## Abstract

A numerical method using the multiple frequencies elliptical whirling orbit model and transient Reynolds-averaged Navier–Stokes (RANS) solution was proposed for prediction of the frequency dependent rotordynamic coefficients of annular gas seals. The excitation signal was the multiple frequencies waveform that acts as the whirling motion of the rotor center. The transient RANS solution combined with mesh deformation method was utilized to solve the leakage flow field in the annular gas seal and obtain the transient response forces on the rotor surface. Frequency dependent rotordynamic coefficients were determined by transforming the dynamic monitoring data of response forces and rotor motions to the frequency domain using the fast fourier transform. The frequency dependent rotordynamic coefficients of three types of annular gas seals, including a labyrinth seal, a fully partitioned pocket damper seal and a hole-pattern seal, were computed using the presented numerical method at thirteen or fourteen frequencies of 20–300 Hz. The obtained rotordynamic coefficients of three types of annular gas seals were all well agreement with the experimental data. The accuracy and availability of the proposed numerical method was demonstrated. The static pressure distributions and leakage flow rate of three types of annular gas seals were also illustrated.

## Introduction

Annular gas seals are widely used in turbomachinery for the control of the leakage flow through rotor-stator clearances from high pressure regions to low pressure regions. This is a necessity to enhance the aerodynamic performance of modern turbomachinery, where the leakage flows between stationary and rotating components are one of the main sources for losses [1]. Although the main function of annular gas seals is to control leakage, seal rotordynamic characteristics has an important influence on the stability response of modern turbomachinery. In response to the constantly increasing demands for aggressive power output, efficiency, and operational life, modern turbomachinery are being designed to run at higher temperature, higher pressure and higher rotational speed. This results in rotordynamic instability of turbomachinery. In order to enhance the stability of the rotor system and ensure operating safety, researchers have focused on solving vibration problems in high performance turbomachinery through the use of bearing dampers and seals [2]. In fact, bearings are often located near the nodes of the whirling mode shapes of the shaft where response amplitude is low and the effective damping cannot be developed. Conversely, seals are usually located near the antinodes where damping is the most effective [2]. Therefore, annular gas seals with large amounts of direct damping and small cross-coupled stiffness coefficients would be the antivibration device of choice for high performance turbomachinery, especially for back-to-back compressor in which the center seal is located at an axial position where the destabilizing force is more effective to excite unstable rotor whirl.

Labyrinth seals, the most common type of annular gas seals, are widely used in high speed turbomachinery where transient rubbing contact is likely. The labyrinth seal is composed of a series of circular blades and annular grooves which present a tortuous path for flow of the process fluid, and the major advantages of the labyrinth seal are its simplicity, reliability, and tolerance to large thermal and pressure variations [3]. The problem with labyrinth seals is that they have certain undesirable rotordynamic characteristics related to instability, which can mainly be attributed to destabilizing follower force modeled with the cross coupling stiffness ($Kxy=-Kyx$) that arises from fluid rotation in the annular plenums of the seal [2]. For high pressure and high rotational speed applications where stability is of a concern, higher performance damper seals often replace conventional labyrinth seals to eliminate synchronous and subsynchronous vibration problems. Presently, two main types of gas damper seals used in industry are hole-pattern or honeycomb seals, developed by Childs et al. [4,5] and pocket damper seals (PDS), patented by Vance and Shultz [6]. These seals possess “textured” (macro roughness) or “pocket” stator surfaces to reduce the impact of undesirable cross-coupled dynamic forces and improve seal damping.

Due to the many influence factors on the rotordynamic characteristics of annular gas seals and their decisive influence on the rotordynamic stability of actual turbomachinery, a large number of experimental and theoretical investigations focused on measuring and predicting the rotordynamic coefficients of annular gas seals have been conducted. At present, mechanical impedance method, dynamic cavity pressure method, and static force deflection method are three main different experimental measurements to determine the rotordynamic coefficients of annual gas seals [2]. The first two methods are used to identify all rotordynamic coefficients of annular gas seals, and the third method usually are performed to determine the direct and cross coupling stiffness, and to verify the results obtained from the other two methods. The mechanical impedance method requires the measurement of the system dynamic forces, stator relative displacement motion, and stator acceleration [2]. The dynamic pressure method requires the measurement of the cavity dynamic pressures in combined with the stator relative displacement motion [2]. The static force deflection method requires the measurements of static force applied to the stator and the relative static displacement of the stator [2]. Ertas et al. [7] investigated the rotordynamic coefficients of the labyrinth seal (LABY), honeycomb seal and fully partitioned pocket damper seal (FPDS) with eight blades and 1:1 clearance ratio, using three different test methods mentioned above.

The bulk-flow models were developed starting with Iwatsubo [8] and have become a standard technique to predict rotordynamic coefficients of annular gas seals. Childs and Scharrer [9] presented the one-control-volume bulk-flow model to predict the cavity pressure, circumferential velocity distribution and rotordynamic coefficients of the labyrinth seal. Scharrer [10] proposed the two-control-volume bulk-flow model for a tooth-on-rotor labyrinth seal. Kleynhans and Childs [11] first developed the two-control-volume bulk-flow model to predict frequency dependent rotordynamic coefficients of honeycomb seals at the rotor centered position. Childs et al. [12] presented the two-control-volume bulk-flow model for Hole-pattern seals (HPS) and developed a new design (various depth pattern) to improve effective damping and reducing crossover frequency of the HPS. Vance and Shultz [6] first developed the one direction bulk-flow model without viscous effects to predict the direct stiffness and damping coefficients of the PDS. Li et al. [13] presented the one-control-volume, turbulent bulk-flow model for the prediction of the direct and cross coupling rotordynamic force coefficients of the PDS. The primary advantage of these bulk-flow models is that they can predict the rotordynamic coefficients with efficient computational time. However, these models lack flow details and rely on the empirical correction, such as flow coefficients and friction factors that may change for varying applications and geometry, to have the analysis predictions match the experimental results.

