected for subsequent calculation and analysis.Figure 4
Grid sensitivity validation.In order to validate the accuracy of the numerical model in current investigation, leakage mass flow rate and windage heating power test data of Millward and Edwards [27] are compared with the simulation results. The experimental investigation was carried out on the Rolls-Royce Internal Air Systems Research facility, installed on the Hucknall Thermal Site. The test configuration was a straight-through labyrinth seal with five stages of teeth under the rotational speed of 13000 rpm. Two groups of experiment data (run 4 and 5) are extracted for numerical simulation method validation. The sealing clearances of both run 4 and 5 were uniform. Specifically, the running sealing clearances of run 4 were 0.75 mm, and that of run 5 was 0.25 mm as described in literature [27]. Other geometric parameters of the experiment configuration are claimed in Figure 5. The mass flow rate and the temperature rise of the air passing through the seal were measured in the experimental work. Further, the windage heating power of the labyrinth seal was derived from the enthalpy rise of the leakage flow. The validation range of pressure ratio is from 1.1 to 1.9. The comparison between the numerical simulation results and experimental data is displayed in Figure 6. The two data are in good agreement, proving that the numerical simulation model has sufficient calculation accuracy. In conclusion, the numerical simulation method in this work is applicable for the leakage flow in the straight-through labyrinth seal.Figure 5
Test configuration [27].Figure 6
Validation of numerical simulation results [27].
(a)
Seal leakage comparison(b)
Windage power comparison


## 2.1. Description of Nonuniform Sealing Clearance

### 2.1.1. Geometric Configuration

A straight-through labyrinth is regarded as the research objective of this investigation, which consists of four stages of labyrinth teeth, as shown in Figure1. The reasons of nonuniform deformation of sealing clearance usually include asymmetric structure and uneven temperature, resulting in the monotonic profile of the labyrinth tip clearance. The labyrinth clearance becomes larger or smaller gradually along the flow direction as shown in Figure 2. Generally, the labyrinth seal has a pretty tiny axial dimension, structured no more than tens of millimeters. In such a limited space, the tip clearances of different stages caused by the nonuniform deformation can be simplified to be linear distribution according to the previous investigation [21–23]. Therefore, in this work, the assumption of the linear clearance profile is also adopted, and the nonuniformity coefficient u is employed to depict the nonuniformity of the labyrinth seal clearance as defined in the authors’ previous investigation [22]. The absolute value of the nonuniformity coefficient u can symbolize the nonuniformity degree of the labyrinth seal clearance, which is defined as the ratio of the maximum tip clearance deviation to twice of the average sealing clearance.
(1)u=cmax−cmin2cav.Figure 1
Geometric model.Figure 2
Nonuniform seal clearance.cmax is the maximum tip clearance, cmin is the minimum tip clearance, and cav is the average tip clearance.As for a labyrinth with four stages of teeth, Equations (2) and (3) provide the calculation method of the average labyrinth tip clearance and the nonuniformity coefficient.
(2)cav=14∑i=14ci,(3)u=c1−c42cav.ci is the tip clearance of the ith stage. u=0 implies a uniform sealing clearance, and u=1 implies a zero minimum labyrinth tip clearance, meaning a situation of rub-impact.The convergent type of sealing clearance is characterized byu>0, while the divergent type of sealing clearance is characterized by u<0. The critical safety-limit parameter for the seal device is the minimum tip clearance. Consequently, the minimum clearance, replacing the mean clearance, together with the nonuniformity coefficient is regarded as the key geometric parameter for clearance nonuniformity description. In this investigation, the minimum tip clearance range is 0.2mm≤cmin≤0.6mm, and the nonuniformity coefficient range is −0.5≤u≤0.5. Table 1 includes other detailed geometric dimensions. The meaning of each parameter has been explained in Figure 1.Table 1
Geometric parameters.
ParameterValueα10°B6 mmH4 mmLi6 mmLo6 mmr254 mmt0.5 mm


### 2.1.2. Dimensionless Parameters

In the labyrinth seal, after the airflow is accelerated by the throttling of the former tip clearance, the kinetic energy of the fluid cannot be completely dissipated into thermal energy in the following cavity. Because of the inertia, part of the kinetic energy is directly transported from the upstream throttle to the downstream throttle. The kinetic energy carry-over effect will weaken the sealing ability. The axial Mach numberMA is an important parameter to describe the axial inertia in the labyrinth seal, which is defined as
(4)MA=VAa.VA is the axial velocity of airflow, and a is the local sound speed.In this study, the pressure coefficientψ is adopted to characterize the dimensionless total pressure [4, 24]. The definition is shown in
(5)ψ=pt−ps,outpt,in−ps,out.pt is total pressure in a tooth cavity, pt,in is total pressure of seal inlet, and ps,out is static pressure of seal outlet.Total pressure drop is an important characterization of the labyrinth seal. For the whole labyrinth seal, the pressure load of labyrinth seal∆ψseal is defined as the pressure coefficient difference between the inlet and the outlet, as shown in Equation (6). For a single stage of the labyrinth tooth, the pressure load of labyrinth tooth ∆ψi is defined as the pressure coefficient difference between the front and rear tooth cavity, regarded as the contribution to the total pressure drop, shown in Equation (7).
(6)∆ψseal=ψinlet−ψoutlet,(7)∆ψi=ψi−ψi+1.ψinlet and ψoutlet are the pressure coefficient of labyrinth seal inlet and outlet, respectively. ψi is the pressure coefficient in the ith teeth cavity.As one of most important characteristic quantity, the flow coefficient [4, 24, 25] is introduced to characterize the dimensionless mass flow rate. The definition is shown in
(8)φ=ṁRTt,in1/2Apt,in.ṁ is leakage mass flow rate, Tt,in is inlet total temperature of seal inlet, R is gas constant, and A is the cross-section area of the labyrinth tip clearance. For labyrinths with nonuniform tip clearances, cross-section area of the labyrinth minimum tip clearance Amin is used in Equation (8) instead of A.Denecke et al. [26] made a dimensional analysis on the leakage flow of rotating labyrinth seal and simplified the parameters affecting flow characteristics according to the results of numerical simulation. Denecke et al. thought that flow characteristics were in connection with axial Reynolds number, effective pressure ratio, circumferential Mach number, geometric parameters, inlet preswirl ratio, specific heat ratio, and Prandtl number. In this study, the inlet preswirl is not the focus of the nonuniform clearance effect, so the air enters the sealing structure along the axial direction in the subsequent numerical simulation, in that situation where the ratio of inlet total pressure to outlet static pressure π=pt,in/ps,out is equivalent to the effective pressure ratio [26]. Specific heat ratio and Prandtl number are considered as temperature-independent since the air would get little temperature increase within the sealing device. As for the geometric parameters, only the seal clearance, including the minimum tip clearance and the nonuniformity coefficient, is investigated to get insight into the influence of the nonuniform clearance. To improve the universality of research results, the minimum tip clearance is normalized by the labyrinth pitch. Nonuniformity coefficient u and dimensionless minimum tip clearance C=cmin/B are employed jointly to depict the sealing clearance. Consequently, flow coefficient and pressure coefficient of the labyrinth seal can be expressed as
(9)φ=fπ,Re,MU,C,u.Definitions of Reynolds number Re and circumferential Mach numberMU are shown in Equations (10) and (11), respectively.
(10)Re=2ρVcminμ=ṁμΠr,(11)MU=Ωra.ρ, μ, and V are the density, the dynamic viscosity, and the velocity of the labyrinth seal inlet, respectively. r is the radius of the labyrinth teeth tip, Π is circumference ratio, Ω is angular velocity of labyrinth rotor, and a is local sound speed.In this investigation, the effect of clearance nonuniformity on the leakage flow is focused on, so the nonuniformity impact factorE is defined to describe the influence of nonuniformity degree on the flow coefficient, shown as
(12)E=φuφ0.φu is the flow coefficient of the nonuniform clearance labyrinth (u≠0), while φ0 is the flow coefficient of the uniform clearance labyrinth (u=0) with the same minimum tip clearance.


## 2.1.1. Geometric Configuration

A straight-through labyrinth is regarded as the research objective of this investigation, which consists of four stages of labyrinth teeth, as shown in Figure1. The reasons of nonuniform deformation of sealing clearance usually include asymmetric structure and uneven temperature, resulting in the monotonic profile of the labyrinth tip clearance. The labyrinth clearance becomes larger or smaller gradually along the flow direction as shown in Figure 2. Generally, the labyrinth seal has a pretty tiny axial dimension, structured no more than tens of millimeters. In such a limited space, the tip clearances of different stages caused by the nonuniform deformation can be simplified to be linear distribution according to the previous investigation [21–23]. Therefore, in this work, the assumption of the linear clearance profile is also adopted, and the nonuniformity coefficient u is employed to depict the nonuniformity of the labyrinth seal clearance as defined in the authors’ previous investigation [22]. The absolute value of the nonuniformity coefficient u can symbolize the nonuniformity degree of the labyrinth seal clearance, which is defined as the ratio of the maximum tip clearance deviation to twice of the average sealing clearance.
(1)u=cmax−cmin2cav.Figure 1
Geometric model.Figure 2
Nonuniform seal clearance.cmax is the maximum tip clearance, cmin is the minimum tip clearance, and cav is the average tip clearance.As for a labyrinth with four stages of teeth, Equations (2) and (3) provide the calculation method of the average labyrinth tip clearance and the nonuniformity coefficient.
(2)cav=14∑i=14ci,(3)u=c1−c42cav.ci is the tip clearance of the ith stage. u=0 implies a uniform sealing clearance, and u=1 implies a zero minimum labyrinth tip clearance, meaning a situation of rub-impact.The convergent type of sealing clearance is characterized byu>0, while the divergent type of sealing clearance is characterized by u<0. The critical safety-limit parameter for the seal device is the minimum tip clearance. Consequently, the minimum clearance, replacing the mean clearance, together with the nonuniformity coefficient is regarded as the key geometric parameter for clearance nonuniformity description. In this investigation, the minimum tip clearance range is 0.2mm≤cmin≤0.6mm, and the nonuniformity coefficient range is −0.5≤u≤0.5. Table 1 includes other detailed geometric dimensions. The meaning of each parameter has been explained in Figure 1.Table 1
Geometric parameters.
ParameterValueα10°B6 mmH4 mmLi6 mmLo6 mmr254 mmt0.5 mm


