Investigation on the Formation and Evolution Mechanism of Flow-Resistance-Increasing Vortex of Aero-Engine Labyrinth Based on Entropy Generation Analysis

: Labyrinth seals are widely employed in the air system of aircraft engines since they reduce the leakages occurring between blades and shrouds, which affect the entropy generation signiﬁcantly. Excessive leakage ﬂow of the labyrinth may be reduced the efﬁciency and performance of the engine. This paper proposes the concept of ﬂow-resistance-increasing vortex (FRIV) on the top of the labyrinth that is based on the ﬂow entropy generation mechanism of the stepped labyrinth and the main ﬂow characteristics that lead to entropy generation. A three-dimensional simulation model of the labyrinth structure was established, and the model was compared and veriﬁed with the experimental data of the reference. The relative dissipation strength and vorticity distribution of the FRIV were theoretically analyzed. It was conﬁrmed that the dissipative intensity distribution was the same as the vorticity distribution, and the correlation coefﬁcient was larger in the labyrinth tip region. Therefore, a parametric study was conducted on design parameters related to the FRIV, including the teeth inclined angle, tooth crest width, step inclined angle, and other parameters. The results are beneﬁcial for the construction of a stronger FRIV to reduce the leakage. This research is of great signiﬁcance for the improvement of engine efﬁciency and for the reduction of fuel consumption in the future.


Introduction
The labyrinth sealing technology is an important sealing technology in the air system of aero-engines, one that is a non-contact dynamic sealing composed of a rotor and a stator. There is a clearance between the tooth top and the static part, which directly affects the leakage flow of the labyrinth teeth. If the leakage flow rate of the labyrinth teeth is greater than the current design, it may directly reduce the efficiency and performance of the engine, and the maximum efficiency may even be reduced by 10% [1]. On the one hand, the kinetic energy of the fluid is consumed by the sudden expansion and contraction of the flow channel and converted to entropy production. On the other hand, when the rotating parts do work on the contact fluid, it will result in energy loss and entropy generation [2]. In other words, the main factors affecting entropy generation include the total pressure loss and the windage heating.
In the literature, a large number of studies on labyrinth seals can be found concerning experimental tests and numerical and analytical models. In terms of theoretical research, Martin first proposed the calculation method of the flow rate of the labyrinth teeth. The labyrinth tooth is regarded as a series of holes, and the fluid is assumed to be in an isothermal flow. However, this method is only suitable for incompressible streams, and the calculation error is large [3]. Then, Vermes revised the calculation formula proposed by Martin and added the correction term [4]. Zimmermann put forward a relational formula for calculating the carry factor or residual energy factor in the leakage flow rate of the labyrinth by means of experiments [5]. These theories are the basis for the research on the leakage of the labyrinth teeth. In terms of numerical research, the processing power of modern computers allows indeed for the simulation of more complex and detailed phenomena than that of the past years. On the basis of the experimental data of Waschka et al., Nayak et al. used a numerical simulation to study the effect of rotational speed on the sealing performance of the labyrinth teeth [6]. Lee et al. reviewed and summarized the literature on the grate seal structure since the 1960s, discussing the effect of sealing parameters on leakage flow [7]. Soemarwoto analyzed the flow mechanism of stepped teeth and stepped helical teeth by numerical simulation, and further studied the effect of tooth height on the vortex [8]. Rhode and Demko conducted a numerical study on the straightthrough grate sealing structure by the finite difference method [9,10]. Stoff calculated the incompressible flow inside the straight-through labyrinth seal structure by the standard k − ε turbulence model and analyzed the relationship between the leakage of the labyrinth structure and the pressure gradient [11]. Rhode and Sobolik et al. modified the standard k − ε turbulence model by introducing the effective viscosity coefficient and studied the compressible flow in the straight-through labyrinth seal structure [12]. Rhode and Hibbs analyzed the effect of dimensional structure changes on the sealing performance of the labyrinth structure [13,14]. Demko and Morrison studied the effect of rotational speed on the flow of the straight-through labyrinth seal structure by numerical method [15]. Rapisarda et al. studied the influence of step position and tooth edge on leakage coefficient by using the numerical method and obtained the influence law of windage heating and vortices on leakage [16,17]. Kali charan Nvayak et al. studied the windage heating characteristics of the grooved stepped tooth-honeycomb bushing labyrinth seal structure by numerical method [18,19]. Kaliraj et al. analyzed static and rotational effects of labyrinth seals at various flows and geometrical dimensions and optimized leak flow by straight and steeped seal configurations [20]. Desando and Rapisarda focused on the implementation of a numerical model for rotating stepped labyrinth seals installed in low-pressure turbines [21]. Ganine presented a simplified coupled transient analysis methodology that allows for the assessment of the aerothermal and thermomechanical responses of engine components together with cooling air mass flow and pressure and temperature distributions in an automatic fully integrated way [22]. In the experimental research, Child provided an overview of the development of vortices in labyrinth seals and their main effects on self-excited instability [23]. Wittig [25]. The results show that the local circumferential velocity and the development of the swirl in the grate flow channel are the key factors affecting the accuracy of the temperature rise calculation [26]. Willenboge et al. studied the influence of Reynolds number and inlet and outlet pressure ratio on the leakage characteristics of stepped grate structure through experiments, which showed that at a high Reynolds number, the flow coefficient may have a local maximum value [27]. Stocker and Rhode et al. studied the effect of the relative position of the wear groove and the axial direction of the labyrinth on the leakage characteristics of the labyrinth seal through experiments [28,29]. Millward and Edward studied the variation law of the heat generated by the wind resistance with the leakage amount and the pressure ratio under different labyrinth tooth structures [30]. Woschka et al. focused on analyzing the effect of rotational speed on head clearance, which showed that rotational speed has a great influence on the head clearance of labyrinth teeth [31]. Braun et al. accurately measured the tooth tip clearance with a static plane labyrinth tooth test bench and showed the effects of pressure ratio and tip clearance on the leakage characteristics of straight and stepped teeth [32]. Min et al. evaluated the leakage characteristics of a stepped labyrinth seal by experiments and computational fluid dynamics [33]. Khan et al. investigated the physical behaviors of heat and mass transfer flow with entropy generation through the effects of embedded parameters [34]. Ramzan et al. reached the conclusion that the entropy generation is increased with the enhancement of radiation parameter, Eckert number, Lewis number, temperature difference parameter, dimensionless constant parameter, Curie temperature, Prandtl number, and concentration difference parameter [35]. Although a large amount of work has been done on the research of labyrinth teeth, hardly anyone has investigated the characteristics of the labyrinth teeth region by the theoretical of entropy generation, which shows energy distribution accurately [36].
Labyrinth sealing technology is becoming increasingly more important in improving the performance of the engine with the advancement of materials and process technology. In practical engineering applications, the flow in the labyrinth seal area is very complicated and there is energy loss. Using the entropy production theory to conduct in-depth research on the flow characteristics of the stepped grate, the energy in this area can be quantitatively analyzed. In this work, flow analysis and reconstruction are performed on the grate region. A method is proposed to construct stronger flow resistance, increasing vortices and reducing leakage by enhancing the labyrinth tip velocity gradient and throttling effect. This research can serve as an important reference for the design and application of aeroengine labyrinth and plays an important role in reducing the kerosene consumption and improving thermal efficiency. The rest of this paper is organized as follows. In Section 2, the flow-resistance-increasing vortex (FRIV) on the top of the labyrinth tooth is introduced, and the leakage characteristics of the labyrinth tooth region are studied on the basis of the theory of entropy production. In Section 3, the labyrinth simulation model is established and checked. In Section 4, the simulation results are analyzed and compared with numerical calculations, which shows that the method proposed in this paper to reduce the leakage of the labyrinth is effective. The last section presents the conclusions of this research.