Recently the progress of computer technologies makes it possible to utilize a computational fluid dynamic (CFD) for predicting the rotordynamic coefficients of annular gas seals. Compared to the experimental measurements and bulk-flow method, the CFD method can offer a great deal of insight for understanding the fluid dynamics of the annular gas seal flows and developing new seal design. Currently, the CFD analysis for the rotordynamic characteristics of the labyrinth seal have been carried out by many authors including Rhode et al. [14], Moore [15] and Hirano et al. [16]. Rao and Saravana [17] used a commercial 3D CFD software to determine the stiffness and damping coefficients of plain annular seals used in high speed cryogenic pumps. Benefiting from the axisymmetric shape and the frequency independent rotordynamic coefficients of the labyrinth seal and plain annular seal, these authors employed a coordinate transformation to transform unsteady problem into a steady one by solving three dimensional, eccentric flow fields in the frame of reference attached to the whirling rotor. Although this steady state approach is suitable for predicting the rotordynamic coefficients of the labyrinth seal and plain annular seal, it cannot be applied to damper seals such as honeycomb seals, hole-pattern seals and pocket damper seals. The reason is that these damper seals possess “textured” or “pocket” stator surfaces and do not have the advantage of an axisymmetric shape. What is more, the rotordynamic coefficients of damper seals are highly frequency dependent. Chochua and Soulas [18] and Yan et al. [19], respectively, presented a transient CFD method, one-dimensional whirling orbit model and circular whirling orbit model, for computations of frequency dependent rotordynamic coefficients of a HPS. These two CFD methods both are based on the transient calculation and mesh deformation theory. It is important to note that the steady state CFD methods for labyrinth seals and transient CFD methods for the HPS only computed four rotordynamic coefficients with the fact that annular gas seals have the same direct force coefficients and opposite sign equal magnitude cross coupling force coefficients in the orthogonal direction. In addition, the steady state CFD methods for labyrinth seals also assume that all rotordynamic coefficients are frequency independent. However, the recent tests [7] have shown that the direct stiffness coefficient of the labyrinth seal is frequency dependent and decreases with the increasing frequency. In some cases, the PDS shows the same-sign unequal-magnitude cross coupling stiffness and damping, such as the diverging configurations of a six-bladed FPDS at 0 rpm and no preswirl [20]. Therefore, above CFD methods are not suitable for computations of the full set (eight) of frequency dependent rotordynamic coefficients of annular gas seals. Furthermore, the transient CFD methods [18,19] for the HPS contain only one frequency in the rotor periodic motion, so they require a separate transient computation for each frequency, which results in a heavy computational effort.

The main objective of the present work is to develop a three-dimensional CFD numerical method through which the full set of the frequency dependent rotordynamic coefficients of annular gas seals could be accurately predicted. This method is based on the multiple frequencies elliptical whirling orbit model and transient Reynolds-averaged Navier–Stokes (RANS) solution. By using the commercial software ANSYS CFX, the transient RANS solution combined with mesh deformation method was utilized to solve the leakage flow field in the annular gas seal and obtained the transient response forces on the rotor surface. Frequency dependent rotordynamic coefficients were determined by transforming the dynamic monitoring data of response forces and rotor motions to the frequency domain using the fast fourier transform (FFT). To evaluate and demonstrate the accuracy of the present numerical method, the computations of rotordynamic coefficients of the published experimental three types of annular gas seals, including a LABY [7], a FPDS [7], and a HPS [5], were conducted at thirteen or fourteen frequencies of 20–300 Hz, and a comparison between experimental data and numerical results was presented. Then, the static pressure distributions and leakage flow rate of three types of annular seals were also discussed.

## Numerical Method

### Computational Model and Mesh.

The seal geometrical parameters and operational conditions utilized as the computational model in this work are based on the experimental results published by Childs and Wade [5] and Ertas et al. [7]. Figure 1 shows the experimental seal geometries of three types of annular gas seals utilized for the CFD analysis in this study. The computational geometrical parameters and operational conditions of the experimental tests [5,7] are listed in Table 1.

Fig. 1
Fig. 1
Close modal
Table 1

Geometry and operational conditions of annual gas seals [5,7]

ParameterLABYFPDSHPS
Inlet pressure (bar)6.96.970
Outlet pressure (bar)1131.5
Inlet temperature (°C)141417.4
Rotational speed (rpm)15,00015,00020,200
Preswirl velocity (m · s−1)000
Seal length (mm)6510286
Seal inner diameter (mm)170.6170.6114.74
Seal L/D0.380.60.75
Clearance ratio1:011:011:01
Cavity length/Hole diameter (mm)513.97/6.353.175
Cavity/Hole depth (mm)4.013.1753.302
Number of pockets/holes82668
ParameterLABYFPDSHPS
Inlet pressure (bar)6.96.970
Outlet pressure (bar)1131.5
Inlet temperature (°C)141417.4
Rotational speed (rpm)15,00015,00020,200
Preswirl velocity (m · s−1)000
Seal length (mm)6510286
Seal inner diameter (mm)170.6170.6114.74
Seal L/D0.380.60.75
Clearance ratio1:011:011:01
Cavity length/Hole diameter (mm)513.97/6.353.175
Cavity/Hole depth (mm)4.013.1753.302
Number of pockets/holes82668

