## Abstract

Two types of liquid hole-pattern seals with axially oblique (A-HPS) or circumferentially oblique (C-HPS) hole cavities are designed. To evaluate the leakage and rotordynamic characteristics of the liquid hole-pattern seals, a 3D transient perturbation method is employed, which based on the multifrequency one-dimensional rotor whirling model and the mesh deformation technique. The accuracy and reliability of the proposed numerical approach is demonstrated based on the published experimental data of the leakage and rotordynamic force coefficients for a hole-pattern seal (HPS). Seal leakage and force coefficients are presented and compared for the A-HPS (axially oblique angle $\alpha =\u221230degto30deg$), C-HPS (circumferentially oblique angle $\beta =\u221230degto30deg$), and HPS (α = 0, β = 0) at various rotational speeds (n = 0.05, 2.0, 4.0, and 6.0 krpm). Results reveal that the tilted hole cavity with positive α or β can reduce the seal effective clearance and strengthen the kinetic dissipation in hole cavities, yielding less leakage by 5–10%, especially at higher rotational speeds. The tilted hole cavity with a positive oblique angle $(\alpha =30deg,\beta =30deg)$ results in a moderate growth (by ∼6% for the A-HPS, by ∼15% for C-HPS) in the effective stiffness. Furthermore, the tilted hole cavity shows a very weak influence (<4.0%) on the effective damping, particularly for higher rotational speeds and vibration frequencies. Considering the decreasing leakage and nonworse rotordynamic characteristics, a tilted hole cavity with suitable positive oblique angles $(10deg\u201330deg)$ is beneficial for the liquid hole-pattern seal.

## Introduction

Annular seals are broadly employed in turbomachinery to enhance machine operation efficiency and stability. The annular seal can restrict the leakage through rotor–stator clearances and generate fluid-excited forces, either improving or degrading the dynamic stability of rotor-bearing systems. The seal fluid-excited forces are significant and non-negligible, particularly for the liquid centrifugal pumps which often work against the large pressure drop, high rotational speed, and dense process fluid [1]. Therefore, it is a vital task for the liquid turbomachinery to develop advanced liquid annular seals, minimize the leakage, and stabilize the rotor-bearing system.

The typical liquid annular seals include the labyrinth seal [2], helical groove seal [3], and smooth plain seal. The labyrinth seal and the smooth plain seal are the most common seal concepts because of their simple structures and low manufacturing cost. However, the sealing capacity of the labyrinth seal and the smooth plain seal is poor to meet the high-pressure difference of the modern liquid turbomachinery. The helical groove seal has an outstanding sealing capacity, particularly at high rotational speeds; however, the poor rotordynamic characteristics may induce rotor instability which is seriously inappropriate for the liquid turbomachinery [3–5]. The hole-pattern seal (HPS)—as an advanced damper seal concept—has been effectually employed in the gas turbomachinery, such as the multistage centrifugal compressor, and has been proved effective to minimize the leakage and stabilize the rotor-bearing system [6–8]. Therefore, for the liquid turbomachinery such as the centrifugal pump, hole-pattern seals are of high interest as potential replacements of the conventional labyrinth seals, helical groove seals, and smooth plain seals. Until now, few research works have been devoted to the hole-pattern seal with incompressible liquid working fluid, and the detailed leakage and rotordynamic characteristic of the liquid hole-pattern seals are still unknown and require further assessing.

The conventional hole-pattern seal consists of a hole-pattern stator and a smooth rotor. Numerous radial hole cavities are manufactured on the stator surface utilizing the milling cutter or electric spark. Many works in the literature [9–14] have shown that the hole geometry factors (hole diameter and depth) are the key design factors for hole-pattern seals. In 1985, Childs and Kim [9] firstly tested three hole-pattern seals with cross-shaped, prismatic, and cylindrical hole cavities. The experimentally observed data revealed that the hole-pattern seal with cylindrical hole cavities possesses the least leakage and largest damping coefficients. In another scrutiny, Childs et al. [11] experimentally examined the influences of the hole depth on the leakage and rotordynamic performance of hole-pattern seals. Three hole-pattern seals with various hole depths but the same hole diameter (*H* = 1.9, 3.3, 6.6 mm; *D* = 3.175 mm) were tested. The test results indicated that the hole-pattern seal with a shallower hole has the best leakage and rotordynamic characteristics. Migliorini et al. [12,13] investigated the impact of the hole aspect ratio (the ratio of the hole depth to diameter) on the leakage and friction factors of hole-pattern seals. It was revealed that the shape of the hole cavity controls the vortex formation in the hole cavity, thus affecting the jet flow in the clearance and the overall seal leakage and friction factors.