## 2.1.2. Dimensionless Parameters

In the labyrinth seal, after the airflow is accelerated by the throttling of the former tip clearance, the kinetic energy of the fluid cannot be completely dissipated into thermal energy in the following cavity. Because of the inertia, part of the kinetic energy is directly transported from the upstream throttle to the downstream throttle. The kinetic energy carry-over effect will weaken the sealing ability. The axial Mach numberMA is an important parameter to describe the axial inertia in the labyrinth seal, which is defined as
(4)MA=VAa.VA is the axial velocity of airflow, and a is the local sound speed.In this study, the pressure coefficientψ is adopted to characterize the dimensionless total pressure [4, 24]. The definition is shown in
(5)ψ=pt−ps,outpt,in−ps,out.pt is total pressure in a tooth cavity, pt,in is total pressure of seal inlet, and ps,out is static pressure of seal outlet.Total pressure drop is an important characterization of the labyrinth seal. For the whole labyrinth seal, the pressure load of labyrinth seal∆ψseal is defined as the pressure coefficient difference between the inlet and the outlet, as shown in Equation (6). For a single stage of the labyrinth tooth, the pressure load of labyrinth tooth ∆ψi is defined as the pressure coefficient difference between the front and rear tooth cavity, regarded as the contribution to the total pressure drop, shown in Equation (7).
(6)∆ψseal=ψinlet−ψoutlet,(7)∆ψi=ψi−ψi+1.ψinlet and ψoutlet are the pressure coefficient of labyrinth seal inlet and outlet, respectively. ψi is the pressure coefficient in the ith teeth cavity.As one of most important characteristic quantity, the flow coefficient [4, 24, 25] is introduced to characterize the dimensionless mass flow rate. The definition is shown in
(8)φ=ṁRTt,in1/2Apt,in.ṁ is leakage mass flow rate, Tt,in is inlet total temperature of seal inlet, R is gas constant, and A is the cross-section area of the labyrinth tip clearance. For labyrinths with nonuniform tip clearances, cross-section area of the labyrinth minimum tip clearance Amin is used in Equation (8) instead of A.Denecke et al. [26] made a dimensional analysis on the leakage flow of rotating labyrinth seal and simplified the parameters affecting flow characteristics according to the results of numerical simulation. Denecke et al. thought that flow characteristics were in connection with axial Reynolds number, effective pressure ratio, circumferential Mach number, geometric parameters, inlet preswirl ratio, specific heat ratio, and Prandtl number. In this study, the inlet preswirl is not the focus of the nonuniform clearance effect, so the air enters the sealing structure along the axial direction in the subsequent numerical simulation, in that situation where the ratio of inlet total pressure to outlet static pressure π=pt,in/ps,out is equivalent to the effective pressure ratio [26]. Specific heat ratio and Prandtl number are considered as temperature-independent since the air would get little temperature increase within the sealing device. As for the geometric parameters, only the seal clearance, including the minimum tip clearance and the nonuniformity coefficient, is investigated to get insight into the influence of the nonuniform clearance. To improve the universality of research results, the minimum tip clearance is normalized by the labyrinth pitch. Nonuniformity coefficient u and dimensionless minimum tip clearance C=cmin/B are employed jointly to depict the sealing clearance. Consequently, flow coefficient and pressure coefficient of the labyrinth seal can be expressed as
(9)φ=fπ,Re,MU,C,u.Definitions of Reynolds number Re and circumferential Mach numberMU are shown in Equations (10) and (11), respectively.
(10)Re=2ρVcminμ=ṁμΠr,(11)MU=Ωra.ρ, μ, and V are the density, the dynamic viscosity, and the velocity of the labyrinth seal inlet, respectively. r is the radius of the labyrinth teeth tip, Π is circumference ratio, Ω is angular velocity of labyrinth rotor, and a is local sound speed.In this investigation, the effect of clearance nonuniformity on the leakage flow is focused on, so the nonuniformity impact factorE is defined to describe the influence of nonuniformity degree on the flow coefficient, shown as
(12)E=φuφ0.φu is the flow coefficient of the nonuniform clearance labyrinth (u≠0), while φ0 is the flow coefficient of the uniform clearance labyrinth (u=0) with the same minimum tip clearance.


## 2.2. Numerical Simulation Method

### 2.2.1. Numerical Model

The commercial solver ANSYS CFX is adopted for the numerical simulation. Due to the rotational periodical characteristics of geometry and flow structure, a periodic symmetry model of three-dimensional labyrinth seal flow is adopted in the numerical simulation, in which the hexahedral structured grids are shown in Figure3. The numerical setup is depicted in Table 2. Steady state analysis is conducted to simulate the seal flow. The working medium in the fluid domain is compressible ideal air. The heat transfer option is selected as the total energy model in the ANSYS CFX preprocessor. The Shear Stress Transport (SST) turbulent model is used to close the Reynolds stress terms. According to the recommendation of SST turbulent model, the maximum wall y+ is near 1, so the grids around the labyrinth rotor and stator are refined locally to satisfy the computation requirement. The convection terms discretization and the turbulence terms discretization are all carried out by high resolution scheme.Figure 3
Mesh.Table 2
Numerical method and boundary conditions.
Analysis typeSteady stateFluidAir ideal gasHeat transferTotal energyTurbulence modelSSTDiscretization schemeHigh resolutionGridPeriodic hexahedral elementsWallAdiabaticInlet total temperature300 KInlet turbulence intensity5%Pressure ratio1.03≤π≤4Reynolds number500≤Re≤25000Circumferential Mach number0.23≤MU≤0.92Dimensionless minimum tip clearance0.033≤C≤0.100Nonuniformity coefficient−0.5≤u≤0.5Table2 also includes the boundary conditions. The labyrinth rotor and stator are assigned as adiabatic wall. The rotating speed of the labyrinth rotor is within the range of 3000rpm≤Ω≤12000rpm corresponding to the circumferential Mach number scope of 0.23≤MU≤0.92. The total temperature of 300 K and the turbulence intensity of 5% are set for the labyrinth inlet. As for the pressure boundaries, the total pressure is assigned to the inlet, and the back pressure (static pressure) is assigned to the outlet. Moreover, a negative feedback method [22] is adopted in order to accurately control the pressure ratio independently with the Reynolds number. The pressure ratio scope of 1.03≤π≤4 and the Reynolds number scope of 500≤Re≤25000 can cover the most common operating state of aeroengine labyrinth seals. The influence of the sealing clearance is the core of this study. Typical labyrinth clearance profiles in the aeroengine are usually within the dimensionless minimum tip clearance scope of 0.033≤C≤0.100 (corresponding to 0.2mm≤cmin≤0.6mm) and the clearance nonuniformity coefficient scope of −0.5≤u≤0.5.


### 2.2.2. Validation of Simulation Results

A grid sensitivity analysis has also been performed to define the mesh size. Three sets of grids with different element numbers are established. Element numbers of coarse, medium, and refined meshes are 21.2 thousand, 44.6 thousand, and 91.4 thousand, respectively. The calculations are carried out under the condition of uniform sealing clearance, that isu=0. The grid independence validation is carried out under the uniform sealing clearance situation. The dimensionless tip clearance C is 0.067 (corresponding to a sealing clearance of 0.4 mm). In the verification case, the pressure ratio varies from 1.03 to 4, and the circumferential Mach number is 10000 (corresponding to a rotating speed of 3000 rpm). The computational convergence is accepted when the residuals of continuity equation, momentum equation, and energy equation converge to the order of 2×10−6.The results of flow coefficients are depicted in Figure 4. The results illustrate that the maximum flow coefficient deviation of different grids is no more than 1.5%. Especially, less than 0.2% difference is observed between refine grid and medium grid. The flow coefficient results of medium grids can be convinced as independent from the mesh size. Considering the compromise between simulation accuracy and calculation cost, the medium grids are selected for subsequent calculation and analysis.Figure 4
Grid sensitivity validation.In order to validate the accuracy of the numerical model in current investigation, leakage mass flow rate and windage heating power test data of Millward and Edwards [27] are compared with the simulation results. The experimental investigation was carried out on the Rolls-Royce Internal Air Systems Research facility, installed on the Hucknall Thermal Site. The test configuration was a straight-through labyrinth seal with five stages of teeth under the rotational speed of 13000 rpm. Two groups of experiment data (run 4 and 5) are extracted for numerical simulation method validation. The sealing clearances of both run 4 and 5 were uniform. Specifically, the running sealing clearances of run 4 were 0.75 mm, and that of run 5 was 0.25 mm as described in literature [27]. Other geometric parameters of the experiment configuration are claimed in Figure 5. The mass flow rate and the temperature rise of the air passing through the seal were measured in the experimental work. Further, the windage heating power of the labyrinth seal was derived from the enthalpy rise of the leakage flow. The validation range of pressure ratio is from 1.1 to 1.9. The comparison between the numerical simulation results and experimental data is displayed in Figure 6. The two data are in good agreement, proving that the numerical simulation model has sufficient calculation accuracy. In conclusion, the numerical simulation method in this work is applicable for the leakage flow in the straight-through labyrinth seal.Figure 5
Test configuration [27].Figure 6
Validation of numerical simulation results [27].
(a)
Seal leakage comparison(b)
Windage power comparison