Figure 2 shows the computational models of three types of annular gas seals. To obtain the full set of the frequency dependent rotordynamic coefficients of annular gas seals using the multiple frequencies elliptical whirling orbit model, the transient analysis combined with mesh deformation method are necessary. Therefore, in this work, unlike Moore's [15] steady state computational model of a full 360 deg seal geometry with a given rotor eccentricity, the calculation of the unsteady flow field inside the whirling annular gas seal requires a full 360 deg computational model with a concentric rotor as shown in Fig. 2. Figure 3 shows the computational meshes of three types of annular gas seals. The commercial software ANSYS ICEM CFD was used to generate the full 360 deg seal model and multiblock structured mesh for calculation. In order to investigate the effect of mesh density and to determine how fine a mesh density is necessary for the computational domain to accurately predict the rotordynamic coefficients, two types of meshes (coarse and fine mesh) were generated for both of the LABY and FPDS. For the LABY (Fig. 3(a)), the size of the coarse mesh and fine mesh are $4.53 × 106$ nodes with $4.29 × 106$ elements and $7.79×106$ nodes with $7.38×106$ elements, respectively. For the FPDS (Fig. 3(b)), the size of the coarse mesh and fine mesh are $4.10×106$ nodes with $3.89×106$ elements and $8.86×106$ nodes with $8.32×106$ elements, respectively. Based on the grid density analysis published by Chochua and Soulas [18] and Yan et al. [19], $5.19×106$ elements are enough to accurately predict the rotordynamic coefficients of the HPS used in this paper. As a result, the final size of the HPS mesh (Fig. 3(c)) adopted in the present numerical study is $5.98×106$ with $5.32×106$ elements. O-type grids were generated in the regions of the annular cavities in the LABY, pocket cavities in the FPDS and holes in the HPS. Compared to the coarse mesh, the fine mesh was placed more nodes in the radial and circumferential direction.

Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

### Rotor Motion.

To obtain the entire set of the frequency dependent rotordynamic coefficients of three types of annular gas seals, the first step in the present numerical analysis is to develop the equations of whirling motion for the rotor surface. Figure 4 shows the axial view of the rotor and stator for the FPDS. In the ideal status, the rotor center $C$ is concentric with the seal stator center $O$, and the rotor spins about the seal stator center $O$ with the rotational speed $ω$ as shown in Fig. 4. The spin velocity $ω$ is performed to the motion model of the rotor in the steady solution to obtain the initial flow field for the transient analysis. In the real industrial applications, a perfectly concentric rotor never occurs, and the rotor center $C$ commonly whirls around the center $O$ of the seal with the whirling speed $Ω$. If the rotor vibrates with a single frequency, the most common whirling motions of the rotor are circular orbits, elliptical orbits and one-dimensional orbits [2]. Figure 5 shows the elliptic orbit whirling model for the rotor vibration with a single frequency. Here, it is assumed that the rotor is whirling around the center of the seal stator $O$ in a periodic elliptic orbit, and that spinning around the center of the rotor $C$, therefore the speed of the rotor surface is the sum of spinning speed $ω$ and whirling speed $Ω$. The major axis of the elliptic orbit reflects the direction of the excitation. Although the vibration model that has a single frequency is always used in the rotordynamic analysis of a seal system [18,19], the multiple frequencies vibration model is more suitable for the real rotor precession phenomenon. What is more, compared to the single vibration model, the multiple frequencies vibration model can significantly reduce the computational time, especially for the calculation of frequency dependent rotordynamic coefficients in a large frequency range. Therefore, to determine the full set of the frequency dependent rotordynamic coefficients of annular gas seals over a wide frequency range, the multiple frequencies periodic elliptical whirling orbit is adopted as the whirling motion model of the rotor in the present numerical analysis. The rotor whirling motion Eqs. (1) and (2) are defined as harmonic functions with specific vibration amplitudes $a$ and $b$, and multiple angular frequencies $Ωi=2πfi$. The vibration amplitudes for each frequency component are constant values and depend on the sealing clearance, $a=0.01·S$ and $b=0.005·S$. The vibration frequencies $fi$ were set in the test frequency range of 20–300 Hz with the fundamental frequency $f1=20 Hz$ and the number of frequencies is $N=13$ for the LABY and FPDS, $N=14$ for the HPS. These values were obtained based on the small motion theory [21] (If the motion of the rotor about a centered position is small compared to the sealing clearance, and the seal force can be modeled by the linear reaction force/motion model) and the experiment results of Refs. [5] and [7]. Figure 6 shows the multiple frequencies elliptical whirling orbit of the rotor with a peak vibration amplitude $3.7×10-5$ m which is near 12% of the seal clearance and captures the linear, small motion characteristics.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal
Fig. 6
Fig. 6
Close modal
Equations of motion for $x$-direction excitation:
$X=a·∑i=1Ncos(Ωit), Y=b·∑i=1Nsin(Ωit)$
(1)
Equations of motion for $y$-direction excitation:
$X=b·∑i=1Ncos(Ωit), Y=a·∑i=1Nsin(Ωit)$
(2)

### Numerical Method.

The present numerical simulation was conducted to solve the compressible RANS equations using the commercial software ANSYS-CFX [22]. As previously mentioned, to obtain the entire set of the frequency dependent rotordynamic coefficients of three types of annular gas seals, the numerical methods combined with mesh deformation and transient analysis are necessary. Before performing the transient solutions, the corresponding steady solutions are necessary for the initial flow fields of the transient solutions. Table 2 lists the detailed numerical approaches for CFD analysis in the current study. This transient calculation assumed the fluid was an ideal air and the entire flow was turbulent. The standard $k-ɛ$ turbulence model was used to model the turbulence characteristics of the flow. The scalable logarithmic wall function was applied to describe the near wall flow conditions. The high resolution scheme and the second order backward Euler scheme were applied for the spatial discretization and the transient term discretization, respectively. In addition, the static and rotating walls were all defined to be adiabatic and smooth. The numerical simulations were conducted at thirteen (fourteen for the HPS) different frequencies with the physical time step 0.0001 s and boundary conditions listed in Table 1. As shown in Fig. 2, the inlet and outlet boundary were placed at the upstream of the inlet extension and the downstream of the outlet extension, respectively. Total pressure, total temperature and turbulence quantities (turbulence intensity = 5%) were defined at the inlet boundary, while the averaged static pressure was specified at the outlet of the seal. The swirl at the inlet is zero and same as the experimental conditions. For the rotor surface; in addition to its rotational speed, the periodic elliptical whirling motion for $x$ or $y$ direction excitation described by Eqs. (1) and (2) was also added to it in the transient solution.

