Conceptual Flutter Analysis of Stepped Labyrinth Seals

2020 ◽  
Vol 142 (7) ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega ◽  
Michele Greco
Keyword(s):  
Natural Frequency ◽  
Fundamental Mode ◽  
Theoretical Background ◽  
Analytical Models ◽  
Second Step ◽  
Pressure Side ◽  
Labyrinth Seals ◽  
Acoustic Frequency ◽  
Flutter Analysis ◽  
The Stability

Abstract A simple nondimensional model to describe the flutter onset of two-fin straight labyrinth seals (Corral and Vega, 2018, “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models—Part I: Theoretical Background,” ASME J. Turbomach., 140(10), p. 121006) is extended to stepped seals. The effect of the axial displacement of the seal is analyzed first in isolation. It is shown that this fundamental mode is always stable. In a second step, the combination of axial and torsion displacements is used to determine the damping of modes with arbitrary torsion centers. It is concluded that the classical Abbot's criterion stating that seals supported on the low-pressure side of the seal are stable provided that natural frequency of the mode is greater than the acoustic frequency breaks down under certain conditions. An analytical expression for the nondimensional work-per-cycle is derived and new nondimensional parameters controlling the seal stability identified. It is finally concluded that the stability of stepped seals can be assimilated to that of a straight through seal if the appropriate distance of the torsion center to the seal is chosen.

Conceptual Flutter Analysis of Stepped Labyrinth Seals

2019 ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega ◽  
Michele Greco
Keyword(s):  
Natural Frequency ◽  
Fundamental Mode ◽  
Second Step ◽  
Pressure Side ◽  
Dimensional Model ◽  
Labyrinth Seals ◽  
Acoustic Frequency ◽  
Flutter Analysis ◽  
Dimensional Parameters ◽  
The Stability

Abstract A simple non-dimensional model to describe the flutter onset of two-fin straight labyrinth seals [1] is extended to stepped seals. The effect of the axial displacement of the seal is analyzed first in isolation. It is shown that this fundamental mode is always stable. In a second step, the combination of axial and torsion displacements is used to determine the damping of modes with arbitrary torsion centers. It is concluded that the classical Abbot’s criterion stating that seals supported in the low-pressure side of the seal are stable provided that natural frequency of the mode is greater than the acoustic frequency breaks down under certain conditions. An analytical expression for the non-dimensional work-per-cycle is derived and new non-dimensional parameters controlling the seal stability identified. It is finally concluded the stability of stepped seals can be assimilated to that of a straight through seal if the appropriate distance of the torsion center to the seal is chosen.

Higher Order Conceptual Model for Labyrinth Seal Flutter

2021 ◽  
Vol 143 (7) ◽  
Author(s):  
Roque Corral ◽  
Michele Greco ◽  
Almudena Vega
Keyword(s):  
High Pressure ◽  
Frequency Domain ◽  
Theoretical Background ◽  
The Body ◽  
Previous Model ◽  
Analytical Models ◽  
Pressure Side ◽  
Labyrinth Seals ◽  
Numerical Work ◽  
Flutter Analysis

Abstract A simple nondimensional model to describe the flutter onset of two-fin straight labyrinth seals (Corral, R., and Vega, A., 2018, “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models. Part I: Theoretical Background,” ASME J. Turbomach., 140(10), p. 121006) is extended to account for nonisentropic flow perturbations. The isentropic relationship is replaced by the more general integral energy equation of the inter-fin cavity. A new expression for the Corral and Vega stability criterion is derived, which is very consistent with the previous model in the whole design space of the seal but for torsion centers located in the high-pressure side close to the seal. The new model formally depends on more dimensionless parameters since the existing parameter grouping of the previous model does not hold anymore, but this dependency is weak in relative terms. The model blends the limit where the discharge time of the inter-fin cavity is much longer than the vibration period, and the flow is nearly isentropic, and the opposite limit, where the perturbations are isothermic, gracefully. A few numerical examples obtained using a three-dimensional linearized frequency domain solver are included to support the model and show that the trends are correct, but the body of the numerical work will be presented in a separated article. The matching between the work-per-cycle obtained with the model and frequency domain solver is good. It is shown that some weird trends obtained using linearized unsteady simulations are qualitatively consistent with the current model but not with the previous one (Corral, R., and Vega, A., 2018, “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models. Part I: Theoretical Background,” ASME J. Turbomach., 140(10), p. 121006). The largest differences between the new and the previous model are seen when the seal is supported at the high-pressure side.

Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models: Part I — Theoretical Support

2018 ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Control Volume ◽  
Pressure Ratio ◽  
Analytical Models ◽  
Propagation Time ◽  
Unified Framework ◽  
Labyrinth Seals ◽  
Acoustic Frequency ◽  
Dimensional Parameters ◽  
The Stability ◽  
And Shifts

A simple non-dimensional model to describe the flutter onset of labyrinth seals is presented. The linearized equations for a control volume which represents the inter-fin seal cavity, retaining the circumferential unsteady flow perturbations created by the seal vibration, are used. Firstly, the downstream fin is assumed to be choked, whereas in a second step the model is generalized for unchocked exit conditions. An analytical expression for the non-dimensional work-per-cycle is derived. It is concluded that the stability of a two-fin seal, depends on three non-dimensional parameters, which allow explaining seal flutter behaviour in a comprehensive fashion. These parameters account for the effect of the pressure ratio, the cavity geometry, the fin clearance, the nodal diameter, the fluid swirl velocity, the vibration frequency and the torsion center location in a compact and interrelated form. A number of conclusions have been drawn by means of a thorough examination of the work-per-cycle expression, also known as the stability parameter by other authors. It was found that the physics of the problem strongly depends on the non-dimensional acoustic frequency. When the discharge time of the seal cavity is much greater than the acoustic propagation time, the damping of the system is very small and the amplitude of the response at the resonance conditions is very high. The model not only provides a unified framework for the stability criteria derived by Ehrich [1] and Abbot [2], but delivers an explicit expression for the work-per-cycle of a two-fin rotating seal. All the existing and well established engineering trends are contained in the model, despite its simplicity. Finally, the effect of swirl in the fluid is included. It is found that the swirl of the fluid in the inter-fin cavity gives rise to a correction of the resonance frequency and shifts the stability region. The non-dimensionalization of the governing equations is an essential part of the method and it groups physical effects in a very compact form. Part I of the paper details the derivation of the theoretical model and draws some preliminary conclusions. Part II of the corresponding paper analyzes in depth the implications of the model and outlines the extension to multiple cavity seals.

Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models—Part I: Theoretical Support

2018 ◽  
Vol 140 (12) ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega
Keyword(s):  
Gas Turbines ◽  
Control Volume ◽  
Pressure Ratio ◽  
Analytical Models ◽  
Propagation Time ◽  
Labyrinth Seals ◽  
Acoustic Frequency ◽  
Aeroelastic Instability ◽  
Prediction And Prevention ◽  
The Stability

A simple nondimensional model to describe the flutter onset of labyrinth seals is presented. The linearized mass and momentum integral equations for a control volume which represents the interfin seal cavity, retaining the circumferential unsteady flow perturbations created by the seal vibration, are used. First, the downstream fin is assumed to be choked, whereas in a second step the model is generalized for unchoked exit conditions. An analytical expression for the nondimensional work-per-cycle is derived. It is concluded that the stability of a two-fin seal depends on three nondimensional parameters, which allow explaining seal flutter behavior in a comprehensive fashion. These parameters account for the effect of the pressure ratio, the cavity geometry, the fin clearance, the nodal diameter (ND), the fluid swirl velocity, the vibration frequency, and the torsion center location in a compact and interrelated form. A number of conclusions have been drawn by means of a thorough examination of the work-per-cycle expression, also known as the stability parameter by other authors. It was found that the physics of the problem strongly depends on the nondimensional acoustic frequency. When the discharge time of the seal cavity is much greater than the acoustic propagation time, the damping of the system is very small and the amplitude of the response at the resonance conditions is very high. The model not only provides a unified framework for the stability criteria derived by Ehrich (1968, “Aeroelastic Instability in Labyrinth Seals,” ASME J. Eng. Gas Turbines Power, 90(4), pp. 369–374) and Abbot (1981, “Advances in Labyrinth Seal Aeroelastic Instability Prediction and Prevention,” ASME J. Eng. Gas Turbines Power, 103(2), pp. 308–312), but delivers an explicit expression for the work-per-cycle of a two-fin rotating seal. All the existing and well-established engineering trends are contained in the model, despite its simplicity. Finally, the effect of swirl in the fluid is included. It is found that the swirl of the fluid in the interfin cavity gives rise to a correction of the resonance frequency and shifts the stability region. The nondimensionalization of the governing equations is an essential part of the method and it groups physical effects in a very compact form. Part I of the paper details the derivation of the theoretical model and draws some preliminary conclusions. Part II of the corresponding paper analyzes in depth the implications of the model and outlines the extension to multiple cavity seals.

Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models: Part II — Physical Interpretation

2018 ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Control Volume ◽  
Pressure Ratio ◽  
Analytical Models ◽  
Propagation Time ◽  
Unified Framework ◽  
Labyrinth Seals ◽  
Acoustic Frequency ◽  
Dimensional Parameters ◽  
The Stability ◽  
And Shifts