All the above research works were carried out for hole-pattern seals with radial hole cavities, and aimed to improve the seal performance by optimizing the hole shape (such as circular, elliptical hole shape) [15], hole’s diameter and depth [16,17], and inlet flow conditions (such as anti-swirl) [18]. However, the effects of the tilted hole cavities on the seal performance were completely neglected. The tilted hole cavity can also affect the leakage and rotordynamic characteristics by altering the vortex formation in the hole cavities, just like the effect of the slanted tooth for the labyrinth seal [19]. Further investigation on the tilted hole-pattern seal should be carried out.

In the present work, two types of liquid hole-pattern seals possess axially oblique (A-HPS) or circumferentially oblique (C-HPS) hole cavities are designed. The leakage and rotordynamic force coefficients are calculated and analyzed for the tilted hole-pattern seals with various axially oblique angle $(\alpha =\u221230degto30deg)$ and circumferentially oblique angle $(\beta =\u221230degto30deg)$ at different rotational speeds (*n* = 0.05, 2.0, 4.0, and 6.0 krpm). The hole-pattern seal with radial hole cavities also was numerically investigated and compared with the tilted hole-pattern seals. The primary objective of the present work is to evaluate the comprehensive effects of the tilted hole cavities on the performance of the hole-pattern seals, providing the desired theoretical basis and recommendations for the design and production of liquid hole-pattern seals.

## Computational Model and Mesh

Two types of tilted hole-pattern seals assessed in the present work are designed based on the experimental liquid hole-pattern seal explained in Ref. [20]. Figure 1 shows the schematic representation of the experimental hole-pattern seal with radial hole cavities (HPS) and two tilted hole-pattern seals (A-HPS, C-HPS). As listed in Table 1, the seal clearance, rotor diameter, seal axial length, and hole cavities pattern of the designed A-HPS and C-HPS are identical to those of the experimental HPS. The only difference is the axially oblique or circumferentially oblique axis of the hole cavities.

Parameter | Value |
---|---|

Inlet pressure (MPa) | 1.0 |

Inlet temperature (°C) | 50 |

Outlet pressure (MPa) | 0.5 |

Rotational speed (krpm) | 0.05, 2.0, 4.0, 6.0 |

Inlet preswirl velocity (m/s) | 0 |

Seal length (mm) | 150 |

Rotor diameter (mm) | 240 |

Radial clearance (mm) | 0.57 |

Total number of holes | 225 |

Hole diameter (mm) | 16 |

Hole depth (mm) | 8 |

Circumferentially oblique angle | −30 to 30 deg |

Axially oblique angle | −30 to 30 deg |

Parameter | Value |
---|---|

Inlet pressure (MPa) | 1.0 |

Inlet temperature (°C) | 50 |

Outlet pressure (MPa) | 0.5 |

Rotational speed (krpm) | 0.05, 2.0, 4.0, 6.0 |

Inlet preswirl velocity (m/s) | 0 |

Seal length (mm) | 150 |

Rotor diameter (mm) | 240 |

Radial clearance (mm) | 0.57 |

Total number of holes | 225 |

Hole diameter (mm) | 16 |

Hole depth (mm) | 8 |

Circumferentially oblique angle | −30 to 30 deg |

Axially oblique angle | −30 to 30 deg |

In Fig. 2, the axially oblique angle *α* and the circumferentially oblique angle *β* are schematically illustrated. In a more detailed, *α* represents the angle between the radial direction and the axis of the axially oblique holes cavity. Here, the positive value of *α* means that the hole cavity tilts in the leakage flow direction (from seal inlet to outlet). The circumferentially oblique angle *β* represents the angle between the radial direction and the axis of the circumferentially oblique holes cavity. The positive value of *β* indicates that the hole cavity tilts in the rotation direction.