Table 2

Numerical method

Solution typeTransient
FluidAir (ideal gas)
Computational methodTime marching method
Mesh motionMesh deformation
Discretization schemeHigh resolution
Turbulence model$k-ɛ$, scalable log wall function
Frequency/Hz20, 40,…, 260, 280
Eccentric ratio (for each frequency component)0.01/0.005
Timestep/s0.0001
Solution typeTransient
FluidAir (ideal gas)
Computational methodTime marching method
Mesh motionMesh deformation
Discretization schemeHigh resolution
Turbulence model$k-ɛ$, scalable log wall function
Frequency/Hz20, 40,…, 260, 280
Eccentric ratio (for each frequency component)0.01/0.005
Timestep/s0.0001

As to the steady solutions, the desired convergent target of each numerical simulation is that the root mean square (RMS) residuals [22] of the momentum and mass equations, energy equations, and turbulence equations reach a value of 10−6 or even lower, the overall imbalance of the mass is less than 0.1%, and the response forces $(Fx,Fy)$ on the rotor surface are lower than 0.1 N. For the transient solutions, the desired convergent target of each numerical simulation is that the RMS residuals of the momentum and mass equations, energy equations, and turbulence equations reach (or even lower than) 10−5, at the same time the response forces $(Fx,Fy)$ on the rotor surface approach periodic vibration and the difference of the response forces between two adjacent vibration periods needs to be less than 0.4%.

### Rotordynamic Coefficients Solution Technique.

This section provides a frequency domain reduction method for calculating the rotordynamic coefficients of an annular gas seal given the time varying rotor motion and response force. For a small motion of the rotor about a centered position within an annular gas seal, neglecting the added mass terms, Childs [21] suggested the following reaction-force/seal-motion model defined in Eq. (3), which relates the response forces due to the rotor motion and velocity through direct force coefficients $(Kxx,Kyy,Cxx,Cyy)$ and cross coupling force coefficients $(Kxy,Kyx,Cxy,Cyx)$.
$-[FxFy]=[KxxKxyKyxKyy]·[XY]+[CxxCxyCyxCyy]·[X·Y·]$
(3)
For the multiple frequencies elliptical whirling orbit model in the present paper, the response forces $(Fx,Fy)$, rotor motions $(X,Y)$, and rotor velocity $(X·,Y·)$ in Eq. (3) are multifrequency signals that contain the same frequency components, but different initial phases. Therefore, to obtain the rotordynamic coefficients for each frequency component, it is necessary to transform Eq. (3) into the frequency domain expression given by Eqs. (4)(5) using the FFT.
$-Fx=(Kxx+jΩCxx)·Dx+(Kxy+jΩCxy)·Dy$
(4)
$-Fy=(Kyy+jΩCyy)·Dy+(Kyx+jΩCyx)·Dx$
(5)

where $j=-1$, $Dx$ and $Dy$ are the frequency domain components of the relative displacement between the seal's rotor and stator in $x$ and $y$ direction. The multifrequency signals of response forces $(Fx,Fy)$ and rotor motions $(Dx,Dy)$ can both be obtained through the transient solutions in this paper. Therefore, Eqs. (4)(5) provides two equations with four unknowns $Kxx+jΩCxx$, $Kxy+jΩCxy$, $Kyy+jΩCyy$, $Kyx+jΩCyx$. To provide enough equations to solve four unknowns in Eqs. (4)(5), two independent transient solutions are conducted in orthogonal directions ($x$ and $y$ excitation shown in Fig. 5), yielding four independent equations and four unknown quantities defined in the following Eqs. (6)(9)

Equations of force/motion for $x$-direction excitation:
$-Fxx=(Kxx+jΩCxx)·Dxx+(Kxy+jΩCxy)·Dxy$
(6)
$-Fxy=(Kyy+jΩCyy)·Dxy+(Kyx+jΩCyx)·Dxx$
(7)
Equations of force/motion for $y$-direction excitation:
$-Fyy=(Kyy+jΩCyy)·Dyy+(Kyx+jΩCyx)·Dyx$
(8)
$-Fyx=(Kxx+jΩCxx)·Dyx+(Kxy+jΩCxy)·Dyy$
(9)

Equations (6)(7) give force/motion relations for $x$-direction excitation, where Eq. (6) represents the direct equation of force/motion and Eq. (7) represents the cross coupling equation of force/motion. Eqs. (8)(9) show the equations of force/motion for $y$-direction excitation. For the response force $Fij$ and rotor motion $Dij$, the first subscript indicates the excitation direction as shown in Fig. 5 and the second one indicates the direction of the response force. For force coefficients $(Kij,Cij)$, the first subscript indicates the direction of the response force, and the second one indicates the direction of the rotor motion.

Note that each force/motion equation (Eqs. (6)(9)) has two unknowns: direct coefficients and cross coupling coefficients which possess a real part (determines the dynamic stiffness coefficients) and an imaginary part (gives rise to the dynamic damping coefficients). Eqs. (10)(13) define the direct and cross-coupled force impedances in the frequency domain.