## 2.2.1. Numerical Model

The commercial solver ANSYS CFX is adopted for the numerical simulation. Due to the rotational periodical characteristics of geometry and flow structure, a periodic symmetry model of three-dimensional labyrinth seal flow is adopted in the numerical simulation, in which the hexahedral structured grids are shown in Figure3. The numerical setup is depicted in Table 2. Steady state analysis is conducted to simulate the seal flow. The working medium in the fluid domain is compressible ideal air. The heat transfer option is selected as the total energy model in the ANSYS CFX preprocessor. The Shear Stress Transport (SST) turbulent model is used to close the Reynolds stress terms. According to the recommendation of SST turbulent model, the maximum wall y+ is near 1, so the grids around the labyrinth rotor and stator are refined locally to satisfy the computation requirement. The convection terms discretization and the turbulence terms discretization are all carried out by high resolution scheme.Figure 3
Mesh.Table 2
Numerical method and boundary conditions.
Analysis typeSteady stateFluidAir ideal gasHeat transferTotal energyTurbulence modelSSTDiscretization schemeHigh resolutionGridPeriodic hexahedral elementsWallAdiabaticInlet total temperature300 KInlet turbulence intensity5%Pressure ratio1.03≤π≤4Reynolds number500≤Re≤25000Circumferential Mach number0.23≤MU≤0.92Dimensionless minimum tip clearance0.033≤C≤0.100Nonuniformity coefficient−0.5≤u≤0.5Table2 also includes the boundary conditions. The labyrinth rotor and stator are assigned as adiabatic wall. The rotating speed of the labyrinth rotor is within the range of 3000rpm≤Ω≤12000rpm corresponding to the circumferential Mach number scope of 0.23≤MU≤0.92. The total temperature of 300 K and the turbulence intensity of 5% are set for the labyrinth inlet. As for the pressure boundaries, the total pressure is assigned to the inlet, and the back pressure (static pressure) is assigned to the outlet. Moreover, a negative feedback method [22] is adopted in order to accurately control the pressure ratio independently with the Reynolds number. The pressure ratio scope of 1.03≤π≤4 and the Reynolds number scope of 500≤Re≤25000 can cover the most common operating state of aeroengine labyrinth seals. The influence of the sealing clearance is the core of this study. Typical labyrinth clearance profiles in the aeroengine are usually within the dimensionless minimum tip clearance scope of 0.033≤C≤0.100 (corresponding to 0.2mm≤cmin≤0.6mm) and the clearance nonuniformity coefficient scope of −0.5≤u≤0.5.


## 2.2.2. Validation of Simulation Results

A grid sensitivity analysis has also been performed to define the mesh size. Three sets of grids with different element numbers are established. Element numbers of coarse, medium, and refined meshes are 21.2 thousand, 44.6 thousand, and 91.4 thousand, respectively. The calculations are carried out under the condition of uniform sealing clearance, that isu=0. The grid independence validation is carried out under the uniform sealing clearance situation. The dimensionless tip clearance C is 0.067 (corresponding to a sealing clearance of 0.4 mm). In the verification case, the pressure ratio varies from 1.03 to 4, and the circumferential Mach number is 10000 (corresponding to a rotating speed of 3000 rpm). The computational convergence is accepted when the residuals of continuity equation, momentum equation, and energy equation converge to the order of 2×10−6.The results of flow coefficients are depicted in Figure 4. The results illustrate that the maximum flow coefficient deviation of different grids is no more than 1.5%. Especially, less than 0.2% difference is observed between refine grid and medium grid. The flow coefficient results of medium grids can be convinced as independent from the mesh size. Considering the compromise between simulation accuracy and calculation cost, the medium grids are selected for subsequent calculation and analysis.Figure 4
Grid sensitivity validation.In order to validate the accuracy of the numerical model in current investigation, leakage mass flow rate and windage heating power test data of Millward and Edwards [27] are compared with the simulation results. The experimental investigation was carried out on the Rolls-Royce Internal Air Systems Research facility, installed on the Hucknall Thermal Site. The test configuration was a straight-through labyrinth seal with five stages of teeth under the rotational speed of 13000 rpm. Two groups of experiment data (run 4 and 5) are extracted for numerical simulation method validation. The sealing clearances of both run 4 and 5 were uniform. Specifically, the running sealing clearances of run 4 were 0.75 mm, and that of run 5 was 0.25 mm as described in literature [27]. Other geometric parameters of the experiment configuration are claimed in Figure 5. The mass flow rate and the temperature rise of the air passing through the seal were measured in the experimental work. Further, the windage heating power of the labyrinth seal was derived from the enthalpy rise of the leakage flow. The validation range of pressure ratio is from 1.1 to 1.9. The comparison between the numerical simulation results and experimental data is displayed in Figure 6. The two data are in good agreement, proving that the numerical simulation model has sufficient calculation accuracy. In conclusion, the numerical simulation method in this work is applicable for the leakage flow in the straight-through labyrinth seal.Figure 5
Test configuration [27].Figure 6
Validation of numerical simulation results [27].
(a)
Seal leakage comparison(b)
Windage power comparison


## 3. Results and Analysis

### 3.1. Effects of Nonuniformity Cause

During the operation process of aeroengines, both the rotor and the stator of labyrinth seal can experience different extents of radial deformations. The nonuniformity of labyrinth sealing clearance is the combined results of both deformations. Taking the convergent type clearance as an example, Figure7 shows the structural diagram of nonuniform sealing clearances with equal stator radius (ESR) and equal rotor radius (ERR). The clearance nonuniformity of ESR is considered to be caused by rotor deformation, while the clearance nonuniformity of ERR is considered to be caused by stator deformation. The flow structure, pressure distribution, and leakage flow in cases of ESR and ERR are compared to clarify the effect of the nonuniformity cause.Figure 7
Convergent type clearances caused by rotor deformation and stator deformation.Figure8 compares the flow structure between the ESR and the ERR. The results indicate that the flow structure in the straight-through labyrinth seal is not sensitive to the nonuniformity cause. In cases of u=−0.1, u=−0.2, and u=−0.3, there is not obvious difference for the streamline distributions between the ESR and the ERR. As for the pressure distribution, the pressure coefficients of tooth cavities and pressure loads of labyrinth teeth are depicted in Figure 9. It is not surprising that the pressure coefficient decreases gradually along the flow direction due to the total pressure drop effect. However, the pressure load of labyrinth tooth present an opposite trend between the convergent type and the divergent type, in which the tooth pressure load of the convergent type grows up and that of the divergent type falls down from inlet to outlet. Nevertheless, the pressure coefficients of tooth cavities and pressure loads of labyrinth teeth are almost the same between the ESR and the ERR, proving again that the flow sensitivity of straight-through labyrinth seals to the clearance nonuniformity cause is very weak.Figure 8
Flow structure in the labyrinth seal (Re=10000, π=1.3, MU=0.23, and C=0.067).Figure 9
Pressure distributions of tooth cavities and pressure loads of labyrinth teeth (Re=10000, π=1.3, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=−0.1(c)
u=0.2(d)
u=−0.2(e)
u=0.3(f)
u=−0.3As one of most important quantity to describe leakage flow, the flow coefficient is also discussed including the ESR and the ERR. Figure10 compares variations of flow coefficient with pressure ratio under different nonuniform clearances (u=0.1, u=0.2, and u=0.3). From the results, when the minimum tip clearance is the same, characteristic lines of flow coefficient to pressure ratio gradually move upward with the increase of the absolute value of the nonuniformity coefficient, which indicates that under the same operating environment (constant circumferential Mach number, constant pressure ratio, and constant Reynolds number), the leakage flow increases with the rise of tip clearance nonuniformity. This is because except for the minimum tip clearance, the rest tip clearances increase with the rise of tip clearance nonuniformity, regardless of convergent type clearance or the divergent type clearance. When tip clearances of straight-through labyrinths become larger, the equivalent circulation area increases. At the same time, the kinetic energy carry-over factor also increases [6], resulting weaker total pressure dissipation in the tooth cavities. As for the effect of nonuniformity cause, when nonuniformity coefficient u is the same, there is no significant difference in the characteristics of flow coefficient to pressure ratio between the ESR and the ERR, regardless of the convergent type or the divergent type. For u=0.1, u=0.2, and u=0.3, the maximum deviation between labyrinth seals with equal stator radius and equal rotor radius is less than 0.8%. Therefore, in the follow-up of this study, only the labyrinth seals with equal stator radius, as shown in Figure 2, are discussed about the influence of nonuniform tip clearance.Figure 10
Variations of flow coefficient with pressure ratio under different nonuniform clearances (Re=10000, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)