For the sake of assessing the leakage characteristics of hole-pattern seals, the periodic computational models and multiblock structured meshes with O-types grids in hole cavities are generated for the radial and tilted hole-pattern seals (see Fig. 3). All seal meshes consist of 1.2 × 0^{6} − 1.4 × 0^{6} elements with 20–25 nodes applied in the seal clearance, adequate to ensure the grid independence.

In order to predict the rotordynamic characteristics of hole-pattern seals, the full −360 deg concentric computational models and meshes of the radial and tilted hole-pattern seals are produced, as shown in Fig. 4. The computational meshes for the radial and tilted and hole-pattern seals are generated via ansys icem with 4.6 × 0^{6} to 5.3 × 0^{6} elements. All these seal meshes possess fine quality and are sufficient to ensure grid independence for predicting the rotordynamic force coefficients.

## Numerical Approach

### Steady Leakage Solution Method.

A steady numerical method is herein exploited to evaluate the leakage characteristic of the tilted hole-pattern seals. The detailed numerical method setups are shown in Table 2. The renormalization group (RNG) *k*–*ɛ* turbulence model with the scalable wall functions is chosen to capture the turbulence characteristics of the leakage flow.

Solution type | Steady |
---|---|

Fluids | Water (isothermal model, 50 °C) |

Inlet boundary condition | Total pressure, turbulence intensity flow direction |

Outlet boundary condition | Average static pressure |

Computational method | Time step marching method |

Discretization scheme | High resolution |

Turbulence model | RNG k–ɛ scalable wall function |

Wall properties | Adiabatic, smooth surface |

Solution type | Steady |
---|---|

Fluids | Water (isothermal model, 50 °C) |

Inlet boundary condition | Total pressure, turbulence intensity flow direction |

Outlet boundary condition | Average static pressure |

Computational method | Time step marching method |

Discretization scheme | High resolution |

Turbulence model | RNG k–ɛ scalable wall function |

Wall properties | Adiabatic, smooth surface |

### Rotordynamic Force Coefficients Solution Method.

A transient computational fluid dynamics (CFD)-based perturbation approach is utilized to calculate the rotordynamic force coefficients of the tilted hole-pattern seals. The present numerical scheme is proposed based on the mesh deformation technique and a multifrequency one-dimensional rotor whirling model, which define the relative vibration motion between the seal rotor and stator. The detailed numerical schemes for the transient approach and the main parameters of the rotor whirling model are shown in Table 3.

Solution type | Transient, time marching method |
---|---|

Rotor motion | Mesh deformation technique |

Discretization scheme | High resolution |

Turbulence model | RNG k–ɛ scalable wall function |

Wall properties | Adiabatic, smooth surface |

Whirling model | Multifrequency, one-dimensional whirling model |

Frequency (Hz) | f = 10, _{o}N = 12 (10, 20, …, 110, 120) |

Vibration amplitude (mm) | A = 0.01s, s = 0.57 |

Time step (s) | 0.0002 |

Solution type | Transient, time marching method |
---|---|

Rotor motion | Mesh deformation technique |

Discretization scheme | High resolution |

Turbulence model | RNG k–ɛ scalable wall function |

Wall properties | Adiabatic, smooth surface |

Whirling model | Multifrequency, one-dimensional whirling model |

Frequency (Hz) | f = 10, _{o}N = 12 (10, 20, …, 110, 120) |

Vibration amplitude (mm) | A = 0.01s, s = 0.57 |

Time step (s) | 0.0002 |

*C*is assumed to periodically vibrate in the

*Y*direction around the stator center

*O*for each single frequency component

*f*, meanwhile, the rotor is spinning around the rotor center

*C*. The multifrequency periodic displacement of the rotor in the

*Y*direction is given by

*A*denotes the vibration amplitude defined as a ratio of the seal radial clearance

*s*(

*A*= 0.01 s), and Ω

_{i}represents the whirling angular velocity defined as a ratio of the fundamental frequency

*f*

_{0}(Ω

_{i}=

*i*· 2

*πf*

_{0}), and

*N*is the number of vibrational frequencies.

