Improving the rotordynamic stability of short labyrinth seals using positive preswirl

3.1. Geometrical parameters

Fig. 3 shows the cross section of the test labyrinth seal. Table 1 gives its detailed dimensions. The test seal includes 10 blades and 9 cavities (#1~#9). In order to study the effect of the blade number on the dynamic characteristics, the labyrinth seal with 5 and 15 blades are also employed in the numerical simulation.

Fig. 3Two-dimensional geometry of the test labyrinth seal

3.2. Numerical model

A commercial CFD software [28], which solves the compressible Reynolds-averaged Navier-Stokes equations, is employed in this paper. The standard $k$-$\epsilon$ is used as the turbulence model with the turbulence intensity of 5 %. The scalable wall function method combines the physical quantity of the wall and the high-level turbulent zone. And the value of $y^{+}$ is controlled within 30-200. In order to investigate the effect of mesh densities for an accurate prediction of the rotordynamic coefficients, coarse and fine mesh grids were tested for the labyrinth seal. The fine mesh placed more nodes in radial and circumferential direction. For the labyrinth seal with 10 blades, the size of the coarse mesh and fine mesh are 2.4×10⁶ nodes and 5.6×10⁶ nodes, respectively. The difference of rotordynamic coefficients predicted by the coarse and fine mesh is under 1.68 %. Therefore, for the studied labyrinth seal in this paper, the mesh with nodes 2.4×10⁶ is accurate enough for the investigation. The grid distribution for the simulation(coarse) is shown in Fig. 4.

Fig. 4Grid distribution

3.3. Boundary conditions

Table 2 lists the detailed parameters for CFD calculation. The fluid (air) is assumed to be ideal gas. The inlet and outlet boundary were placed at the upstream of the inlet extension and the downstream of the outlet extension, respectively. The walls of rotor and stator are defined to be adiabatic, smooth and no slip. The total pressure and temperature are defined at the inlet boundary, while the average static pressure is specified at the outlet. The inlet preswirl ratio is defined as Eq. (6):

where, $V_{\theta }$ is the circumferential velocity at the seal inlet.

The desired convergent target of each numerical simulation is that the RMS residuals of the momentum and mass equations, energy equations, and turbulence equations are below 10^-6.

3.4. Validation

In order to validate the reliability of the CFD simulation, this paper measured the static seal force acting on the rotor with various eccentric ratios, and compared the numerical results with experiments. Fig. 5 shows the experiment facility.

Fig. 5Gas facility, experiment part and testing system

Fig. 6 shows detailed experiment device of the rotor-seal system. Two hydrodynamic bearings are used to support the rotor. The rotor diameter of the test section is 60 mm. The stator is supported by eight springs in vertical and horizontal directions. The labyrinth seal contains 10 blades, and the detailed dimension is shown in Fig. 3 and Table 1. Four inlet ports are arranged symmetrically around the middle plane of the cylindrical stator surface. High-pressure gas enters the seal from four centered inlet ports and discharges into the atmosphere. The inlet pressure is ~6.9 bar, and the rotational speed is zero. In the experiment, the zero calibration for all the test instruments is guaranteed to avoid the errors caused by their deviation.

Fig. 6Experimental device of the rotor-seal system: 1. Rotor, 2. #1 Journal bearing, 3. Turbine oil, 4. Stator, 5. Elastic support, 6. Inlet pipe, 7. #2 Journal bearing, 8. Displacement sensor, 9. Dial indicator, 10. Pressure sensors, 11. Balance disk, 12. Tachometric transducer, 13. Inlet port, 14. Blades, 15. Elastic metal insert

To improve the identification accuracy, the seal housing is supported by eight elastic springs in both horizontal and vertical directions. The dynamic response of the stator can be amplified by the supporting springs. By multiplying the stator displacement with the spring stiffness, the seal force acting on the rotor can be obtained. The stator displacement is measured by averaging 10 tests under the same working condition. Table 3 gives the comparison of the static experimental force and the results obtained by the CFD method. Because of the machining accuracy of the rotor and seal, a weak seal force can be found even the eccentric ratio $\epsilon$ is zero, which is small enough to be neglected. The difference between experiment and CFD simulation is less than 2.9 %, which verifies the accuracy of the current CFD simulation.