Definition of Flow-Resistance-Increasing Vortices
The labyrinth structure is widely used in the air system of the engine, which is mainly composed of the turntable and the stator casing. The working principle of a labyrinth seal consists of separating regions at different pressures through the clearance, i.e., a gap placed between rotating and stationary components [37]. In this gap, the conversion of pressure energy into kinetic energy occurs. The latter can then be dissipated into thermal energy into the cavity downstream of the clearance or transferred to the next gap through the kinetic carryover [38]. This labyrinth seal adopted in the present work is a stepped type, with five labyrinth teeth [39]. It can be seen from Figure 1 that there is a vortex in the area behind the labyrinth teeth and another vortex structure in the labyrinth cavity. The vortex can hinder the airflow of the labyrinth tip, and we define the vortex as a flow-resistance-increasing vortices. The labyrinth geometric parameters and their representative symbols are shown in Table 1.

Theoretical of Entropy Generation
In this work, we focused on the sealing characteristics of the labyrinth; therefore, the system is partially simplified. The flow in the labyrinth is assumed to be steady, viscous, adiabatic, and compressible, and thermal conductivity λ and rotational speed n are not considered. Theoretically, the labyrinth structure of an aero-engine can be regarded as an adiabatic continuous power system. The theoretical deduction process of the entropy generation about aero-engine system is shown as follows, which includes viscous dissipation, heat transfer with temperature difference, metal friction, and mixing.