*F*

_{x},

*F*

_{y}), can be expressed by

*X*,

*Y*) represent the rotor relative displacements. (

*K*,

*C*,

*M*) in order are the direct stiffness, damping, and virtual mass coefficients, respectively. (

*k*,

*c*,

*m*) are the cross-coupling stiffness, damping, and virtual mass coefficient, respectively.

*X*,

*Y*) and fluid response forces (

*F*

_{x},

*F*

_{y}) acting on the rotor surface are monitored during the transient CFD solution processes. Subsequently, the detailed parameters (i.e., amplitude and phase angle) of the rotor relative motions and fluid response forces for each frequency component can be evaluated by employing fast Fourier transforms (FFT). In the frequency domain, Eq. (2) can be restated as

*H*

_{ij}are the force impedances defined in terms of the rotordynamic coefficients by the following relations:

In Eq. (3), the time-varying dynamic parameters monitored during the transient CFD solutions directly provide all the variables except *H*_{ij}. Thus, the force impedances, *H*_{ij}, can be computed from the following relations:

*e*, the fluid response forces on the rotor surface (i.e., (

*F*

_{x},

*F*

_{y})) can be stated in the cylindrical-coordinate system as [22]

*F*

_{r}represents the radial force which controls the seal effective stiffness, and

*F*

_{t}denotes the tangential force which determines the seal effective damping.

In Eqs. (4) and (5), the seal rotordynamic force coefficients are assumed to be frequency-independent for the liquid annular seal. This issue was validated by lubrication theory and experiment data [21]. Once the frequency-dependent force impedances are obtained from Eqs. (6) and (7), then the seal fluid response forces (−*F*_{r}/*e*, −*F*_{t}/*e*) are calculated over a range of whirling angle frequencies Ω given in Table 3. A least-squares regression curve fit is applied to (−*F*_{r}/*e*, −*F*_{t}/*e*) versus Ω, based on the quadratic function relationship defined in Eqs. (8) and (9). Figure 6 shows an example of curve fit plots (i.e., quadratic regression fitting) of the (−*F*_{r}/*e*, −*F*_{t}/*e*) as a function of the stator whirling angle frequency Ω. According to Eqs. (8) and (9), the rotordynamic forces coefficients can be obtained from the curvatures, slopes, and intercepts of the curve fit plots of (−*F*_{r}/*e*, −*F*_{t}/*e*) in terms of Ω.

*K*

_{eff}and damping

*C*

_{eff}are presented which defined by Eqs. (10) and (11):

The more detailed solution procedure for evaluating the rotordynamic forces coefficients is provided in Ref. [22].

## Numerical Method Validations

To validate the accuracy of the proposed steady and transient numerical methods, the leakage and rotordynamic force coefficients of an experimental hole-pattern seal are calculated and compared with the published experimental data [20]. The seal geometries and operation conditions are the same as those of the test [20].

Figure 7 illustrates the predicted and measured leakage for the experimental hole-pattern seal [20] at different rotational speeds. According to the plotted results, the tendency of the seal leakage versus rotational speed is correctly predicted, and the predicted magnitude of the seal leakage also shows a good agreement with the experimental data for all the rotational speeds (<3% prediction error).

In general, the numerically predicted results agree well with experimental data for most coefficients of the seal rotordynamic force. The exceptional cases are related to the direct stiffness and direct damping at the highest rotational speed in which both two coefficients are underestimated by about 10%.

The results in Figs. 8 and 9 suggest that the proposed steady and transient numerical approaches have a reasonable accuracy to predict the leakage and rotordynamic force coefficients for the experimental hole-pattern seal with radial hole cavities. Therefore, it is believed that the present numerical approach can be exploited to reasonably predict those for the tilted hole-pattern seals (i.e., the structure generally possesses the geometrical parameters identical to those of the experimental hole-pattern seal, except for the tilted hole cavities).

## Results and Discussions

To assess the performance of the present two types of tilted hole-pattern seals (i.e., A-HPS and C-HPS), the seal leakage and rotordynamic forces coefficients were presented and compared for the tilted hole-pattern seals with various oblique angles $(\alpha =\u221230degto30deg;\beta =\u221230degto30deg)$ at four different rotational speeds (*n* = 0.05, 2.0, 4.0, 6.0 krpm). The hole-pattern seal with radial hole cavities is also introduced to evaluate the influence of the tilted hole cavity.