Force impedances in $x$ direction:
$Hxx=Kxx+j(ΩCxx)$
(10)
$Hxy=Kxy+j(ΩCxy)$
(11)
Force impedances in $y$ direction:
$Hyy=Kyy+j(ΩCyy)$
(12)
$Hyx=Kyx+j(ΩCyx)$
(13)
Substituting Eqs. (10)(13) into Eqs. (6)(9), the final equation of force/motion in matrix form is obtained by Eq. (14)
$-[FxxFyxFxyFyy]=[HxxHxyHyxHyy]·[DxxDyxDxyDyy]$
(14)

where $Fij$ and $Dij$ are the FFT of the time dependent monitored response forces and rotor displacements. $Fij$, $Dij$, and $Hij$ are all complex variables. In Eq. (14), the data obtained from the CFD analysis directly provides all of the variables except for four force impedances $Hij$. Frequency dependent force impedances in Eqs. (15)(18) are determined from Eq. (14). Once the force impedances are obtained, rotordynamic coefficients are obtained by separating Eqs. (10)(13) into real and imaginary parts and solving for $Kij$ and $Cij$, as defined in Eqs. (19)(22).

Direct and cross-coupled force impedance for $x$ direction:
$Hxx=(-Fxx)·Dyy-(-Fyx)·DxyDxx·Dyy-Dyx·Dxy$
(15)
$Hxy=(-Fxx)·Dyx-(-Fyx)·DxxDxy·Dyx-Dyy·Dxx$
(16)
Direct and cross-coupled force impedance for $y$ direction:
$Hyy=(-Fyy)·Dxx-(-Fxy)·DyxDyy·Dxx-Dyx·Dxy$
(17)
$Hyx=(-Fyy)·Dxy-(-Fxy)·DyyDxy·Dyx-Dyy·Dxx$
(18)
Direct stiffness and damping coefficients for $x$ direction:
${Kxx=Re(Hxx)Cxx=Im(Hxx)/Ω$
(19)
Cross-coupled stiffness and damping coefficients for $x$ direction:
${Kxy=Re(Hxy)Cxy=Im(Hxy)/Ω$
(20)
Direct stiffness and damping coefficients for $y$ direction:
${Kyy=Re(Hyy)Cyy=Im(Hyy)/Ω$
(21)
Cross-coupled stiffness and damping coefficients for $y$ direction:
${Kyx=Re(Hyx)Cyx=Im(Hyx)/Ω$
(22)

### Dynamic Monitoring Data.

As mentioned above, determination of the frequency dependent rotordynamic coefficients requires the dynamic monitoring data of the time varying response force and rotor motion to calculate the force impedances from Eqs. (15)(18). Figures 7 and 8 show examples of dynamic monitoring data for $x$ excitation. Figure 7 shows the time varying rotor motion of the HPS. Figure 8 shows the time varying response force of three types of annular gas seals. Both the $x$ and $y$ direction time varying rotor motion and response force are shown along with their representative Fourier transforms. As shown in Fig. 7, the peak spectral vibration (rotor motion) amplitudes in the $x$ and $y$ direction for each frequency component are approximate to the target values 2.0 $μm$ ($0.01·S$) and 1.0 $μm$ ($0.005·S$), where the little errors come from the calculation of the mesh deformation and the FFT. Figure 8 shows that three types of annular gas seals have very different behavior for the dependence of the response force versus vibration frequency. For the LABY and FPDS, the amplitude of the direct response force ($Fx$) is maximized at the highest frequency 260 Hz and drops in value towards the lower frequency. However, the amplitude of the cross coupling response force ($Fy$) of the FPDS is minimized at the frequency 100 Hz and increases towards the lower and higher frequency, and that of the LABY is less frequency dependent. For the HPS, the amplitude of the direct and cross coupling response forces both are less frequency dependent. This suggests that the rotordynamic characteristics of three types of annular gas seals are different.

Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal

## Results and Discussions

### Rotordynamic Coefficients Results.

In order to compare the rotordynamic coefficients obtained from the present numerical method with the experimental data [5,7], the graphs presented here are displayed in form of the averaged direct damping and stiffness coefficients($Kavg$, $Cavg$) defined in Eqs. (23)(24), and the effective stiffness and damping coefficients ($Keff$, $Ceff$) defined in Eqs. (25)(26). The full set of frequency dependent rotordynamic coefficients of three types of annular gas seals are presented in tabular form in the Appendix of this paper.
$Kavg=(Kxx+Kyy)/2$
(23)
$Cavg=(Cxx+Cyy)/2$
(24)
$Keff=Kxx+Cxy·Ω$
(25)
$Ceff=Cxx-Kxy/Ω$
(26)

Figure 9 illustrates predictions and experimental data of the averaged direct stiffness and damping, the cross coupling stiffness and the effective damping versus vibration frequency for the LABY. The rotordynamic coefficients are generally in good agreement with the experimental data. The direct stiffness and damping coefficients are modestly under predicted. The cross coupling stiffness coefficients are over predicted by the present numerical method. The effective damping coefficients are well predicted at higher frequencies, but over predicted at lower frequencies (below 60 Hz). Here, the prediction error is partly attributed to the geometric differences between test seal (Fig. 1(a)) and numerical model (Fig. 2(a)): the test seal stator possess a long smooth surface with no seal tooth at the upstream of inlet and the downstream of outlet. It is important to note the interesting conclusion that both numerical results and experimental data show that the direct damping and cross coupling stiffness coefficients are independent of frequency, while the direct stiffness coefficient decreases with the increasing vibration frequency, following a trend indicative of a parabolic decay. There is not a clear difference between the solutions of the coarse mesh and fine mesh. Therefore, mesh independent tests with the coarse mesh ($4.53×106$ nodes with $4.29×106$ elements) for predictions of the rotordynamic coefficients of the LABY was demonstrated. Table 5 gives all the rotordynamic coefficients in the $x$ and $y$ direction for the LABY. The direct stiffness and damping coefficients in the $x$ and $y$ direction are the same ($Kxx≈Kyy$, $Cxx≈Cyy$). The cross coupling stiffness and damping coefficients are equal in magnitude with opposite sign ($Kxy≈-Kyx$, $Cxy≈-Cyx$).