Theoretical of Entropy Generation
In this work, we focused on the sealing characteristics of the labyrinth; therefore, the system is partially simplified. The flow in the labyrinth is assumed to be steady, viscous, adiabatic, and compressible, and thermal conductivity  and rotational speed n are not considered. Theoretically, the labyrinth structure of an aero-engine can be regarded as an adiabatic continuous power system. The theoretical deduction process of the entropy generation about aero-engine system is shown as follows, which includes viscous dissipation, heat transfer with temperature difference, metal friction, and mixing.
The entropy equation of the open system [40] is as follows:

Geometric Parameters Symbol Initial Value Unit
Step where S g is the entropy generation; T r is the wall temperature; Q is the amount of heat exchange; s 2 , s 1 is the specific entropy; m 1 , m 2 is the mass flow; and S CV is the total entropy of the stable flow system. For steady flow, adiabatic systems are as follows: For the unit mass working medium, introduce Equation (2) into Equation (1), and the entropy generation equation is simplified to Symmetry 2022, 14, 881

of 22
The energy equation is as follows [41]: where h 1 denotes import specific enthalpy, h 2 denotes export specific enthalpy, h * 1 denotes total import enthalpy, h * 2 denotes total export enthalpy, T * 1 denotes total inlet temperature, T * 2 denotes total outlet temperature, and c p denotes constant pressure specific heat. For a labyrinth in an air system, the entropy generation of the system can be expressed as follows [42]: where T 1 denotes inlet static temperature, T 2 denotes outlet static temperature, P 1 denotes inlet static pressure, P 2 denotes outlet static pressure, P * 1 denotes inlet total pressure, P * 2 denotes outlet total pressure, T * 1 denotes inlet total temperature, T * 2 denotes outlet total temperature, c p denotes constant pressure specific heat, and R denotes general gas constant.
For the windage heating, it is mainly caused by friction: where M denotes the frictional force, ω denotes the rotational angular velocity, and w s denotes the frictional power.
The dimensionless parameters such as the discharge coefficient and windage heating can be expressed as follows [43]: mc p ∆T * (10) .
The dissipative entropy generation in the flow originates from the dissipation generated by the viscosity of the fluid, resulting in the irreversible conversion of mechanical energy to internal energy. The equation of the relative dissipation intensity of the fluid micro-element is as follows [44]: Symmetry 2022, 14, 881 6 of 22 Flow vorticity can be written as where V x , V y , V z are the flow velocities in the x, y, and z directions, respectively. In order to investigate the correlation between the vorticity and the relative dissipation intensity distribution, the correlation coefficient τ is defined as follows [45]: Introduce the vorticity transport equation [46]: where (ω•∇)V denotes the stretching and bending of the vortex line caused by the velocity gradient of the flow field, which causes the velocity gradient in the flow field to change significantly and results in a change in the absolute value of the vorticity; ω(∇•V) denotes the change in the magnitude of the vorticity caused by the volume change of the fluid micelle, which causes the absolute value of the vorticity to change; and ∇ρ × ∇p/ρ 2 denotes the effect of baroclinic moment on vorticity due to non-parallel pressure gradients and density gradients. For a positive pressure fluid, whose density is only a function of pressure, and thus the density and pressure terms have the same gradient, this term is equal to 0, where v∇ 2 ω denotes the viscous diffusion effect of vorticity. It can be seen that the first two items in the equation are the main reasons for the generation of the FRIV.
In conclusion, the local entropy generation of the labyrinth can be expressed as It can be seen that the major local entropy increase comes from the windage heating and the total pressure loss in the flow of the labyrinth.

Simulation Model and Verification of Labyrinth
In this section, the labyrinth model is established for simulation and was checked with the known experimental data. In the following sections, the simulation results and theoretical calculations are compared and analyzed.

Numerical Approach and Computational Meshing
In this paper, UG parametric modeling was adopted, and the step labyrinth and bushing are both axisymmetric. ANSYS CFX (ver. 19.0, ANSYS Inc., Canonsburg, PA, USA, 2018) [47], a commercial software program, was used for CFD analysis. Figure 2 shows the diagram of meshes, Figure 3 shows the diagram of local mesh refinement in the tip of the labyrinth that is critical for investigation [48]. Nevertheless, we ensured that the 3D was practically close to the 2D domain by setting the smallest width as far as we could and applied symmetry conditions to lateral faces. In order to reduce the amount of calculation, a two-dimensional axisymmetric (rotational) model was adopted; this method is recommended for 2D calculations according to the CFX manual [49]. ANSYS ICEM is used for unstructured meshing. ANSYS Fluent is used for the calculation, the inlet and outlet set as the pressure control, and the wall set as non-slip adiabatic. The 3D unsteady compressible N-S equation is solved by the finite volume method. The turbulence is modeled with the k − ε model with a second-order upwind discretization scheme in time and space.
tip of the labyrinth that is critical for investigation [48]. Nevertheless, we ensured that the 3D was practically close to the 2D domain by setting the smallest width as far as we could and applied symmetry conditions to lateral faces. In order to reduce the amount of calculation, a two-dimensional axisymmetric (rotational) model was adopted; this method is recommended for 2D calculations according to the CFX manual [49]. ANSYS ICEM is used for unstructured meshing. ANSYS Fluent is used for the calculation, the inlet and outlet set as the pressure control, and the wall set as non-slip adiabatic. The 3D unsteady compressible N-S equation is solved by the finite volume method. The turbulence is modeled with the k  − model with a second-order upwind discretization scheme in time and space.