### 3.2. Effects of Nonuniformity Type

#### 3.2.1. Subcritical State

The axial inertia of the airflow in the labyrinth seal can be reflected by the axial Mach number. Under typical subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067), Figure 11 compares the axial Mach number distributions in the labyrinths with different clearance nonuniformity type (convergent type and divergent type). For labyrinths with convergent type clearance (u>0), the axial Mach number increases gradually along the flow direction. The maximum axial Mach number appears at the last tip clearance. The axial Mach number at the front three tip clearance is small, and the axial inertia is weak, which is conducive to the dissipation from kinetic energy to thermal energy inside the tooth cavities. For labyrinths with divergent type clearance (u<0), the axial Mach number decreases gradually along the flow direction. The maximum axial Mach number appears at the first tip clearance. The airflow with high axial velocity passes through the teeth cavity and directly to the next tip clearance. The kinetic energy carry-over effect of divergent type (u<0) gets more significant than that of convergent type (u>0). When u is the same, the axial Mach number at the front three tip clearances of convergent type clearance labyrinths is generally smaller than that of divergent type clearance labyrinths. This shows that the airflow inside divergent type clearance labyrinth seals has greater axial inertia.Figure 11
Axial Mach number distributions of nonuniform tip clearance labyrinths in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).The pressure load of labyrinth tooth can also be influenced by the clearance nonuniformity significantly. Figure12 compares the teeth pressure loads between the convergent type and the divergent type under different clearance nonuniformity degree (−0.5≤u≤0.5). For the labyrinths with uniform clearance (u=0), the teeth pressure loads are almost uniform, which is approximately between 0.17 and 0.18. However, for the labyrinths with nonuniform clearance (u≠0), the teeth pressure loads of four stages also exhibit a nonuniform profile. It is remarkable that the teeth pressure load varies monotonously from the first stage to the fourth stage. But an obvious difference between the two nonuniformity types is that the teeth pressure load of convergent type (u>0) increases monotonically, and the teeth pressure load of divergent type (u<0) decreases monotonically along the flow direction. Comparing the convergent type and the divergent type, an interesting phenomenon is found that the teeth pressure load of third stage is all similar in cases of u=0.1, u=0.2, u=0.3, u=0.4, and u=0.5 despite the opposite distribution trends. Moreover, the teeth pressure load difference between the first stage and the last stage (maximum deviation) gradually rises with the increase of the clearance nonuniformity degree, that is, the absolute value of the nonuniformity coefficient u, as shown in Table 3. The results indicate that the maximum teeth pressure load deviation of the divergent type (u<0) increases at a faster rate than that of the convergent type (u>0). By contrast, the divergent nonuniformity type contributes more to the nonuniform teeth pressure load profile in the subcritical state.Figure 12
Teeth pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=0.2(c)
u=0.3(d)
u=0.4(e)
u=0.5Table 3
Maximum deviation of different teeth pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).
u0.10.20.30.40.5∆ψmax−∆ψminu>00.080.150.200.240.26u<00.080.180.270.350.41


#### 3.2.2. Critical State

In the typical critical state (Re=10000, π=4, MU=0.23, C=0.067, and u=0), Figure 13 shows the axial Mach number distribution of a uniform tip clearance labyrinth. When the air flows through a labyrinth seal, due to the large total pressure loss, the total pressure of the last stage tip clearance is the lowest among all the stages. As mass flow rate is continuous, the axial velocity at the last stage tip clearance is the largest, so the last stage tip clearance enters the critical state first as shown in Figure 13.Figure 13
Axial Mach number distributions of uniform tip clearance labyrinth in the critical state (Re=10000, π=4, MU=0.23, C=0.067, and u=0).Under the critical state (Re=10000, π=4, MU=0.23, and C=0.067), the axial Mach number distributions in the labyrinths with different clearance nonuniformity type (convergent type and divergent type) are compared in Figure 14. For convergent type clearance labyrinths (u>0), the critical section is the last stage tip clearance all the cases, so the maximum flow capacity of the labyrinth seal is limited by the last stage tip clearance. With the increase of nonuniformity coefficient, the axial Mach number at the front three tip clearances gradually reduces. For the divergent clearance labyrinth (u<0), when the nonuniformity coefficient is small, such as the case of u=−0.1, the difference between the four tip clearances is not obvious. Although the area of the last stage tip clearance is the largest, it is still the first one to enter the critical state. As the clearance nonuniformity degree increases gradually, such as u=−0.3, u=−0.4, and u=−0.5, the critical section switches from the last stage tip clearance to the first stage tip clearance. The maximum flow capacity of the labyrinth seal is limited by the first stage tip clearance. Similar to the subcritical state, the kinetic energy carry-over effect of divergent type (u<0) gets more significant than that of convergent type (u>0).Figure 14
Axial Mach number distributions of nonuniform tip clearance labyrinths in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).In the critical state (Re=10000, π=4, MU=0.23, and C=0.067), the teeth pressure loads under different clearance nonuniformity degree (-0.5≤u≤0.5) are also compared between the convergent type and the divergent type in Figure 15. Different from the subcritical state, the teeth pressure loads in the critical state is not uniform, even if the seal clearance is uniform (u=0). The teeth pressure load of the last stage is significantly larger than those of the front three stages. For labyrinths with convergent type clearance (u>0), the distributions of teeth pressure loads in the teeth cavities are similar to that of the labyrinth with uniform clearance (u=0). Contrast to the uniform clearance labyrinth, the teeth pressure loads of the front three stages are lower, while that of the last stage is higher. For labyrinths with divergent type clearance (u<0), the variation of the teeth pressure load is no longer monotonic along the flow direction. The largest teeth pressure load is still the last stage when the clearance nonuniformity is not obvious (u=−0.1), but with the increase of the nonuniformity degree (u=−0.2, u=−0.3, u=−0.4, and u=−0.5), the distributions of teeth pressure loads in the teeth cavities become different from those of labyrinths with uniform clearance and convergent clearance. Although the first stage tip clearance is the critical section, the second stage develops into the key teeth for total pressure drop and undertakes the largest single teeth pressure load. Starting at the second stage, the teeth pressure load decreases monotonically along the flow direction. Similar to the subcritical state, the teeth pressure load maximum deviation gradually rises with the increase of the absolute value of the nonuniformity coefficient u, as shown in Table 4, meaning the clearance nonuniformity can promote the teeth pressure load deviation among the four stage. Opposed to the subcritical state, the maximum teeth pressure load deviation of the convergent type (u>0) increases at a faster rate than that of the divergent type (u<0). By contrast, the convergent nonuniformity type contributes more to the nonuniform teeth pressure load profile in the critical state.Figure 15
Teeth pressure loads in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=0.2(c)
u=0.3(d)
u=0.4(e)
u=0.5Table 4
Maximum deviation of different teeth pressure loads in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).
u0.10.20.30.40.5∆ψmax−∆ψminu>00.160.190.210.220.22u<00. 040.060.150.180.18


### 3.3. Effects of Nonuniformity Degree

#### 3.3.1. Pressure Coefficient

Due to nonuniform tip clearances, the law of kinetic energy dissipation will be different from that of uniform clearance labyrinths. The distribution of total pressure in the tooth cavities will also change with clearance nonuniformity degree. Figure16 shows pressure coefficient distributions in the teeth cavities of subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067) and critical state (Re=10000, π=4, MU=0.23, and C=0.067), respectively. The pressure coefficient in the teeth cavities reduces gradually along the flow direction, bur the decrease tendency is not the same under different clearance nonuniformity coefficient. When aerodynamic boundary parameters and other geometric structure parameters are all the same, the pressure coefficients in the teeth cavities of the convergent type are higher than that of the divergent type with the equal clearance nonuniformity degree u. Compared with uniform clearance labyrinth seals, the difference of total pressure distributions may change the axial aerodynamic load of nonuniform clearance labyrinth seals.Figure 16
Total pressure distributions of different teeth cavities.
(a)
Subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067)(b)
Critical state (Re=10000, π=4, MU=0.23, and C=0.067)Although the ratio of inlet total pressure to outlet static pressure is the same, pressure coefficients at the labyrinth outlet with different nonuniform coefficients are not the same. According to Figure16, the outlet pressure coefficient of the convergent type clearance labyrinth gets larger with the increase of nonuniformity degree, meaning that the proportion of total pressure dissipation from inlet to outlet decreases. Situations for divergent type clearance labyrinths are just the opposite. Figure 17 shows the labyrinth seal pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067) and critical state (Re=10000, π=4, MU=0.23, and C=0.067), respectively. With the increase of the absolute value of the nonuniformity coefficient u, seal pressure load of convergent type gradually falls down, while that of divergent type gradually grows up, regardless of subcritical state or critical state. The seal pressure load represents the ability to realize the total pressure drop, which is one of important aspect to evaluate the seal performance. In terms of total pressure drop, when the dimensionless minimum tip clearance is the same, convergent clearance contributes less to the total pressure drop with the increase of the clearance nonuniformity, while divergent clearance contributes more to the total pressure drop with the increase of the clearance nonuniformity.Figure 17
Labyrinth seal pressure loads.
(a)
Subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067)(b)
Critical state (Re=10000, π=4, MU=0.23, and C=0.067)