Fig. 9
Fig. 9
Close modal

Figure 10 shows predictions and experimental data for the averaged direct stiffness and damping, the cross coupling stiffness and the effective damping versus vibration frequency for the FPDS. The rotordynamic coefficients of the FPDS are well predicted by the present numerical method. The direct stiffness coefficients are modestly over predicted, and the direct damping and effective damping coefficients are modestly under predicted. The cross coupling stiffness coefficients are very well predicted. Figure 10(d) shows that the “crossover” frequency $Ωco$, at which $Ceff$ changes sign (Below this frequency, the effective damping is negative; above it the seal has positive effective damping), are over predicted about 10 Hz. Refining the mesh slightly changes the values of the rotordynamic coefficients, especially for the cross coupling stiffness at higher frequencies. The cross coupling stiffness varies less than 5% with increasing mesh density. Therefore, the coarse mesh ($4.10×106$ nodes with $3.89×106$ elements) is effectively mesh independent for the computations of the rotordynamic coefficients of the FPDS. Table 6 shows that for the FPDS, the direct stiffness and damping coefficients also are the same in the $x$ and $y$ direction ($Kxx≈Kyy$, $Cxx≈Cyy$), and the cross coupling stiffness and damping coefficients are equal in magnitude with opposite sign ($Kxy≈-Kyx$, $Cxy≈-Cyx$).

Fig. 10
Fig. 10
Close modal

Figure 11 compares the rotordynamic coefficients of the HPS obtained from the present numerical method, Chochua's one-direction whirling model [18], Yan's circular whirling model [19], and Child's experiment and one dimensional constant temperature bulk-flow model (ISOT) [5]. Compared to Chochua's CFD results and the ISOT code predictions, the present CFD results are more agreeable with the experiment data, with the direct damping and cross coupling stiffness coefficients modestly under predicted at higher frequencies of 100–280 Hz. As shown in Fig. 11, the present numerical results and Yan's results [19] are almost the same. This suggests that the present multiple frequencies elliptical whirling model has the same prediction precision with the Yan's single frequency circular whirling model. It is important to note that compared to Chochua's and Yan's numerical methods based on the single frequency whirling model, the present numerical method has a different rotor motion based on the multiple frequencies whirling model. Unlike the single frequency whirling models, which require a separate transient solution for each frequency, performing the transient solution using the present multiple frequencies whirling model yields results for multiple frequencies; therefore requiring only one transient solution for the fourteen frequencies of 20–280 Hz. Although not presented in Fig. 11, Table 7 shows $Kxx≈Kyy$, $Cxx≈Cyy$, and $Kxy≈-Kyx$, $Cxy≈-Cyx$.

Fig. 11
Fig. 11
Close modal

### Seal Leakage.

Table 3 compared the steady and transient solutions of the seal leakage rates obtained by the coarse and the fine mesh computational analyses for the LABY and FPDS. The leakage rate varies less than 0.1% with increasing mesh density as shown in Table 3, therefore, the coarse meshes for the LABY and the FPDS both are effectively mesh independent for the seal leakage. Table 3 also shows that the effect of rotor vibration on the leakage rate ($γ=(mtr-mst)/mst$) is very small and can be ignored for the LABY and the FPDS. Table 4 shows the leakage rate of HPS computed using different methods, including ISOT bulk-flow code [5], Chochuua's [18], and Yan's [19] CFD methods, and the present CFD method, and their comparison to the experimental data from Childs [5]. Compared to the other numerical results, the present numerical results are more agreeable with the experimental data. The negative effect of rotor vibration $γ=-4.64%$ in Table 4 suggests that the rotor vibration significantly decreases the leakage rate and changes the leakage characteristics of the HPS.

Table 3

Leakage rate of the LABY and FPDS

SealMesh type$m·$ (kg/s) Steady solution$m·$ (kg/s) Transient solution$γ$ (%) Effect of vibration
LABYCoarse mesh0.102860.102930.07
Fine mesh0.102820.102870.05
FPDSCoarse mesh0.102130.102220.09
Fine mesh0.102140.102250.11
SealMesh type$m·$ (kg/s) Steady solution$m·$ (kg/s) Transient solution$γ$ (%) Effect of vibration
LABYCoarse mesh0.102860.102930.07
Fine mesh0.102820.102870.05
FPDSCoarse mesh0.102130.102220.09
Fine mesh0.102140.102250.11
Table 4

Leakage rate of the HPS

$m·$ (kg/s) Steady solution$m·$ (kg/s) Transient solution$γ$ (%) Effect of vibration
$m·$ (kg/s) Test date [5]$m·$ (kg/s) ISOT [5]$m·$ (kg/s) Chochuua [18]Yan [19]PresentYan [19]PresentYan [19]Present
0.4090.4370.4380.4650.4310.4290.429−7.74−4.64
$m·$ (kg/s) Steady solution$m·$ (kg/s) Transient solution$γ$ (%) Effect of vibration
$m·$ (kg/s) Test date [5]$m·$ (kg/s) ISOT [5]$m·$ (kg/s) Chochuua [18]Yan [19]PresentYan [19]PresentYan [19]Present
0.4090.4370.4380.4650.4310.4290.429−7.74−4.64