### Leakage.

Figure 9 exhibits the dependency of the leakage to the axially oblique angle *α* in the case of A-HPS. Compared with the radial hole-pattern seal (*α* = 0 deg), the A-HPS with positive *α* presents a slightly smaller (<7.5%) leakage; however, the A-HPS with negative *α* has a slightly larger (<7.5%) leakage. For lower rotational speeds, the leakage exhibits a stronger dependence on *α*, and sharply decreases by growing of *α*. At the lowest rotational speed (*n* = 2.0 krpm), the axially oblique hole cavity results in a non-negligible influence (>15%) on the leakage characteristics of the A-HPS; nevertheless, at high rotational speed conditions, the influence is fairly negligible (<4%).

Figure 10 demonstrates the plots of the leakage as a function of the circumferentially oblique angle *β* for the C-HPS. Compared to the radial hole-pattern seal (*β* = 0 deg), the C-HPS with positive *β* can result in a moderate decrease (5.3–10%) in the seal leakage. However, the C-HPS with negative *β* leads to a moderate increase (5–10%) in the seal leakage. At higher levels of the rotational speed, the seal leakage shows a stronger dependence on *β*, and harshly lessens with the growth of *β*.

Based on the previous investigation [13], for the gas hole-pattern seal with compressible gas as a working fluid, the sealing capability of the hole-pattern seal mainly relies on the effective clearance and the kinetic dissipation intensity in seal cavities. Therefore, the flow field, effective clearance, axial velocity pattern, and turbulence eddy dissipation contours distributions in hole cavities are of our concern to explain the influences of the tilted hole cavity on the leakage characteristics.

Figure 11 shows the axial velocity contour distributions and flow field on the meridian plane of the last cavity for both A-HPS and HPS at *n* = 2.0 krpm. As shown in Figs. 11(a) and 11(b), when the jet fluid flows from the hole cavity to the downstream clearance, a small vortex is formed in the seal clearance (region A), which results in a negative axial velocity region and noticeably blocks the axial flow in the seal clearance. The smallest effective clearance (∼75% *C _{r}*) indicates the lowest effective flow areas, resulting in the least leakage in the case of A-HPS with $\alpha =30deg$. For the case of A-HPS with $\alpha =\u221230deg$, as presented in Fig. 11(c), no vortex is observed in the region A, and the effective flow areas are the largest, yielding the largest leakage. Moreover, the most complicated flow field in the hole cavity is observed for the case of A-HPS with $\alpha =30deg$. Several vortexes are observed at the bottom of the hole cavity. A vortex in region B results in a negative axial velocity region, markedly blocking the axial flow, which also account for the less leakage for A-HPS with $\alpha =30deg$.

The comparison of the kinetic dissipation in the hole cavity for A-HPSs and HPS is also of interest. To this end, the turbulence eddy dissipation contour distributions (which represents the conversion of the fluid kinetic energy into heat) on the meridian plane of the last cavity for A-HPSs and HPS at *n* = 2.0 krpm are shown in Fig. 12. For all three hole-pattern seals, the turbulence eddy dissipation exhibits a similar distribution. A larger turbulence eddy dissipation is detectable in the seal clearance and downstream region of the hole cavity. By increasing *α* from $\u221230deg$ to $30deg$, the turbulence eddy dissipation moderately grows in the hole cavity.

Figure 13 demonstrates the static pressure as a function of the axial distance in the seal clearance for both C-HPS and HPS at *n* = 6.0 krpm. For both C-HPS and HPS, when the jet fluid in the clearance passes through the hole cavity region, the static pressure first reduces due to the growth of the flow area and then increases at the downstream region due to the kinetic dissipation in the hole cavity. The larger growth in the static pressure indicates the stronger kinetic dissipation. As demonstrated in Fig. 13, compared to the HPS, the increase in the static pressure is apparently larger in the case of C-HPS with $\beta =20deg$, but smaller in the case of C-HPS with $\beta =\u221220deg$. The given static pressure distribution in Fig. 13 shows that the kinetic dissipation in the cavity for C-HPS with positive *β* would be stronger than that of the C-HPS with negative *β*. This leads to a smaller leakage for the C-HPS with positive *β*, as displayed in Fig. 10.