4.1. Static characteristics

The cross-coupled stiffness coefficient $k$ is one of the main criteria to evaluate the system stability. For a steady flow, the $k$ is the function of the tangential seal reaction force, as shown in Eq. (4). Fig. 7 shows the tangential seal reaction force $F_t$ vs. preswirl ratios for the labyrinth seal with different blade numbers. With the increasing preswirl ratio, the tangential seal reaction force of the labyrinth seal with 5 blades exhibits an opposite trend compared with that of the labyrinth seal with 15 blades. For the labyrinth seal with 5 blades, the tangential seal reaction force decreases with the increasing preswirl ratio, and corresponding cross-coupled stiffness coefficient $k$ becomes to be negative. From the stability viewpoint, the increasing preswirl ratio tends to enhance the stability. This result shows a contrary trend to the conventional conclusion. The tangential seal reaction force of the labyrinth seal with 15 blades increases with the rising preswirl ratio. The cross-coupled stiffness coefficient changes in signs from negative to positive, which processes the decreasing stability. One notable phenomenon for the labyrinth seal with 10 blades is that it shows little dependency on preswirl ratios and keeps an almost constant value. Above results show the tangential seal reaction force is greatly correlated with the blade number. The seal reaction force from each seal cavity is extracted to clarify the phenomenon in following section.

Figs. 8-10 show the tangential seal reaction forces inside each cavity for the labyrinth seal with 5, 10, 15 blades, respectively. For #1 cavity, the increasing preswirl decreases the tangential seal reaction force. However, the effect of preswirl on the seal reaction force inside #2~#14 cavity shows an opposite trend, and the increasing preswirl ratio tends to produce a larger seal reaction force.

Fig. 7Tangential seal reaction forces Ft vs. preswirl ratios

Fig. 8Tangential seal reaction forces Ft vs. seal cavities (5 blades)

Fig. 9Tangential seal reaction forces Ft vs. seal cavities (10 blades)

Fig. 10Tangential seal reaction forces Ft vs. seal cavities (15 blades)

For the labyrinth seal with 5, 10, 15 blades, the preswirl mainly influences the fluid flow in #1 cavity, and it has a relatively minor effect on the flow in other seal cavities. Hence, when the number of seal cavities is relatively small (such as 5 blades in this paper), the changing trend of the tangential seal reaction force acting on the whole rotor largely depends on the force variation inside #1 cavity.

When the cavity number becomes relatively large (such as 15 blades in this paper), the tangential force inside #2~#14 cavities dominates the whole seal. The labyrinth seal with 10 blades shows little dependency on the preswirl ratio. This can be attributed to the counteracting effect of the flow forces between #1 cavity and other cavities.

Fig. 11 shows the distribution of high-pressure spots for the labyrinth seal with various blade numbers ($\lambda =$0). $\theta$ is the circumferential angle of high pressure spots in spatial polar coordinate system. All high-pressure spots inside #1 cavity locate at the first quadrant (divergent gap), while $\theta$ in other cavities becomes more than 150°. Along the leakage direction, the high-pressure spot for each cavity continuously moves toward the rotational direction.

Fig. 11High pressure spots vs. seal cavities without inlet preswirl

Fig. 12Effects of high pressure spots on the tangential seal reaction force

Fig. 12 depicts the direction and magnitude of the tangential seal reaction force due to the changing position of high-pressure spots affects. As the high-pressure spot locates in the first and second quadrants, the tangential seal reaction force shows a negative sign, which is consistent with the results for the labyrinth seal with 5 and 10 blades shown in Fig. 7. The shifting of high pressure spots inside cavities (except for the #1 cavity) from the second quadrant to the third quadrant results in a sign change of the tangential seal reaction force from negative to positive, which agrees with the variation of tangential seal reaction forces in Fig. 10. The main reason for the variation of the seal reaction force is that the circumferential location of the high-pressure spot for each seal cavity varies along the leakage direction.