Boundary Conditions and Mesh Independent
Inlet temperature and outlet static pressure and temperature were used as boundary conditions to simulate the operating conditions of the test. Table 2 shows the boundary conditions.
Define the inlet and outlet pressure ratio as where converted flow is calculated as follows:

Boundary Conditions
Value Unit Outlet pressure 101,325 Pa Import and export pressure ratio 1.5 tip of the labyrinth that is critical for investigation [48]. Nevertheless, we ensured that the 3D was practically close to the 2D domain by setting the smallest width as far as we could and applied symmetry conditions to lateral faces. In order to reduce the amount of calculation, a two-dimensional axisymmetric (rotational) model was adopted; this method is recommended for 2D calculations according to the CFX manual [49]. ANSYS ICEM is used for unstructured meshing. ANSYS Fluent is used for the calculation, the inlet and outlet set as the pressure control, and the wall set as non-slip adiabatic. The 3D unsteady compressible N-S equation is solved by the finite volume method. The turbulence is modeled with the k  − model with a second-order upwind discretization scheme in time and space.

Boundary Conditions and Mesh Independent
Inlet temperature and outlet static pressure and temperature were used as boundary conditions to simulate the operating conditions of the test. Table 2 shows the boundary conditions.
Define the inlet and outlet pressure ratio as where converted flow is calculated as follows:

Boundary Conditions
Value Unit Outlet pressure 101,325 Pa Import and export pressure ratio 1.5

Boundary Conditions and Mesh Independent
Inlet temperature and outlet static pressure and temperature were used as boundary conditions to simulate the operating conditions of the test. Table 2 shows the boundary conditions.

Boundary Conditions Value Unit
Outlet pressure 101,325 Pa Import and export pressure ratio 1.5 Inlet temperature 300 K Outlet temperature 300 K Define the inlet and outlet pressure ratio as where converted flow is calculated as follows: There are two parameters that affect the number of grids and the calculation results in the mesh settings: global minimum mesh size and the minimum mesh size for local refinement of tooth tips. The mesh-independent calculation will be performed for different values of the parameters.
The global minimum mesh sizes were set as 2, 0.6, and 0.2 mm for three working conditions, and the minimum mesh size for local refinement of tooth tips was set as 0.03 mm. The grid numbers corresponding to the three working conditions were 180,000, 190,000, and 360,000. It can be seen from Figure 4 that the flow rate of the grate remained stable with global minimum mesh size varying from 0.2 to 2 mm. In this research, the global minimum mesh size was set as 0.6 mm.
stable with global minimum mesh size varying from 0.2 to 2 mm. In this research, the global minimum mesh size was set as 0.6 mm.
The minimum mesh size for local refinement of tooth tips was set as 0.1, 0.03, 0.02, and 0.01 mm for four working conditions, and the global minimum mesh size was set as 0.6 mm. The grid numbers corresponding to the four working conditions were 180,000, 200,000, 320,000, and 710,000. It can be seen from Figure 5, that when the minimum size of the refined mesh of the tooth tip was reduced to 0.03 mm, the flow rate of the labyrinth did not change significantly. Therefore, the minimum size of the local refined mesh of the tooth tip region in the CFD model was set as 0.03 mm, and the number of grids was around 200,000.   The minimum mesh size for local refinement of tooth tips was set as 0.1, 0.03, 0.02, and 0.01 mm for four working conditions, and the global minimum mesh size was set as 0.6 mm. The grid numbers corresponding to the four working conditions were 180,000, 200,000, 320,000, and 710,000. It can be seen from Figure 5, that when the minimum size of the refined mesh of the tooth tip was reduced to 0.03 mm, the flow rate of the labyrinth did not change significantly. Therefore, the minimum size of the local refined mesh of the tooth tip region in the CFD model was set as 0.03 mm, and the number of grids was around 200,000.
There are two parameters that affect the number of grids and the calculation results in the mesh settings: global minimum mesh size and the minimum mesh size for local refinement of tooth tips. The mesh-independent calculation will be performed for different values of the parameters.
The global minimum mesh sizes were set as 2, 0.6, and 0.2 mm for three working conditions, and the minimum mesh size for local refinement of tooth tips was set as 0.03 mm. The grid numbers corresponding to the three working conditions were 180,000, 190,000, and 360,000. It can be seen from Figure 4 that the flow rate of the grate remained stable with global minimum mesh size varying from 0.2 to 2 mm. In this research, the global minimum mesh size was set as 0.6 mm.
The minimum mesh size for local refinement of tooth tips was set as 0.1, 0.03, 0.02, and 0.01 mm for four working conditions, and the global minimum mesh size was set as 0.6 mm. The grid numbers corresponding to the four working conditions were 180,000, 200,000, 320,000, and 710,000. It can be seen from Figure 5, that when the minimum size of the refined mesh of the tooth tip was reduced to 0.03 mm, the flow rate of the labyrinth did not change significantly. Therefore, the minimum size of the local refined mesh of the tooth tip region in the CFD model was set as 0.03 mm, and the number of grids was around 200,000.