### Rotordynamic Force Coefficients.

Two tilted hole-pattern seals with the largest positive oblique angle (A-HPS with *α* = 30 deg and C-HPS with *β* = 30 deg) are chosen to assess the rotordynamic characteristics, which have been demonstrated better leakage characteristics than radial hole-pattern seal in Figs. 9 and 10.

Figure 14 illustrates the plots of the dimensionless rotordynamic force coefficients versus the rotational speed for the HPS, A-HPS, and C-HPS. As it is seen, the direct stiffness $K*$ decreases with the growth of the rotational speed and crosses over to negative at a critical rotational speed. The A-HPS and C-HPS possess the obviously larger magnitudes of the negative direct stiffness compared with the HPS in the presence of the rotational speed (*n* = 2–6 krpm). The negative direct stiffness leads to a significant reduction of the natural frequency of the long-flexible rotor system; as a result, it would be inappropriate for the multiple-stage high-pressure liquid pump. As demonstrated in Fig. 14(b), the A-HPS and C-HPS possess a slightly larger direct damping $C*$ than the HPS. For the largest rotational speed case with *n* = 6.0 krpm, compared to the HPS, the direct damping $C*$ increases by 6% and 13% in order for the A-HPS and C-HPS. As shown in Fig. 14(c), the present three types of hole-pattern seals almost exhibit similar dimensionless cross-coupled stiffness $k*$ which increases with the growth of the rotational speed. The plotted results in Fig. 14(d) reveal that the smallest and largest cross-coupled damping $c*$ are detected for the HPS and A-HPS. Furthermore, the discrepancies of the results of these two with those of the C-HPS are more apparent for higher rotational speeds (*n* ≥ 2.0 krpm).

Figure 15 displays the plots of the dimensionless effective stiffness (*K*_{eff}) versus the vibration frequency at various rotational speeds for the tilted and radial hole-pattern seals. For all three types of hole-pattern seals, the effective stiffness lessens with the increase of the vibration frequency, commonly following a trend of the quadratic curve. Obviously, the effective stiffness is negative with larger magnitudes at higher vibration frequencies. Additionally, the effective stiffness grows with the rotational speed, and the undesired negative effective stiffness is observed for all considered rotational speeds. Compared to the HPS, the tilted hole-pattern seal with the positive oblique angle has a moderately larger effective stiffness. The effective stiffness values of the C-HPS and A-HPS in order are 15% and 6% greater than that of the HPS.

Figure 16 shows the plots of the dimensionless effective damping (*C*_{eff}) in terms of the vibration frequency at various rotational speeds for the tilted and radial hole-pattern seals. For all three types of hole-pattern seals, the effective damping linearly increases with the vibration frequency and decreases with the rotational speed. All hole-pattern seals possess the desired positive effective damping throughout the whole frequency range for the lower rotational speeds (*n* = 0.05 and 2 krpm); however, the effective damping crossover to the negative value at the crossover frequency (10–30 Hz) at the higher levels of the rotational speed (*n* = 4 and 6 krpm). Generally, the tilted hole cavity has a very weak influence (<4.0%) on the effective damping of the hole-pattern seal.

The present results demonstrate that the tilted hole-pattern seals with a positive *α* or *β* (10–30 deg) can result in a moderate reduction of the seal leakage and the nonworse rotordynamic characteristics (about 5–10%), which is beneficial for the liquid pump. However, some potential drawbacks of the tilted hole-pattern seal also should be acknowledged for industrial applications. For instance, the C-HPS can only be used in the turbomachinery with one rotation direction to reduce the leakage. Additional costs of machining are needed to use the split-seal stator design instead of a single one-piece seal, where the tilted hole cavities on the split-stator of the seal can be manufactured by milling cutter or electric spark.