### Static Pressure Distributions.

Figure 12 shows the static pressure contours on the rotor surface and the cross section of three types of annular gas seals at the moment when the rotor center $C$ moves to the point P (shown in Fig. 6) for $x$-direction excitation. As shown in Fig. 6, at the point P, the rotor displacement in $x$-direction reaches the maximum value and that in $y$-direction is zero ($X=Dmax$, $Y=0$), and the rotor is moving in the positive $y$-direction ($X·=0,Y·>0$). For the LABY, the higher static pressure on the upper and left surface relative to the lower and right surface indicates a response force $Fx>0,Fy<0$, which results in a negative stiffness coefficient in $x$-direction and a positive damping coefficient in $y$-direction, as illustrated in Fig. 9. For the FPDS and HPS, the higher static pressure on the upper and right surface relative to the lower and left surface indicates a response force $Fx<0,Fy<0$, which results in a positive stiffness in $x$-direction and a positive damping coefficient in $y$-direction, as illustrated in Figs. 10 and 11. Figure 12 also shows that the circumferential pressure difference on the rotor surface of the FPDS and HPS are quite larger than that in the LABY, which suggests that the damper seal (FPDS and HPS) have the larger damping capacity and can produce much more effective rotordynamic damping. Comparing the pressure distributions in Fig. 12 shows that the damping mechanisms of damper seals can be visualized as follows. Due to the “textured” or “pocket” stator surface which makes the seal cavities are unconnected and forms isolate regions around the circumference, the larger pressure variations across the seal diameter can exist in the damper seal when the rotor is off-center (the volumes of isolate regions are changed), and the dynamically varying pressure forces always oppose the rotor vibration velocity. Therefore, the compressibility of the working fluid is more important to the damper seal. The “textured” or “pocket” stator surface also blocks the destabilizing circumferential swirl of the working fluid and reduces the destabilizing cross coupling stiffness.

Fig. 12
Fig. 12
Close modal

## Conclusions

A numerical method based on the multiple frequencies elliptical whirling orbit model and transient RANS solutions were proposed to predict the frequency dependent rotordynamic coefficients of annular gas seals in this work. The transient solution and mesh deformation technology were utilized in the presented numerical method. The entire set of frequency dependent rotordynamic coefficients of three types of experiment annular gas seals, including a LABY [7], a FPDS [7], and a HPS [5], were computed using the development numerical method. Comparison of the experimental data and numerical results demonstrates the accuracy and reliability of the present numerical method for predicting the frequency dependent rotordynamic coefficients of annuals gas seals in turbomachinery.

The LABY has frequency independent of direct damping, cross coupling stiffness and damping coefficients. However, the direct stiffness coefficient of the LABY is frequency dependent and decreases with the increasing frequency following a trend indicative of a parabolic decay. The rotordynamic coefficients are strong frequency dependency for the FPDS and HPS. The rotor vibration significantly decreases the leakage rate of the HPS, but has little or no effect on the leakage rate of the LABY and FPDS. In addition, the present numerical method can readily be used to investigate the unsteady fluid dynamics of the annual gas seal flows and discover new damping mechanisms to develop new damper seal with excellent rotordynamic properties.

## Acknowledgment

The authors are grateful for the Grants from the National Natural Science Foundation (51106122), Specialized Research Fund for the Doctoral Program of Higher Education (20100201110013) and the Fundamental Research Funds for the Central Universities of China.

### Nomenclature

Nomenclature
$a$ =

major axis of an ellipse (m)

$b$ =

minor axis of an ellipse (m)

$Cxx$ =

direct damping in $x$ direction (N-s/m2)

$Cxy$ =

cross coupling damping in $x$ direction (N-s/m2)

$Cyy$ =

direct damping in $y$ direction (N-s/m2)

$Cyx$ =

cross coupling damping in $y$ direction (N-s/m2)

$Cavg$ =

average direct damping (N-s/m2)

$Ceff$ =

effective damping (N-s/m2)

$Dxx$ =

rotor motion in $x$ direction for $x$ direction excitation (m)

$Dxy$ =

rotor motion in $y$ direction for $x$ direction excitation (m)

$Dyy$ =

rotor motion in $y$ direction for $y$ direction excitation (m)

$Dyx$ =

rotor motion in $x$ direction for $y$ direction excitation (m)

$Fxx$ =

rotor response force in $x$ direction for $x$ direction excitation (N)

$Fxy$ =

rotor response force in $y$ direction for $x$ direction excitation (N)

$Fyy$ =

rotor response force in $y$ direction for $y$ direction excitation (N)

$Fyx$ =

rotor response force in $x$ direction for $y$ direction excitation (N)

$Hxx$ =

direct force impedance in $x$ direction (N/m)

$Hxy$ =

cross coupling force impedance in $x$ direction (N/m)

$Hyy$ =

direct force impedance in $y$ direction (N/m)

$Hyx$ =

cross coupling force impedance in $y$ direction (N/m)

$Kxx$ =

direct stiffness in $x$ direction (N/m)

$Kxy$ =

cross coupling stiffness in $x$ direction (N/m)

$Kyy$ =

direct stiffness in $y$ direction (N/m)

$Kyx$ =

cross coupling stiffness in $y$ direction (N/m)

$Kavg$ =

averaged direct stiffness (N-s/m2)

$Keff$ =

effective direct stiffness (N-s/m2)

$L$ =

seal cavity length (m)

$n$ =

rotational speed (rpm)

$R$ =

$S$ =

sealing clearance (m)

$Pk$ =

dynamic cavity pressure (Pa)

$γc$ =

hole area ratio

$f$ =

rotor vibration frequency (Hz)

$ω$ =

angular frequency of rotor spin (rad/s)

$Ω$ =

angular frequency of rotor whirl (rad/s)

$π$ =

circumference ratio

$θk$ =

angular location of cavity (deg)

$θ1$ =

start angle of cavity (deg)

$θ2$ =

end angle of cavity (deg)

### Appendix

Tables 5, 6 and 7 contain the entire set (eight) of rotordynamic coefficients in the x and y direction for the LABY, the FPDS and the HPS, respectively.