Model Checking
In order to verify the accuracy of the parametric model, the calculation results of the simulation model in this paper were compared with the experimental data provided by known references [49]. The geometrical parameters of the labyrinth provided by the reference [50] are shown in Table 3. The data were input into the simulation model for calculating the leakage flow of the labyrinth under different inlet and outlet pressure ratios, and then we converted the leakage into the converted flow for comparison. The comparison results are shown in Table 4. In the present research, we briefly discussed the influence of various boundary conditions on the converted flow. It can be seen from Figure 6 that the converted flow was gradually increased with the increase of the pressure ratio. In addition, we can quantify that the conversion flow increased with the inlet temperature increase, and the conversion flow increased with the inlet pressure decrease from Equation (2).  It can be seen from Table 4 that there were some differences in the calculation results obtained using the various turbulence models. In the calculated operating conditions, the maximum relative error of the converted flow was 4.29% which used the standard k  − turbulence model. Therefore, the standard k  − turbulence model was selected as the turbulence model for subsequent related calculations.
The results show that the calculation results of this paper are in fantastic agreement with the experimental results of the reference (Figure 7). In the working conditions, the maximum relative error of the converted flow rate was no more than 5%. The simulation model of the labyrinth established in this paper has high reliability. It can be seen from Table 4 that there were some differences in the calculation results obtained using the various turbulence models. In the calculated operating conditions, the maximum relative error of the converted flow was 4.29% which used the standard k − ε turbulence model. Therefore, the standard k − ε turbulence model was selected as the turbulence model for subsequent related calculations.
The results show that the calculation results of this paper are in fantastic agreement with the experimental results of the reference (Figure 7). In the working conditions, the maximum relative error of the converted flow rate was no more than 5%. The simulation model of the labyrinth established in this paper has high reliability.
It can be seen from Table 4 that there were some differences in the calculation results obtained using the various turbulence models. In the calculated operating conditions, the maximum relative error of the converted flow was 4.29% which used the standard k  − turbulence model. Therefore, the standard k  − turbulence model was selected as the turbulence model for subsequent related calculations.
The results show that the calculation results of this paper are in fantastic agreement with the experimental results of the reference (Figure 7). In the working conditions, the maximum relative error of the converted flow rate was no more than 5%. The simulation model of the labyrinth established in this paper has high reliability.

Results and Discussion
In this section, the relationship between the FRIV and the relative dissipation intensity was first analyzed, being based on the theoretical research and simulation results.

Results and Discussion
In this section, the relationship between the FRIV and the relative dissipation intensity was first analyzed, being based on the theoretical research and simulation results. Then, the influence of the labyrinth tooth inclination angle, tooth tip width, step shape, and other parameters on the FRIV entropy generation were investigated. Finally, a more effective method to utilize the FRIV was proposed that is based on the research results.

The Formation Mechanism of FRIV
The flow field in the labyrinth region is shown in Figure 8. It can be seen that a high-speed airflow was formed in labyrinth region when the fluid flowed through the labyrinth teeth, which was mainly because of the strong throttling effect of the incoming flow at the tip. When the airflow passed through the labyrinth teeth, it suddenly expanded and was spread at a certain angle. Under the action of the flow resistance of the fluid in the labyrinth cavity and the frictional resistance of the wall surface, a vortex was formed in the labyrinth cavity. Figure 9 shows the vorticity cloud map of the labyrinth, and Figure 10 shows the relative dissipation intensity distribution cloud map. Comparing and analyzing the two figures, we see that the distribution of the relative dissipative intensity was the same as the vorticity distribution, and it was mainly distributed on the wall and the tooth tip.
In order to express the correlation of the vorticity with the relative dissipative intensity distribution, we illustrated in Figure 11 the correlation coefficient cloud diagram, which was obtained by simulation. From the simulation results, it can be seen that there was a large correlation coefficient in the labyrinth tip. This result is consistent with the conclusion obtained from Equation (16), which indicates that there was a larger vortex. This vortex can be explained by the first term in Equation (17), which impeded the airflow at the tip and reduced leakage. When we input the vortex intensity ω obtained by simulation into Equation (18), it can be seen that the entropy generation increased exponentially.

Influence of Labyrinth Tip width Variation on Entropy Generation
The labyrinth teeth with the labyrinth tip widths of 0.2, 0.4, 0.6, and 0.8 mm were selected for comparative research. Figure 12 shows the flow field for different labyrinth tip widths. Figure 13 shows the distribution of entropy generation for different labyrinth tip widths. effective method to utilize the FRIV was proposed that is based on the research results.