#### 3.3.2. Flow Coefficient

When the minimum tip clearance is the same, the average tip clearance of nonuniform clearance labyrinths is larger than that of uniform clearance labyrinths. Obviously, the labyrinth with smaller average tip clearance has stronger sealing ability if the minimum tip clearance is the same. Therefore, on the premise of the same minimum tip clearance, it is easy to infer that the labyrinth seals with nonuniform tip clearance have more leakage flow than that of the labyrinth seals with uniform tip clearance. Thereby, the nonuniformity impact factor is bound to be greater than 1. However, the growth rate of the leakage flow will be different under different nonuniformity degree. The variation of the nonuniformity impact factor with the clearance nonuniformity coefficient is analyzed to clarify the effect of clearance nonuniformity on the leakage flow.Figures18–21 show the variations of nonuniformity impact factor with nonuniformity coefficient under different Reynolds numbers, pressure ratios, circumferential Mach numbers, and dimensionless minimum tip clearances (500≤Re≤25000, 1.03≤π≤4, 0.23≤MU≤0.92, 0.033≤C≤0.100, and −0.5≤u≤0.5), respectively. If minimum tip clearances are the same, two common points can be found. One is that when absolute values of nonuniformity coefficients are the same, nonuniformity impact factors of divergent type clearance labyrinths are larger than that of convergent type clearance labyrinths, meaning the seal performance of the convergent type is better than that of the divergent type. The other is that when clearance nonuniformity is small, characteristic lines of nonuniformity impact factor have little dispersion. Within the range of −0.1<u<0.1, under the same nonuniformity coefficient, deviations of nonuniformity impact factor under different operation conditions (Reynolds number, pressure ratio, circumferential Mach number, and dimensionless minimum tip clearance) are no more than 1.2%, which indicates that the effect of nonuniformity degree on the leakage flow is irrelevant to the operation conditions. In addition, other variation characteristics of the nonuniformity impact factor are also described in detail.Figure 18
Variations of nonuniformity impact factor under different Reynolds number (π=1.1, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 19
Variations of nonuniformity impact factor under different pressure ratios (Re=10000, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 20
Variations of nonuniformity impact factor under different circumferential Mach numbers (Re=10000, π=1.1, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 21
Variations of nonuniformity impact factor under different dimensionless minimum tip clearances (Re=10000, π=1.1, MU=0.23).
(a)
convergent clearance (u>0)(b)
divergent clearance (u<0)Variations of the nonuniformity impact factor at different Reynolds numbers (500≤Re≤25000) are compared in Figure 18. Except for the cases of small Reynolds number (Re=500 and Re=1000), there is little deviation between variation curves of nonuniformity impact factor when Reynolds numbers are different. Under the actual operating conditions of aeroengines, the Reynolds number is usually much larger than 500 and 1000, so the influence of Reynolds number on the characteristic of nonuniformity impact factor to nonuniformity coefficient is not obvious. In the high Reynolds number range, the influence of Reynolds number can be ignored when analyzing the effect of clearance nonuniformity on the leakage flow.Variations of nonuniformity impact factor at different pressure ratios (1.03≤π≤4) are compared in Figure 19. With the increase of pressure ratio, characteristic lines of nonuniformity impact factor gradually move down, meaning that the rise of leakage flow caused by nonuniform clearance is suppressed. When the sealing flow enters the critical state (π=2.5, π=3, and π=4), pressure ratio has little influence on the variation curve of nonuniformity impact factor. Therefore, the leakage flow of labyrinths in the subcritical state (small pressure ratio) will be affected more significantly by the nonuniform clearance. And the influence of pressure ratio can be ignored when the seal flow enters the critical state (large pressure ratio).Variations of nonuniformity impact factor at different circumferential Mach numbers (0.23≤MU≤0.92) are compared in Figure 20. With the increase of circumferential Mach number, the characteristic lines of nonuniformity impact factor gradually move up, meaning that the rise of leakage flow caused by nonuniform clearance is aggravated. The results indicate that the labyrinth seal located at high radius location in the aeroengines will be affected more obviously by the clearance nonuniformity. Meanwhile, when the aeroengines operate at high speed, the leakage flow of labyrinth will be also affected more significantly by the nonuniform clearance. Because the higher radius and the higher rotating speed contribute the larger circumferential Mach numbers.Variations of nonuniformity impact factor at different dimensionless minimum tip clearances (0.033≤C≤0.100) are compared in Figure 21. With the increase of dimensionless minimum tip clearance, the characteristic lines of nonuniformity impact factor gradually move down, meaning that the rise of leakage flow caused by nonuniform clearance is suppressed. For labyrinth seals with the same tooth pitch in aeroengines, the leakage flow of labyrinth seals clearance with smaller sealing clearance will be affected more significantly by the nonuniform clearance.


## 3.1. Effects of Nonuniformity Cause

During the operation process of aeroengines, both the rotor and the stator of labyrinth seal can experience different extents of radial deformations. The nonuniformity of labyrinth sealing clearance is the combined results of both deformations. Taking the convergent type clearance as an example, Figure7 shows the structural diagram of nonuniform sealing clearances with equal stator radius (ESR) and equal rotor radius (ERR). The clearance nonuniformity of ESR is considered to be caused by rotor deformation, while the clearance nonuniformity of ERR is considered to be caused by stator deformation. The flow structure, pressure distribution, and leakage flow in cases of ESR and ERR are compared to clarify the effect of the nonuniformity cause.Figure 7
Convergent type clearances caused by rotor deformation and stator deformation.Figure8 compares the flow structure between the ESR and the ERR. The results indicate that the flow structure in the straight-through labyrinth seal is not sensitive to the nonuniformity cause. In cases of u=−0.1, u=−0.2, and u=−0.3, there is not obvious difference for the streamline distributions between the ESR and the ERR. As for the pressure distribution, the pressure coefficients of tooth cavities and pressure loads of labyrinth teeth are depicted in Figure 9. It is not surprising that the pressure coefficient decreases gradually along the flow direction due to the total pressure drop effect. However, the pressure load of labyrinth tooth present an opposite trend between the convergent type and the divergent type, in which the tooth pressure load of the convergent type grows up and that of the divergent type falls down from inlet to outlet. Nevertheless, the pressure coefficients of tooth cavities and pressure loads of labyrinth teeth are almost the same between the ESR and the ERR, proving again that the flow sensitivity of straight-through labyrinth seals to the clearance nonuniformity cause is very weak.Figure 8
Flow structure in the labyrinth seal (Re=10000, π=1.3, MU=0.23, and C=0.067).Figure 9
Pressure distributions of tooth cavities and pressure loads of labyrinth teeth (Re=10000, π=1.3, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=−0.1(c)
u=0.2(d)
u=−0.2(e)
u=0.3(f)
u=−0.3As one of most important quantity to describe leakage flow, the flow coefficient is also discussed including the ESR and the ERR. Figure10 compares variations of flow coefficient with pressure ratio under different nonuniform clearances (u=0.1, u=0.2, and u=0.3). From the results, when the minimum tip clearance is the same, characteristic lines of flow coefficient to pressure ratio gradually move upward with the increase of the absolute value of the nonuniformity coefficient, which indicates that under the same operating environment (constant circumferential Mach number, constant pressure ratio, and constant Reynolds number), the leakage flow increases with the rise of tip clearance nonuniformity. This is because except for the minimum tip clearance, the rest tip clearances increase with the rise of tip clearance nonuniformity, regardless of convergent type clearance or the divergent type clearance. When tip clearances of straight-through labyrinths become larger, the equivalent circulation area increases. At the same time, the kinetic energy carry-over factor also increases [6], resulting weaker total pressure dissipation in the tooth cavities. As for the effect of nonuniformity cause, when nonuniformity coefficient u is the same, there is no significant difference in the characteristics of flow coefficient to pressure ratio between the ESR and the ERR, regardless of the convergent type or the divergent type. For u=0.1, u=0.2, and u=0.3, the maximum deviation between labyrinth seals with equal stator radius and equal rotor radius is less than 0.8%. Therefore, in the follow-up of this study, only the labyrinth seals with equal stator radius, as shown in Figure 2, are discussed about the influence of nonuniform tip clearance.Figure 10
Variations of flow coefficient with pressure ratio under different nonuniform clearances (Re=10000, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)


## 3.2. Effects of Nonuniformity Type

### 3.2.1. Subcritical State

The axial inertia of the airflow in the labyrinth seal can be reflected by the axial Mach number. Under typical subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067), Figure 11 compares the axial Mach number distributions in the labyrinths with different clearance nonuniformity type (convergent type and divergent type). For labyrinths with convergent type clearance (u>0), the axial Mach number increases gradually along the flow direction. The maximum axial Mach number appears at the last tip clearance. The axial Mach number at the front three tip clearance is small, and the axial inertia is weak, which is conducive to the dissipation from kinetic energy to thermal energy inside the tooth cavities. For labyrinths with divergent type clearance (u<0), the axial Mach number decreases gradually along the flow direction. The maximum axial Mach number appears at the first tip clearance. The airflow with high axial velocity passes through the teeth cavity and directly to the next tip clearance. The kinetic energy carry-over effect of divergent type (u<0) gets more significant than that of convergent type (u>0). When u is the same, the axial Mach number at the front three tip clearances of convergent type clearance labyrinths is generally smaller than that of divergent type clearance labyrinths. This shows that the airflow inside divergent type clearance labyrinth seals has greater axial inertia.Figure 11
Axial Mach number distributions of nonuniform tip clearance labyrinths in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).The pressure load of labyrinth tooth can also be influenced by the clearance nonuniformity significantly. Figure12 compares the teeth pressure loads between the convergent type and the divergent type under different clearance nonuniformity degree (−0.5≤u≤0.5). For the labyrinths with uniform clearance (u=0), the teeth pressure loads are almost uniform, which is approximately between 0.17 and 0.18. However, for the labyrinths with nonuniform clearance (u≠0), the teeth pressure loads of four stages also exhibit a nonuniform profile. It is remarkable that the teeth pressure load varies monotonously from the first stage to the fourth stage. But an obvious difference between the two nonuniformity types is that the teeth pressure load of convergent type (u>0) increases monotonically, and the teeth pressure load of divergent type (u<0) decreases monotonically along the flow direction. Comparing the convergent type and the divergent type, an interesting phenomenon is found that the teeth pressure load of third stage is all similar in cases of u=0.1, u=0.2, u=0.3, u=0.4, and u=0.5 despite the opposite distribution trends. Moreover, the teeth pressure load difference between the first stage and the last stage (maximum deviation) gradually rises with the increase of the clearance nonuniformity degree, that is, the absolute value of the nonuniformity coefficient u, as shown in Table 3. The results indicate that the maximum teeth pressure load deviation of the divergent type (u<0) increases at a faster rate than that of the convergent type (u>0). By contrast, the divergent nonuniformity type contributes more to the nonuniform teeth pressure load profile in the subcritical state.Figure 12
Teeth pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=0.2(c)
u=0.3(d)
u=0.4(e)
u=0.5Table 3
Maximum deviation of different teeth pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).
u0.10.20.30.40.5∆ψmax−∆ψminu>00.080.150.200.240.26u<00.080.180.270.350.41