Table 5

Rotordynamic coefficients of the LABY

Stiffness (kN/m)Damping (N s/m)
Frequency (Hz)KxxKxyKyyKyxCxxCxyCyyCyx
20−86.59−170.18−82.59170.24384.69130.71334.73−136.76
40−95.51−160.13−88.26168.44314.81120.62296.18−125.51
60−115.05−175.31−125.37169.70264.16137.73283.32−133.14
80−111.00−168.43−118.09172.24282.36115.40251.14−128.72
100−162.55−167.08−152.04168.60288.44148.48304.26−137.01
120−174.60−177.08−190.30173.31266.06137.39264.25−142.86
140−214.56−160.57−203.41168.99305.79145.81294.27−145.83
160−276.05−172.23−281.49168.28288.16160.83299.54−157.40
180−302.42−165.15−305.90167.12303.69153.50290.70−159.75
200−394.21−153.85−382.16156.71325.03175.79331.35−170.19
220−458.05−163.01−470.93155.47317.67180.09317.28−182.15
240−533.79−132.40−521.15140.06357.01186.71353.18−188.37
260−656.89−124.58−660.26123.55367.57207.31373.90−203.02
Stiffness (kN/m)Damping (N s/m)
Frequency (Hz)KxxKxyKyyKyxCxxCxyCyyCyx
20−86.59−170.18−82.59170.24384.69130.71334.73−136.76
40−95.51−160.13−88.26168.44314.81120.62296.18−125.51
60−115.05−175.31−125.37169.70264.16137.73283.32−133.14
80−111.00−168.43−118.09172.24282.36115.40251.14−128.72
100−162.55−167.08−152.04168.60288.44148.48304.26−137.01
120−174.60−177.08−190.30173.31266.06137.39264.25−142.86
140−214.56−160.57−203.41168.99305.79145.81294.27−145.83
160−276.05−172.23−281.49168.28288.16160.83299.54−157.40
180−302.42−165.15−305.90167.12303.69153.50290.70−159.75
200−394.21−153.85−382.16156.71325.03175.79331.35−170.19
220−458.05−163.01−470.93155.47317.67180.09317.28−182.15
240−533.79−132.40−521.15140.06357.01186.71353.18−188.37
260−656.89−124.58−660.26123.55367.57207.31373.90−203.02
Table 6

Rotordynamic coefficients of the FPDS

Stiffness (MN/m)Damping (kN s/m)
Frequency (Hz)KxxKxyKyyKyxCxxCxyCyyCyx
20−0.101.22−0.13−1.254.51−1.243.590.92
400.071.230.10−1.213.61−1.063.660.85
600.221.190.16−1.143.42−0.883.620.83
800.301.110.37−1.073.35−0.773.220.80
1000.611.000.60−0.993.19−0.713.350.76
1200.760.900.71−0.923.22−0.703.120.72
1400.990.821.09−0.852.96−0.682.990.66
1601.350.781.27−0.782.93−0.632.970.60
1801.510.731.55−0.712.81−0.562.730.54
2001.890.651.91−0.642.64−0.492.700.50
2202.170.572.09−0.582.60−0.452.570.46
2402.410.512.49−0.532.41−0.422.400.42
2602.810.462.76−0.482.31−0.402.350.38
Stiffness (MN/m)Damping (kN s/m)
Frequency (Hz)KxxKxyKyyKyxCxxCxyCyyCyx
20−0.101.22−0.13−1.254.51−1.243.590.92
400.071.230.10−1.213.61−1.063.660.85
600.221.190.16−1.143.42−0.883.620.83
800.301.110.37−1.073.35−0.773.220.80
1000.611.000.60−0.993.19−0.713.350.76
1200.760.900.71−0.923.22−0.703.120.72
1400.990.821.09−0.852.96−0.682.990.66
1601.350.781.27−0.782.93−0.632.970.60
1801.510.731.55−0.712.81−0.562.730.54
2001.890.651.91−0.642.64−0.492.700.50
2202.170.572.09−0.582.60−0.452.570.46
2402.410.512.49−0.532.41−0.422.400.42
2602.810.462.76−0.482.31−0.402.350.38
Table 7

Rotordynamic coefficients of the HPS

Stiffness (MN/m)Damping (kN s/m)
Frequency (Hz)KxxKxyKyyKyxCxxCxyCyyCyx
203.5916.942.96−16.9838.76−21.5937.0722.27
405.8015.854.95−15.6933.92−19.4834.0618.85
607.2914.317.46−14.7431.89−17.9131.2918.68
8011.4712.5910.73−12.2928.08−16.3728.3716.38
10013.5210.7913.85−11.1324.16−14.4524.5814.29
12017.349.0716.95−8.8921.87−12.6721.4612.97
14019.627.4719.71−7.5117.72−10.8918.2810.57
16021.996.0422.11−6.1116.13−9.3115.689.53
18024.434.8524.23−4.7513.05−7.9513.257.76
20025.353.9125.72−4.1911.42−6.8311.276.85
22027.453.1827.27−2.949.65−5.869.665.82
24027.952.6028.26−2.658.07−5.058.184.92
26029.362.1029.29−1.937.24−4.337.054.37
28030.031.8330.05−1.715.85−3.755.973.61
Stiffness (MN/m)Damping (kN s/m)
Frequency (Hz)KxxKxyKyyKyxCxxCxyCyyCyx
203.5916.942.96−16.9838.76−21.5937.0722.27
405.8015.854.95−15.6933.92−19.4834.0618.85
607.2914.317.46−14.7431.89−17.9131.2918.68
8011.4712.5910.73−12.2928.08−16.3728.3716.38
10013.5210.7913.85−11.1324.16−14.4524.5814.29
12017.349.0716.95−8.8921.87−12.6721.4612.97
14019.627.4719.71−7.5117.72−10.8918.2810.57
16021.996.0422.11−6.1116.13−9.3115.689.53
18024.434.8524.23−4.7513.05−7.9513.257.76
20025.353.9125.72−4.1911.42−6.8311.276.85
22027.453.1827.27−2.949.65−5.869.665.82
24027.952.6028.26−2.658.07−5.058.184.92
26029.362.1029.29−1.937.24−4.337.054.37
28030.031.8330.05−1.715.85−3.755.973.61

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