The Formation Mechanism of FRIV
The flow field in the labyrinth region is shown in Figure 8. It can be seen that a highspeed airflow was formed in labyrinth region when the fluid flowed through the labyrinth teeth, which was mainly because of the strong throttling effect of the incoming flow at the tip. When the airflow passed through the labyrinth teeth, it suddenly expanded and was spread at a certain angle. Under the action of the flow resistance of the fluid in the labyrinth cavity and the frictional resistance of the wall surface, a vortex was formed in the labyrinth cavity.  Figure 9 shows the vorticity cloud map of the labyrinth, and Figure 10 shows the relative dissipation intensity distribution cloud map. Comparing and analyzing the two figures, we see that the distribution of the relative dissipative intensity was the same as the vorticity distribution, and it was mainly distributed on the wall and the tooth tip.  teeth, which was mainly because of the strong throttling effect of the incoming flow at the tip. When the airflow passed through the labyrinth teeth, it suddenly expanded and was spread at a certain angle. Under the action of the flow resistance of the fluid in the labyrinth cavity and the frictional resistance of the wall surface, a vortex was formed in the labyrinth cavity.  Figure 9 shows the vorticity cloud map of the labyrinth, and Figure 10 shows the relative dissipation intensity distribution cloud map. Comparing and analyzing the two figures, we see that the distribution of the relative dissipative intensity was the same as the vorticity distribution, and it was mainly distributed on the wall and the tooth tip.  In order to express the correlation of the vorticity with the relative dissipative intensity distribution, we illustrated in Figure 11 the correlation coefficient cloud diagram, which was obtained by simulation. From the simulation results, it can be seen that there was a large correlation coefficient in the labyrinth tip. This result is consistent with the conclusion obtained from Equation (16), which indicates that there was a larger vortex. This vortex can be explained by the first term in Equation (17), which impeded the airflow which was obtained by simulation. From the simulation results, it can be seen that there was a large correlation coefficient in the labyrinth tip. This result is consistent with the conclusion obtained from Equation (16), which indicates that there was a larger vortex. This vortex can be explained by the first term in Equation (17), which impeded the airflow at the tip and reduced leakage. When we input the vortex intensity  obtained by simulation into Equation (18), it can be seen that the entropy generation increased exponentially. Figure 11. Correlation coefficient distribution.

Influence of Labyrinth Tip width Variation on Entropy Generation
The labyrinth teeth with the labyrinth tip widths of 0.2, 0.4, 0.6, and 0.8 mm were selected for comparative research. Figure 12 shows the flow field for different labyrinth tip widths. Figure 13 shows the distribution of entropy generation for different labyrinth tip widths.
It can be seen from Figure 12 that with the increase of labyrinth width, the direction of airflow velocity was gradually reversed, and the strength of the FRIV was weakened.
It can be seen from Equation (14) that  may be decreased with the increase of the labyrinth tip width, and the relative dissipative strength decreased with the increase of the tip width. It can be seen from Equation (15) that the vortex strength decreased with the increase of the tip width and the influence of the vortex became weaker when the tip width was larger. When the tip width was small, the gas flow direction changed rapidly after the gas in the upper labyrinth cavity flowed through the tip, where the velocity gradient was large. This result was consistent with the conclusion in Equation (17) that the vortex was stretched and bent due to the velocity gradient of the flow field, which led to changes in the magnitude and direction of the vorticity. In addition, the major flow in the tip area may be affected by the extrusion of the vortex on the rear side of the labyrinth tooth tip. When the tooth tip width of the labyrinth was smaller, the vortex on the rear side of the tooth tip was stronger, which strengthened the influence on the flow in the tooth tip area and strengthened the compression effect of the tooth tip gap. Further, the flow field data (as shown in Figure 12) obtained by the simulation was output and substituted into the entropy generation of Equations (15) and (18). The relationship between the entropy generation and the airflow position can be obtained. It can be seen from Figure 14 that the results are consistent with the variation of entropy generation with tooth tip width (as shown in Figure 13).  It can be seen from Figure 12 that with the increase of labyrinth width, the direction of airflow velocity was gradually reversed, and the strength of the FRIV was weakened. It can be seen from Equation (14) that φ may be decreased with the increase of the labyrinth tip width, and the relative dissipative strength decreased with the increase of the tip width. It can be seen from Equation (15) that the vortex strength decreased with the increase of the tip width and the influence of the vortex became weaker when the tip width was larger. When the tip width was small, the gas flow direction changed rapidly after the gas in the upper labyrinth cavity flowed through the tip, where the velocity gradient was large. This result was consistent with the conclusion in Equation (17) that the vortex was stretched and bent due to the velocity gradient of the flow field, which led to changes in the magnitude and direction of the vorticity. In addition, the major flow in the tip area may be affected by the extrusion of the vortex on the rear side of the labyrinth tooth tip. When the tooth tip width of the labyrinth was smaller, the vortex on the rear side of the tooth tip was stronger, which strengthened the influence on the flow in the tooth tip area and strengthened the compression effect of the tooth tip gap. Further, the flow field data (as shown in Figure 12) obtained by the simulation was output and substituted into the entropy generation of Equations (15) and (18). The relationship between the entropy generation and the airflow position can be obtained. It can be seen from Figure 14 that the results are consistent with the variation of entropy generation with tooth tip width (as shown in Figure 13). In conclusion, in the labyrinth tip, the entropy generation was the largest; with the increase of tooth tip width, the entropy generation of the FRIV gradually decreased.