### 3.2.2. Critical State

In the typical critical state (Re=10000, π=4, MU=0.23, C=0.067, and u=0), Figure 13 shows the axial Mach number distribution of a uniform tip clearance labyrinth. When the air flows through a labyrinth seal, due to the large total pressure loss, the total pressure of the last stage tip clearance is the lowest among all the stages. As mass flow rate is continuous, the axial velocity at the last stage tip clearance is the largest, so the last stage tip clearance enters the critical state first as shown in Figure 13.Figure 13
Axial Mach number distributions of uniform tip clearance labyrinth in the critical state (Re=10000, π=4, MU=0.23, C=0.067, and u=0).Under the critical state (Re=10000, π=4, MU=0.23, and C=0.067), the axial Mach number distributions in the labyrinths with different clearance nonuniformity type (convergent type and divergent type) are compared in Figure 14. For convergent type clearance labyrinths (u>0), the critical section is the last stage tip clearance all the cases, so the maximum flow capacity of the labyrinth seal is limited by the last stage tip clearance. With the increase of nonuniformity coefficient, the axial Mach number at the front three tip clearances gradually reduces. For the divergent clearance labyrinth (u<0), when the nonuniformity coefficient is small, such as the case of u=−0.1, the difference between the four tip clearances is not obvious. Although the area of the last stage tip clearance is the largest, it is still the first one to enter the critical state. As the clearance nonuniformity degree increases gradually, such as u=−0.3, u=−0.4, and u=−0.5, the critical section switches from the last stage tip clearance to the first stage tip clearance. The maximum flow capacity of the labyrinth seal is limited by the first stage tip clearance. Similar to the subcritical state, the kinetic energy carry-over effect of divergent type (u<0) gets more significant than that of convergent type (u>0).Figure 14
Axial Mach number distributions of nonuniform tip clearance labyrinths in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).In the critical state (Re=10000, π=4, MU=0.23, and C=0.067), the teeth pressure loads under different clearance nonuniformity degree (-0.5≤u≤0.5) are also compared between the convergent type and the divergent type in Figure 15. Different from the subcritical state, the teeth pressure loads in the critical state is not uniform, even if the seal clearance is uniform (u=0). The teeth pressure load of the last stage is significantly larger than those of the front three stages. For labyrinths with convergent type clearance (u>0), the distributions of teeth pressure loads in the teeth cavities are similar to that of the labyrinth with uniform clearance (u=0). Contrast to the uniform clearance labyrinth, the teeth pressure loads of the front three stages are lower, while that of the last stage is higher. For labyrinths with divergent type clearance (u<0), the variation of the teeth pressure load is no longer monotonic along the flow direction. The largest teeth pressure load is still the last stage when the clearance nonuniformity is not obvious (u=−0.1), but with the increase of the nonuniformity degree (u=−0.2, u=−0.3, u=−0.4, and u=−0.5), the distributions of teeth pressure loads in the teeth cavities become different from those of labyrinths with uniform clearance and convergent clearance. Although the first stage tip clearance is the critical section, the second stage develops into the key teeth for total pressure drop and undertakes the largest single teeth pressure load. Starting at the second stage, the teeth pressure load decreases monotonically along the flow direction. Similar to the subcritical state, the teeth pressure load maximum deviation gradually rises with the increase of the absolute value of the nonuniformity coefficient u, as shown in Table 4, meaning the clearance nonuniformity can promote the teeth pressure load deviation among the four stage. Opposed to the subcritical state, the maximum teeth pressure load deviation of the convergent type (u>0) increases at a faster rate than that of the divergent type (u<0). By contrast, the convergent nonuniformity type contributes more to the nonuniform teeth pressure load profile in the critical state.Figure 15
Teeth pressure loads in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=0.2(c)
u=0.3(d)
u=0.4(e)
u=0.5Table 4
Maximum deviation of different teeth pressure loads in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).
u0.10.20.30.40.5∆ψmax−∆ψminu>00.160.190.210.220.22u<00. 040.060.150.180.18


## 3.2.1. Subcritical State

The axial inertia of the airflow in the labyrinth seal can be reflected by the axial Mach number. Under typical subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067), Figure 11 compares the axial Mach number distributions in the labyrinths with different clearance nonuniformity type (convergent type and divergent type). For labyrinths with convergent type clearance (u>0), the axial Mach number increases gradually along the flow direction. The maximum axial Mach number appears at the last tip clearance. The axial Mach number at the front three tip clearance is small, and the axial inertia is weak, which is conducive to the dissipation from kinetic energy to thermal energy inside the tooth cavities. For labyrinths with divergent type clearance (u<0), the axial Mach number decreases gradually along the flow direction. The maximum axial Mach number appears at the first tip clearance. The airflow with high axial velocity passes through the teeth cavity and directly to the next tip clearance. The kinetic energy carry-over effect of divergent type (u<0) gets more significant than that of convergent type (u>0). When u is the same, the axial Mach number at the front three tip clearances of convergent type clearance labyrinths is generally smaller than that of divergent type clearance labyrinths. This shows that the airflow inside divergent type clearance labyrinth seals has greater axial inertia.Figure 11
Axial Mach number distributions of nonuniform tip clearance labyrinths in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).The pressure load of labyrinth tooth can also be influenced by the clearance nonuniformity significantly. Figure12 compares the teeth pressure loads between the convergent type and the divergent type under different clearance nonuniformity degree (−0.5≤u≤0.5). For the labyrinths with uniform clearance (u=0), the teeth pressure loads are almost uniform, which is approximately between 0.17 and 0.18. However, for the labyrinths with nonuniform clearance (u≠0), the teeth pressure loads of four stages also exhibit a nonuniform profile. It is remarkable that the teeth pressure load varies monotonously from the first stage to the fourth stage. But an obvious difference between the two nonuniformity types is that the teeth pressure load of convergent type (u>0) increases monotonically, and the teeth pressure load of divergent type (u<0) decreases monotonically along the flow direction. Comparing the convergent type and the divergent type, an interesting phenomenon is found that the teeth pressure load of third stage is all similar in cases of u=0.1, u=0.2, u=0.3, u=0.4, and u=0.5 despite the opposite distribution trends. Moreover, the teeth pressure load difference between the first stage and the last stage (maximum deviation) gradually rises with the increase of the clearance nonuniformity degree, that is, the absolute value of the nonuniformity coefficient u, as shown in Table 3. The results indicate that the maximum teeth pressure load deviation of the divergent type (u<0) increases at a faster rate than that of the convergent type (u>0). By contrast, the divergent nonuniformity type contributes more to the nonuniform teeth pressure load profile in the subcritical state.Figure 12
Teeth pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=0.2(c)
u=0.3(d)
u=0.4(e)
u=0.5Table 3
Maximum deviation of different teeth pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067).
u0.10.20.30.40.5∆ψmax−∆ψminu>00.080.150.200.240.26u<00.080.180.270.350.41