Taking the labyrinth seal with 15 blades as an example, the variation of high-pressure spots for #1, #3, #7, #13 cavity is shown in Fig. 13. The pressure high spot shifts toward the inlet preswirl direction. Additionally, the inlet preswirl has a more dramatic effect on the seal cavity close to the inlet. Combining with Fig. 11, it’s obvious that the high-pressure spots of #1 cavity locate at the first and fourth quadrants. While the high-pressure spots of #2~14 cavities locate at the third and fourth quadrants.

Fig. 13Position of high-pressure spots vs. preswirl ratios

Fig. 14 presents the effect of various high-pressure spots on the tangential seal reaction force. Fig. 14(a) depicts the transition of the high-pressure spot in different quadrants. Fig. 14(b) shows the projection of the seal reaction force in the tangential direction ($t$ axis). According to the distribution of high-pressure spots, the cavities, for which the high pressure spot locates at the first and fourth quadrant, are classified as Type A Cavity (e.g. # 1 cavity). And the other cavities, for which the high-pressure spot locates at the second and third quadrant, are classified as Type B Cavity (e.g. #2-#14 cavities). When the preswirl ratio is positive ($\lambda >$ 0), the tangential seal reaction force in the Type A cavities shifts to be negative, while that in the Type B cavities converts to be positive. When the inlet preswirl ratio is negative ($\lambda <$ 0), the tangential seal reaction force inside both Type A and B cavities show the opposite result. This phenomenon is consistent with the variation trend of tangential seal reaction forces shown in Fig. 10.

Fig. 14Effects of high pressure spot shifts caused by preswirls on the tangential seal reaction force

a) Hight pressure spot position

b) Projection of seal reaction force in $t$ axis

4.2. Dynamic characteristics

The effective damping coefficient $C_{eff}$ is a main parameter to evaluate the system stability [32, 33], which is defined as follows:

where $F_t$, $e$, $C$, $k$, denote the tangential seal reaction force, radius of whirling orbit, direct damping coefficient, cross-coupled stiffness coefficient, respectively. The direct damping coefficient $C$ reveals the capacity for reducing the rotor whirling motion. The cross-coupled stiffness coefficient $k$ shows the destabilization effect. As a result, the effective damping coefficient $C_{eff}$ could be employed to depict the resultant effect.

Fig. 15 presents the tangential seal reaction force inside seal cavities (15 blades) with whirling frequencies 0 Hz, 50 Hz, 100 Hz and 150 Hz. With the increasing whirling frequency, the seal reaction force in the tangential direction for each cavity becomes toward negative gradually. This is mainly because the increasing direct damping effect can suppress the whirling motion.

Fig. 15Distribution of the tangential seal reaction force inside seal cavities with different whirling frequencies (15 blades)

Fig. 16 shows the distribution of tangential seal reaction forces inside the cavities with a whirling frequency 50 Hz. It’s obvious that the responses of the seal reaction force inside the cavities working in dynamic conditions are similar with that working in the static condition.

Fig. 17 shows the effective damping coefficient $C_{eff}$ versus inlet preswirl ratios with a whirling frequency 50 Hz. The labyrinth seal with different blade numbers differs greatly in the response to preswirl ratios. The increasing preswirl ratios enhances the stability of the labyrinth seal with 5 blades, while the labyrinth seal with 15 blades shows the opposite trend. The labyrinth seal with 10 blades shows the lowest preswirl dependency.

Fig. 16The tangential seal reaction force in seal cavities under different preswirl ratios (50 Hz)

Fig. 17Effective damping coefficients for different blades numbers vs. preswirl ratios (50 Hz)

Under static and dynamic conditions, the stability of the labyrinth seal with 5 blades increases with the rising preswirl ratio, which is contrary to the traditional viewpoint. The main reason is that the circumferential position of the high pressure spot inside the #1 cavity is different from the others, which results in the different effect of the preswirl on the tangential force. When the blade number is small (e.g. 5 blades), the effect of preswirl on #1 cavity plays a dominant role, and the high pressure spot inside #1 cavity appears in the divergence gap. Meanwhile, the positive preswirl is conducive to improving the overall stability of the seal system. When the blade number is large, the effect of preswirl on #1 cavity is reduced by the other cavities, and the high pressure spot mainly appears in the convergence gap. The overall tangential seal reaction force is dominated by the downstream cavities, and the negative preswirl is beneficial to improve the system stability.