Influence of Labyrinth Inclination on Entropy Generation
The labyrinth inclination angle α from 50° to 120° is divided into 8 groups for comparative research. Figure 15 shows the effect of different Labyrinth inclination angles on the vortex strength, Figure 16 shows the effect of different labyrinth inclination angles on the flow field.
It can be seen from Figure 16 that when the labyrinth teeth were inclined forward, the FRIV on the top of the teeth increased significantly compared to the vertical position In conclusion, in the labyrinth tip, the entropy generation was the largest; with the increase of tooth tip width, the entropy generation of the FRIV gradually decreased.

Influence of Labyrinth Inclination on Entropy Generation
The labyrinth inclination angle α from 50° to 120° is divided into 8 groups for comparative research. Figure 15 shows the effect of different Labyrinth inclination angles on the vortex strength, Figure 16 shows the effect of different labyrinth inclination angles on the flow field.
It can be seen from Figure 16 that when the labyrinth teeth were inclined forward, the FRIV on the top of the teeth increased significantly compared to the vertical position In conclusion, in the labyrinth tip, the entropy generation was the largest; with the increase of tooth tip width, the entropy generation of the FRIV gradually decreased.

Influence of Labyrinth Inclination on Entropy Generation
The labyrinth inclination angle α from 50 • to 120 • is divided into 8 groups for comparative research. Figure 15 shows the effect of different Labyrinth inclination angles on the vortex strength, Figure 16 shows the effect of different labyrinth inclination angles on the flow field. of the labyrinth teeth. The flow-resistance-increasing vortices almost occupied the clearance, which is beneficial to the sealing of the labyrinth. When the labyrinth teeth were inclined backward, the FRIV on the top of the labyrinth gradually weakened, being transferred to the low radius of the labyrinth back, forming two vortexes in the labyrinth cavity.
In terms of the cloud image data shown in Figures 15 and 16, the trend diagram of the vortex intensity with the labyrinth inclination angle was obtained, as shown in Figure 17.  It can be seen from Figure 16 that when the labyrinth teeth were inclined forward, the FRIV on the top of the teeth increased significantly compared to the vertical position of the labyrinth teeth. The flow-resistance-increasing vortices almost occupied the clearance, which is beneficial to the sealing of the labyrinth. When the labyrinth teeth were inclined backward, the FRIV on the top of the labyrinth gradually weakened, being transferred to the low radius of the labyrinth back, forming two vortexes in the labyrinth cavity. In terms of the cloud image data shown in Figures 15 and 16, the trend diagram of the vortex intensity with the labyrinth inclination angle was obtained, as shown in Figure 17. Figure 17 illustrates the labyrinth vortex and the thickness of vortex as a function of labyrinth inclination. It can be seen from Figure 17 that with the labyrinth inclination decreasing, the vortex increased, and the thickness of the vortex increased, indicating that the FRIV of the labyrinth was gradually strengthened. The FRIV occupied the radial clearance of the labyrinth and hindered the flow. When the labyrinth inclination angle was less than 60 • , the growth trend of the vortex slowed down, and the thickness of the vortex remained stable. This result is consistent with the conclusion from the qualitative analysis of Equation (17). When the inclination angle became smaller, the velocity and direction of the flow field changed more greatly, which caused the velocity gradient in the flow field to change significantly and led to the change of the absolute value of the vorticity. As a result, the FRIV effect was enhanced accordingly.