## 3.2.2. Critical State

In the typical critical state (Re=10000, π=4, MU=0.23, C=0.067, and u=0), Figure 13 shows the axial Mach number distribution of a uniform tip clearance labyrinth. When the air flows through a labyrinth seal, due to the large total pressure loss, the total pressure of the last stage tip clearance is the lowest among all the stages. As mass flow rate is continuous, the axial velocity at the last stage tip clearance is the largest, so the last stage tip clearance enters the critical state first as shown in Figure 13.Figure 13
Axial Mach number distributions of uniform tip clearance labyrinth in the critical state (Re=10000, π=4, MU=0.23, C=0.067, and u=0).Under the critical state (Re=10000, π=4, MU=0.23, and C=0.067), the axial Mach number distributions in the labyrinths with different clearance nonuniformity type (convergent type and divergent type) are compared in Figure 14. For convergent type clearance labyrinths (u>0), the critical section is the last stage tip clearance all the cases, so the maximum flow capacity of the labyrinth seal is limited by the last stage tip clearance. With the increase of nonuniformity coefficient, the axial Mach number at the front three tip clearances gradually reduces. For the divergent clearance labyrinth (u<0), when the nonuniformity coefficient is small, such as the case of u=−0.1, the difference between the four tip clearances is not obvious. Although the area of the last stage tip clearance is the largest, it is still the first one to enter the critical state. As the clearance nonuniformity degree increases gradually, such as u=−0.3, u=−0.4, and u=−0.5, the critical section switches from the last stage tip clearance to the first stage tip clearance. The maximum flow capacity of the labyrinth seal is limited by the first stage tip clearance. Similar to the subcritical state, the kinetic energy carry-over effect of divergent type (u<0) gets more significant than that of convergent type (u>0).Figure 14
Axial Mach number distributions of nonuniform tip clearance labyrinths in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).In the critical state (Re=10000, π=4, MU=0.23, and C=0.067), the teeth pressure loads under different clearance nonuniformity degree (-0.5≤u≤0.5) are also compared between the convergent type and the divergent type in Figure 15. Different from the subcritical state, the teeth pressure loads in the critical state is not uniform, even if the seal clearance is uniform (u=0). The teeth pressure load of the last stage is significantly larger than those of the front three stages. For labyrinths with convergent type clearance (u>0), the distributions of teeth pressure loads in the teeth cavities are similar to that of the labyrinth with uniform clearance (u=0). Contrast to the uniform clearance labyrinth, the teeth pressure loads of the front three stages are lower, while that of the last stage is higher. For labyrinths with divergent type clearance (u<0), the variation of the teeth pressure load is no longer monotonic along the flow direction. The largest teeth pressure load is still the last stage when the clearance nonuniformity is not obvious (u=−0.1), but with the increase of the nonuniformity degree (u=−0.2, u=−0.3, u=−0.4, and u=−0.5), the distributions of teeth pressure loads in the teeth cavities become different from those of labyrinths with uniform clearance and convergent clearance. Although the first stage tip clearance is the critical section, the second stage develops into the key teeth for total pressure drop and undertakes the largest single teeth pressure load. Starting at the second stage, the teeth pressure load decreases monotonically along the flow direction. Similar to the subcritical state, the teeth pressure load maximum deviation gradually rises with the increase of the absolute value of the nonuniformity coefficient u, as shown in Table 4, meaning the clearance nonuniformity can promote the teeth pressure load deviation among the four stage. Opposed to the subcritical state, the maximum teeth pressure load deviation of the convergent type (u>0) increases at a faster rate than that of the divergent type (u<0). By contrast, the convergent nonuniformity type contributes more to the nonuniform teeth pressure load profile in the critical state.Figure 15
Teeth pressure loads in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).
(a)
u=0.1(b)
u=0.2(c)
u=0.3(d)
u=0.4(e)
u=0.5Table 4
Maximum deviation of different teeth pressure loads in the critical state (Re=10000, π=4, MU=0.23, and C=0.067).
u0.10.20.30.40.5∆ψmax−∆ψminu>00.160.190.210.220.22u<00. 040.060.150.180.18


## 3.3. Effects of Nonuniformity Degree

### 3.3.1. Pressure Coefficient

Due to nonuniform tip clearances, the law of kinetic energy dissipation will be different from that of uniform clearance labyrinths. The distribution of total pressure in the tooth cavities will also change with clearance nonuniformity degree. Figure16 shows pressure coefficient distributions in the teeth cavities of subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067) and critical state (Re=10000, π=4, MU=0.23, and C=0.067), respectively. The pressure coefficient in the teeth cavities reduces gradually along the flow direction, bur the decrease tendency is not the same under different clearance nonuniformity coefficient. When aerodynamic boundary parameters and other geometric structure parameters are all the same, the pressure coefficients in the teeth cavities of the convergent type are higher than that of the divergent type with the equal clearance nonuniformity degree u. Compared with uniform clearance labyrinth seals, the difference of total pressure distributions may change the axial aerodynamic load of nonuniform clearance labyrinth seals.Figure 16
Total pressure distributions of different teeth cavities.
(a)
Subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067)(b)
Critical state (Re=10000, π=4, MU=0.23, and C=0.067)Although the ratio of inlet total pressure to outlet static pressure is the same, pressure coefficients at the labyrinth outlet with different nonuniform coefficients are not the same. According to Figure16, the outlet pressure coefficient of the convergent type clearance labyrinth gets larger with the increase of nonuniformity degree, meaning that the proportion of total pressure dissipation from inlet to outlet decreases. Situations for divergent type clearance labyrinths are just the opposite. Figure 17 shows the labyrinth seal pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067) and critical state (Re=10000, π=4, MU=0.23, and C=0.067), respectively. With the increase of the absolute value of the nonuniformity coefficient u, seal pressure load of convergent type gradually falls down, while that of divergent type gradually grows up, regardless of subcritical state or critical state. The seal pressure load represents the ability to realize the total pressure drop, which is one of important aspect to evaluate the seal performance. In terms of total pressure drop, when the dimensionless minimum tip clearance is the same, convergent clearance contributes less to the total pressure drop with the increase of the clearance nonuniformity, while divergent clearance contributes more to the total pressure drop with the increase of the clearance nonuniformity.Figure 17
Labyrinth seal pressure loads.
(a)
Subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067)(b)
Critical state (Re=10000, π=4, MU=0.23, and C=0.067)


### 3.3.2. Flow Coefficient

When the minimum tip clearance is the same, the average tip clearance of nonuniform clearance labyrinths is larger than that of uniform clearance labyrinths. Obviously, the labyrinth with smaller average tip clearance has stronger sealing ability if the minimum tip clearance is the same. Therefore, on the premise of the same minimum tip clearance, it is easy to infer that the labyrinth seals with nonuniform tip clearance have more leakage flow than that of the labyrinth seals with uniform tip clearance. Thereby, the nonuniformity impact factor is bound to be greater than 1. However, the growth rate of the leakage flow will be different under different nonuniformity degree. The variation of the nonuniformity impact factor with the clearance nonuniformity coefficient is analyzed to clarify the effect of clearance nonuniformity on the leakage flow.Figures18–21 show the variations of nonuniformity impact factor with nonuniformity coefficient under different Reynolds numbers, pressure ratios, circumferential Mach numbers, and dimensionless minimum tip clearances (500≤Re≤25000, 1.03≤π≤4, 0.23≤MU≤0.92, 0.033≤C≤0.100, and −0.5≤u≤0.5), respectively. If minimum tip clearances are the same, two common points can be found. One is that when absolute values of nonuniformity coefficients are the same, nonuniformity impact factors of divergent type clearance labyrinths are larger than that of convergent type clearance labyrinths, meaning the seal performance of the convergent type is better than that of the divergent type. The other is that when clearance nonuniformity is small, characteristic lines of nonuniformity impact factor have little dispersion. Within the range of −0.1<u<0.1, under the same nonuniformity coefficient, deviations of nonuniformity impact factor under different operation conditions (Reynolds number, pressure ratio, circumferential Mach number, and dimensionless minimum tip clearance) are no more than 1.2%, which indicates that the effect of nonuniformity degree on the leakage flow is irrelevant to the operation conditions. In addition, other variation characteristics of the nonuniformity impact factor are also described in detail.Figure 18
Variations of nonuniformity impact factor under different Reynolds number (π=1.1, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 19
Variations of nonuniformity impact factor under different pressure ratios (Re=10000, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 20
Variations of nonuniformity impact factor under different circumferential Mach numbers (Re=10000, π=1.1, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 21
Variations of nonuniformity impact factor under different dimensionless minimum tip clearances (Re=10000, π=1.1, MU=0.23).
(a)
convergent clearance (u>0)(b)
divergent clearance (u<0)Variations of the nonuniformity impact factor at different Reynolds numbers (500≤Re≤25000) are compared in Figure 18. Except for the cases of small Reynolds number (Re=500 and Re=1000), there is little deviation between variation curves of nonuniformity impact factor when Reynolds numbers are different. Under the actual operating conditions of aeroengines, the Reynolds number is usually much larger than 500 and 1000, so the influence of Reynolds number on the characteristic of nonuniformity impact factor to nonuniformity coefficient is not obvious. In the high Reynolds number range, the influence of Reynolds number can be ignored when analyzing the effect of clearance nonuniformity on the leakage flow.Variations of nonuniformity impact factor at different pressure ratios (1.03≤π≤4) are compared in Figure 19. With the increase of pressure ratio, characteristic lines of nonuniformity impact factor gradually move down, meaning that the rise of leakage flow caused by nonuniform clearance is suppressed. When the sealing flow enters the critical state (π=2.5, π=3, and π=4), pressure ratio has little influence on the variation curve of nonuniformity impact factor. Therefore, the leakage flow of labyrinths in the subcritical state (small pressure ratio) will be affected more significantly by the nonuniform clearance. And the influence of pressure ratio can be ignored when the seal flow enters the critical state (large pressure ratio).Variations of nonuniformity impact factor at different circumferential Mach numbers (0.23≤MU≤0.92) are compared in Figure 20. With the increase of circumferential Mach number, the characteristic lines of nonuniformity impact factor gradually move up, meaning that the rise of leakage flow caused by nonuniform clearance is aggravated. The results indicate that the labyrinth seal located at high radius location in the aeroengines will be affected more obviously by the clearance nonuniformity. Meanwhile, when the aeroengines operate at high speed, the leakage flow of labyrinth will be also affected more significantly by the nonuniform clearance. Because the higher radius and the higher rotating speed contribute the larger circumferential Mach numbers.Variations of nonuniformity impact factor at different dimensionless minimum tip clearances (0.033≤C≤0.100) are compared in Figure 21. With the increase of dimensionless minimum tip clearance, the characteristic lines of nonuniformity impact factor gradually move down, meaning that the rise of leakage flow caused by nonuniform clearance is suppressed. For labyrinth seals with the same tooth pitch in aeroengines, the leakage flow of labyrinth seals clearance with smaller sealing clearance will be affected more significantly by the nonuniform clearance.