It is worth mentioning that the present conclusions are obtained for the liquid hole-pattern seal with certain geometrical parameters at certain operation conditions. Therefore, further investigations on the tilted hole-pattern seal with various geometrical parameters, especially for various hole diameters and depths, still should be conducted at higher rotational speeds and pressure drops. In addition, detailed experimental studies also should be carried out to further show the influences of the tilted hole cavity. A test rig for the leakage and rotordynamic characteristics of the liquid annular seals has been developed, and more detailed experimental studies on the liquid hole-pattern seals will be performed in future works.

## Conclusions

To better understand the influences of the tilted hole cavity on the static and rotordynamic characteristics, two types of liquid hole-pattern seals possessing axially oblique (A-HPS) or circumferentially oblique (C-HPS) hole cavities are designed and evaluated in the present work. The leakage flowrates and rotordynamic force coefficients are calculated and compared for the tilted hole-pattern seals with various axially oblique angles (A-HPS, $\alpha =\u221230degto30deg$) as well as circumferentially oblique angles (C-HPS, $\beta =\u221230degto30deg$) at different rotational speeds (*n* = 0.05, 2.0, 4.0, and 6.0 krpm). The major findings are summarized in the following:

The tilted hole cavity can result in a moderate deviation (−15% to 15%) for the seal leakage, and the oblique direction is crucial for the sealing performance. Compared to the radial hole cavity, the tilted hole cavity with the positive oblique angle

*α*or*β*leads to a moderate decrease of about 5–10% in the seal leakage. Furthermore, the tilted hole cavity with the negative*α*or*β*results in a moderate increase of about 5–10% in the seal leakage.The study on the flow field in hole cavities and the turbulence eddy dissipation distribution in the seal clearance reveals that the tilted hole cavities with a positive

*α*or*β*can lessen the effective clearance size and strengthen the kinetic vortex-dissipation in the hole cavities, yielding the lower leakage.The tilted hole cavity with a positive oblique angle $(\alpha =30deg,\beta =30deg)$ results in a moderate increase in the effective stiffness of the hole-pattern seal. The amount of increase for A-HPS and C-HPS would be in order 6% and 15%, respectively. For the meantime, the tilted hole cavity shows a very weak influence, lower than 4.0%, on the effective damping, particularly for higher rotational speeds and vibration frequencies. In order to lessen the leakage loss and enhance the rotor stability, a tilted hole cavity with suitable positive

*α*or*β*(in the range of $10\u201330deg$) would be beneficial to be applied in the primary design phase of the liquid hole-pattern seal.

## Funding Data

The National Natural Science Foundation of China (Grant No. 51776152).

## Nomenclature

*c*=cross-coupling damping coefficient (N s/m)

*f*=rotor vibration frequency (Hz)

*k*=cross-coupling stiffness coefficient (N/m)

*m*=cross-coupling virtual mass coefficient (N s

^{2}/m)*n*=rotor rotational speed (rpm)

*A*=vibration amplitude of rotor whirling orbit (m)

*C*=direct damping coefficient (N s/m)

*D*=hole diameter (mm)

*H*=hole depth (mm)

*K*=direct stiffness coefficient (N/m)

*M*=direct virtual mass coefficient (N s

^{2}/m)*f*_{o}=fundamental frequency (Hz)

*C*_{eff}=effective damping (N s/m)

*C*_{r}=sealing clearance (m)

*F*_{r}=radial response force component (N)

*F*_{t}=tangential response force component (N)

*F*_{x}=response force in the

*x*direction (N)*F*_{y}=response force in the

*y*direction (N)*H*_{xx}=direct force impedance in the

*x*direction (N/m)*H*_{xy}=cross-coupling force impedance in the

*x*direction (N/m)*H*_{yx}=cross-coupling force impedance in the

*y*direction (N/m)*H*_{yy}=direct force impedance in the

*y*direction (N/m)*K*_{eff}=effective stiffness (N/m)

- $c*$ =
dimensionless cross-coupling damping coefficient (−)

- $k*$ =
dimensionless cross-coupling stiffness coefficient (−)

- $C*$ =
dimensionless direct damping coefficient (−)

- $K*$ =
dimensionless direct stiffness coefficient (−)

*α*=axially oblique angle (degree)

*β*=circumferentially oblique angle (degree)

- Ω
_{i}=rotor whirling angular velocity (rad/s)