Influence of Step Inclination Angle Change on Entropy Generation
Taking the labyrinth inclination angle of 60 degrees, the labyrinth teeth with step inclination angles of 60 • , 70 • , 80 • , and 90 • were selected for comparative research. Figures 18 and 19 show the flow field diagram and the entropy generation diagram in the labyrinth with different step inclination angles, respectively.   Figure 17 illustrates the labyrinth vortex and the thickness of vortex as a function of labyrinth inclination. It can be seen from Figure 17 that with the labyrinth inclination decreasing, the vortex increased, and the thickness of the vortex increased, indicating that the FRIV of the labyrinth was gradually strengthened. The FRIV occupied the radial clearance of the labyrinth and hindered the flow. When the labyrinth inclination angle was less than 60°, the growth trend of the vortex slowed down, and the thickness of the vortex remained stable. This result is consistent with the conclusion from the qualitative analysis   Figure 17 illustrates the labyrinth vortex and the thickness of vortex as a function of labyrinth inclination. It can be seen from Figure 17 that with the labyrinth inclination decreasing, the vortex increased, and the thickness of the vortex increased, indicating that the FRIV of the labyrinth was gradually strengthened. The FRIV occupied the radial clearance of the labyrinth and hindered the flow. When the labyrinth inclination angle was less than 60°, the growth trend of the vortex slowed down, and the thickness of the vortex remained stable. This result is consistent with the conclusion from the qualitative analysis  It can be seen from Figure 18 that the incoming flow through the labyrinth tip directly hit the step wall and then changed the flow direction, which flowed to the labyrinth cavity along the inclined angle of the step. Then, the air flow was divided into two parts, and two opposite vortices were formed at the inner step of the labyrinth and in the tooth cavity. As the inclination angle of the step decreased, the vortex at the inner step of the labyrinth was squeezed toward the labyrinth tip and increased. This result is consistent with the conclusion drawn from the second item in Equation (17). Due to the decrease of the inclination angle, the volume change rate of the airflow in the tooth tip became larger, which caused the absolute value of the vorticity to increase. Further, the flow field data in Figure 18 were substituted into the entropy generation Equations (15) and (18), wherein the relationship between the entropy generation and the airflow position as shown in Figure 20 can be obtained. It can be seen from Figure 20 that this result was consistent with the result (Figure 19) of the simulation. In other words, with the increase of the inclination angle of the step, the entropy generation of the FRIV gradually decreased.  Figure 19. Distribution of the entropy generation with different step inclinations. Figure 19. Distribution of the entropy generation with different step inclinations.

Influence of FRIV Strength on Leakage
In this paper, a stronger FRIV was constructed on the basis of the influence of geometric parameter changes on entropy production [51,52], which is defined as the best; the geometric parameters are shown in Table 5. The best entropy generation was studied and compared with the initial FRIV. Figure 21 shows the comparison flow field between the best and the initial. Figure 22 shows the comparison distribution of FRIV entropy generation between the best and the initial.

Influence of FRIV Strength on Leakage
In this paper, a stronger FRIV was constructed on the basis of the influence of geometric parameter changes on entropy production [51,52], which is defined as the best; the geometric parameters are shown in Table 5. The best entropy generation was studied and compared with the initial FRIV. Figure 21 shows the comparison flow field between the best and the initial. Figure 22 shows the comparison distribution of FRIV entropy generation between the best and the initial.

Geometrical Parameters Symbol Initial Value Unit
Step It can be seen from Figure 22 that the best entropy production was larger. From the calculation results in Table 6, the best flow rate through the top of the labyrinth teeth was reduced by 22%, which verifies the fact that the sealing effect of the labyrinth structure became stronger. The calculation results of entropy generation and leakage flow rate illustrate the fact that the method proposed in this study is effective in reducing the leakage of labyrinth teeth.  It can be seen from Figure 22 that the best entropy production was larger. From the calculation results in Table 6, the best flow rate through the top of the labyrinth teeth was reduced by 22%, which verifies the fact that the sealing effect of the labyrinth structure became stronger. The calculation results of entropy generation and leakage flow rate illustrate the fact that the method proposed in this study is effective in reducing the leakage of labyrinth teeth.  It can be seen from Figure 22 that the best entropy production was larger. From the calculation results in Table 6, the best flow rate through the top of the labyrinth teeth was reduced by 22%, which verifies the fact that the sealing effect of the labyrinth structure became stronger. The calculation results of entropy generation and leakage flow rate illustrate the fact that the method proposed in this study is effective in reducing the leakage of labyrinth teeth.

Conclusions
In this paper, the modeling and simulation of the aero-engine labyrinth structure was carried out, and the model was checked according to the known experimental data. The concept of FRIV was proposed, and the influence of related geometric parameters on the entropy generation of FRIV was studied.
(1) The relative dissipative intensity distribution was the same as the vorticity distribution, and there was a region with a large correlation coefficient at the top of the tooth, where the vortex was stronger and the dissipation effect was stronger. (2) When the labyrinth width was small, the flow direction changed rapidly after the gas in the upper labyrinth cavity flowed into the labyrinth tip, which formed a FRIV at the labyrinth tip. With the increase of tooth width, the direction of fluid velocity was gradually reversed. In addition, the entropy generation and the resistance increase effect were weakened. When the labyrinth teeth were inclined forward, the FRIV on the top of the tooth increased significantly, and the entropy generation increased, almost occupying the entire labyrinth clearance. When the labyrinth teeth were inclined backward, the FRIV on the top of the tooth gradually weakened. As the inclination angle of the step decreased, the vortex at the inner step of the labyrinth squeezed toward the tooth tip and increased. In addition, the FRIV was enhanced, the entropy generation increased, and the sealing effect was strengthened. (3) By selecting reasonable labyrinth geometric parameters, the labyrinth tip velocity gradient and throttling effect can be enhanced, and the (ω·∇)V and ω(∇·V) terms in the vorticity transport equation can be increased, which can strengthen FRIV and reduce leakage. The results of this research show that the reasonable labyrinth parameters can reduce leakage by 22%. Therefore, the method of reducing leakage proposed in this paper is effective.  Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data used to support the findings of this study are included within the article.