## 3.3.1. Pressure Coefficient

Due to nonuniform tip clearances, the law of kinetic energy dissipation will be different from that of uniform clearance labyrinths. The distribution of total pressure in the tooth cavities will also change with clearance nonuniformity degree. Figure16 shows pressure coefficient distributions in the teeth cavities of subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067) and critical state (Re=10000, π=4, MU=0.23, and C=0.067), respectively. The pressure coefficient in the teeth cavities reduces gradually along the flow direction, bur the decrease tendency is not the same under different clearance nonuniformity coefficient. When aerodynamic boundary parameters and other geometric structure parameters are all the same, the pressure coefficients in the teeth cavities of the convergent type are higher than that of the divergent type with the equal clearance nonuniformity degree u. Compared with uniform clearance labyrinth seals, the difference of total pressure distributions may change the axial aerodynamic load of nonuniform clearance labyrinth seals.Figure 16
Total pressure distributions of different teeth cavities.
(a)
Subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067)(b)
Critical state (Re=10000, π=4, MU=0.23, and C=0.067)Although the ratio of inlet total pressure to outlet static pressure is the same, pressure coefficients at the labyrinth outlet with different nonuniform coefficients are not the same. According to Figure16, the outlet pressure coefficient of the convergent type clearance labyrinth gets larger with the increase of nonuniformity degree, meaning that the proportion of total pressure dissipation from inlet to outlet decreases. Situations for divergent type clearance labyrinths are just the opposite. Figure 17 shows the labyrinth seal pressure loads in the subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067) and critical state (Re=10000, π=4, MU=0.23, and C=0.067), respectively. With the increase of the absolute value of the nonuniformity coefficient u, seal pressure load of convergent type gradually falls down, while that of divergent type gradually grows up, regardless of subcritical state or critical state. The seal pressure load represents the ability to realize the total pressure drop, which is one of important aspect to evaluate the seal performance. In terms of total pressure drop, when the dimensionless minimum tip clearance is the same, convergent clearance contributes less to the total pressure drop with the increase of the clearance nonuniformity, while divergent clearance contributes more to the total pressure drop with the increase of the clearance nonuniformity.Figure 17
Labyrinth seal pressure loads.
(a)
Subcritical state (Re=10000, π=1.3, MU=0.23, and C=0.067)(b)
Critical state (Re=10000, π=4, MU=0.23, and C=0.067)


## 3.3.2. Flow Coefficient

When the minimum tip clearance is the same, the average tip clearance of nonuniform clearance labyrinths is larger than that of uniform clearance labyrinths. Obviously, the labyrinth with smaller average tip clearance has stronger sealing ability if the minimum tip clearance is the same. Therefore, on the premise of the same minimum tip clearance, it is easy to infer that the labyrinth seals with nonuniform tip clearance have more leakage flow than that of the labyrinth seals with uniform tip clearance. Thereby, the nonuniformity impact factor is bound to be greater than 1. However, the growth rate of the leakage flow will be different under different nonuniformity degree. The variation of the nonuniformity impact factor with the clearance nonuniformity coefficient is analyzed to clarify the effect of clearance nonuniformity on the leakage flow.Figures18–21 show the variations of nonuniformity impact factor with nonuniformity coefficient under different Reynolds numbers, pressure ratios, circumferential Mach numbers, and dimensionless minimum tip clearances (500≤Re≤25000, 1.03≤π≤4, 0.23≤MU≤0.92, 0.033≤C≤0.100, and −0.5≤u≤0.5), respectively. If minimum tip clearances are the same, two common points can be found. One is that when absolute values of nonuniformity coefficients are the same, nonuniformity impact factors of divergent type clearance labyrinths are larger than that of convergent type clearance labyrinths, meaning the seal performance of the convergent type is better than that of the divergent type. The other is that when clearance nonuniformity is small, characteristic lines of nonuniformity impact factor have little dispersion. Within the range of −0.1<u<0.1, under the same nonuniformity coefficient, deviations of nonuniformity impact factor under different operation conditions (Reynolds number, pressure ratio, circumferential Mach number, and dimensionless minimum tip clearance) are no more than 1.2%, which indicates that the effect of nonuniformity degree on the leakage flow is irrelevant to the operation conditions. In addition, other variation characteristics of the nonuniformity impact factor are also described in detail.Figure 18
Variations of nonuniformity impact factor under different Reynolds number (π=1.1, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 19
Variations of nonuniformity impact factor under different pressure ratios (Re=10000, MU=0.23, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 20
Variations of nonuniformity impact factor under different circumferential Mach numbers (Re=10000, π=1.1, and C=0.067).
(a)
Convergent clearance (u>0)(b)
Divergent clearance (u<0)Figure 21
Variations of nonuniformity impact factor under different dimensionless minimum tip clearances (Re=10000, π=1.1, MU=0.23).
(a)
convergent clearance (u>0)(b)
divergent clearance (u<0)Variations of the nonuniformity impact factor at different Reynolds numbers (500≤Re≤25000) are compared in Figure 18. Except for the cases of small Reynolds number (Re=500 and Re=1000), there is little deviation between variation curves of nonuniformity impact factor when Reynolds numbers are different. Under the actual operating conditions of aeroengines, the Reynolds number is usually much larger than 500 and 1000, so the influence of Reynolds number on the characteristic of nonuniformity impact factor to nonuniformity coefficient is not obvious. In the high Reynolds number range, the influence of Reynolds number can be ignored when analyzing the effect of clearance nonuniformity on the leakage flow.Variations of nonuniformity impact factor at different pressure ratios (1.03≤π≤4) are compared in Figure 19. With the increase of pressure ratio, characteristic lines of nonuniformity impact factor gradually move down, meaning that the rise of leakage flow caused by nonuniform clearance is suppressed. When the sealing flow enters the critical state (π=2.5, π=3, and π=4), pressure ratio has little influence on the variation curve of nonuniformity impact factor. Therefore, the leakage flow of labyrinths in the subcritical state (small pressure ratio) will be affected more significantly by the nonuniform clearance. And the influence of pressure ratio can be ignored when the seal flow enters the critical state (large pressure ratio).Variations of nonuniformity impact factor at different circumferential Mach numbers (0.23≤MU≤0.92) are compared in Figure 20. With the increase of circumferential Mach number, the characteristic lines of nonuniformity impact factor gradually move up, meaning that the rise of leakage flow caused by nonuniform clearance is aggravated. The results indicate that the labyrinth seal located at high radius location in the aeroengines will be affected more obviously by the clearance nonuniformity. Meanwhile, when the aeroengines operate at high speed, the leakage flow of labyrinth will be also affected more significantly by the nonuniform clearance. Because the higher radius and the higher rotating speed contribute the larger circumferential Mach numbers.Variations of nonuniformity impact factor at different dimensionless minimum tip clearances (0.033≤C≤0.100) are compared in Figure 21. With the increase of dimensionless minimum tip clearance, the characteristic lines of nonuniformity impact factor gradually move down, meaning that the rise of leakage flow caused by nonuniform clearance is suppressed. For labyrinth seals with the same tooth pitch in aeroengines, the leakage flow of labyrinth seals clearance with smaller sealing clearance will be affected more significantly by the nonuniform clearance.


## 4. Conclusions

In this work, nonuniform clearance effects on pressure distribution and leakage flow in straight-through labyrinths are investigated by numerical simulation, including nonuniformity cause (rotor deformation or stator deformation), nonuniformity type (convergent clearance or divergent clearance), and no-uniformity degree (nonuniformity coefficient). The pressure coefficient and the flow coefficient of labyrinth seals with equal stator radius (ESR) and equal rotor radius (ERR) is compared to clarify the effect of nonuniformity cause. Distributions of axial Mach number and pressure load are discussed for convergent and divergent type clearance labyrinths to analyze the effect of nonuniformity type. The variations of nonuniformity impact factor are also studied under different Reynolds numbers, pressure ratios, circumferential Mach numbers, and dimensionless minimum tip clearances to visualize the effect of nonuniformity degree. This paper draws the following conclusions:(i)
For typical nonuniform clearance labyrinths in aeroengines (u=±0.1, ±0.2, and ±0.3), there is no significant influence of nonuniformity cause (rotor deformation or stator deformation) on the pressure coefficient and the flow coefficient. The maximum flow coefficient deviation between the ESR and the ERR is less than 0.8%(ii)
When absolute values of nonuniformity coefficients are the same, nonuniformity impact factors of divergent type clearance labyrinths are larger than that of convergent type clearance labyrinths, meaning the seal performance of the convergent type is better than that of the divergent type(iii)
When the absolute value of the nonuniformity coefficientu is less than 0.1, the effect of the nonuniformity degree on the leakage flow is almost irrelevant to the operating conditions. In the parameter range of the current study, deviations of nonuniformity impact factor under different operating conditions (Reynolds number, pressure ratio, circumferential Mach number, and dimensionless minimum tip clearance) are no more than 1.2%.(iv)
Under the typical labyrinths operation range in aeroengines (500≤Re≤25000, 1.03≤π≤4, 0.23≤MU≤0.92, and 0.033≤C≤0.100 and −0.5≤u≤0.5), compared with the uniform sealing clearance labyrinth with the same minimum tip clearance, the nonuniform labyrinth has a larger leakage flow. The effect of the nonuniform clearance is more obvious at small pressure ratio, high circumferential Mach number, and small dimensionless minimum tip clearance



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*Source: 9684007-2022-12-20.xml*