A simple non-dimensional model to describe the flutter onset of labyrinth seals is presented. The linearized equations for a control volume which represents the inter-fin seal cavity, retaining the circumferential unsteady flow perturbations created by the seal vibration, are used. Firstly, the downstream fin is assumed to be choked, whereas in a second step the model is generalized for unchocked exit conditions. An analytical expression for the non-dimensional work-per-cycle is derived. It is concluded that the stability of a two-fin seal, depends on three non-dimensional parameters, which allow explaining seal flutter behaviour in a comprehensive fashion. These parameters account for the effect of the pressure ratio, the cavity geometry, the fin clearance, the nodal diameter, the fluid swirl velocity, the vibration frequency and the torsion center location in a compact and interrelated form. A number of conclusions have been drawn by means of a thorough examination of the work-per-cycle expression, also known as the stability parameter by other authors. It was found that the physics of the problem strongly depends on the non-dimensional acoustic frequency. When the discharge time of the seal cavity is much greater than the acoustic propagation time, the damping of the system is very small and the amplitude of the response at the resonance conditions is very high. The model not only provides a unified framework for the stability criteria derived by Ehrich [1] and Abbot [2], but delivers an explicit expression for the work-per-cycle of a two-fin rotating seal. All the existing and well established engineering trends are contained in the model, despite its simplicity. Finally, the effect of swirl in the fluid is included. It is found that the swirl of the fluid in the inter-fin cavity gives rise to a correction of the resonance frequency and shifts the stability region. The non-dimensionalization of the governing equations is an essential part of the method and it groups physical effects in a very compact form. Part I of the paper[3] detailed the derivation of the theoretical model and drew some preliminary conclusions. Part II analyzes in depth the implications of the model and outlines the extension to multiple cavity seals.

Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models—Part II: Physical Interpretation

2018 ◽  
Vol 140 (12) ◽  
Author(s):  
Almudena Vega ◽  
Roque Corral
Keyword(s):  
Pressure Ratio ◽  
Analytical Models ◽  
Design Parameters ◽  
Physical Parameters ◽  
Labyrinth Seals ◽  
Simple Method ◽  
Nodal Diameter ◽  
Swirl Velocity ◽  
Flutter Analysis ◽  
Unsteady Interaction

The dimensionless model presented in part I of the corresponding paper to describe the flutter onset of two-fin rotating seals is exploited to extract valuable engineering trends with the design parameters. The analytical expression for the nondimensional work-per-cycle depends on three dimensionless parameters of which two of them are new. These parameters are simple but interrelate the effect of the pressure ratio, the height, and length of the interfin geometry, the seal clearance, the nodal diameter (ND), the fluid swirl velocity, the vibration frequency, and the torsion center location in a compact and intricate manner. It is shown that nonrelated physical parameters can actually have an equivalent impact on seal stability. It is concluded that the pressure ratio can be stabilizing or destabilizing depending on the case, whereas the swirl of the flow is always destabilizing. Finally, a simple method to extend the model to multiple interfin cavities, neglecting the unsteady interaction among them, is described.

Effective Clearance And Differential Gapping Impact on Seal Flutter Modelling and Validation

2021 ◽  
Author(s):  
Roque Corral ◽  
Michele Greco ◽  
Almudena Vega
Keyword(s):  
Labyrinth Seal ◽  
Analytical Models ◽  
Navier Stokes ◽  
Labyrinth Seals ◽  
Theoretical Support ◽  
Gap Ratio ◽  
Flutter Analysis ◽  
The Impact ◽  
Cv Model ◽  
Additional Loss

Abstract This paper presents an update of the model derived by Corral and Vega (2018, “Conceptual Flutter Analysis of Labyrinth Seal Using Analytical Models. Part I - Theoretical Support”, ASME J. of Turbomach., 140 (12), pp. 121006) for labyrinth seal flutter stability, providing a method of accounting for the effect of dissimilar gaps. The original CV model was intended as a conceptual model for understanding the effect of different geometric parameters on the seal stability comprehensively, providing qualitative trends for seal flutter stability. However, the quantitative evaluation of seal flutter, and the comparison of the CV model with detailed unsteady numerical simulations or experimental data, require including additional physics. The kinetic energy generated in the inlet gap is not dissipated entirely in the inter-fin cavity of straight-through labyrinth seals, and part is recovered in the downstream knife. This mechanism needs to be retained in the seal flutter model. It is concluded that when the theoretical gaps are identical, the impact of the recovery factor on the seal stability can be high. The sensitivity of the seal stability to large changes in the outlet to inlet gap ratio is high as well. It is concluded that fin variations due to rubbing or wearing inducing inlet gaps more open than the exit gaps lead to an additional loss of stability concerning the case of identical gaps. The agreement between the updated model and 3D linearized Navier-Stokes simulations is excellent when the model is informed with data coming from steady RANS simulations of the seal.

scholarly journals Numerical validation of an analytical seal flutter model

2021 ◽  
Vol 5 ◽  
pp. 191-201
Author(s):  
Michele Greco ◽  
Roque Corral
Keyword(s):  
Analytical Model ◽  
Fluid Dynamic ◽  
Stability Limit ◽  
Labyrinth Seal ◽  
Analytical Models ◽  
Mode Shapes ◽  
Labyrinth Seals ◽  
Low Frequencies ◽  
Wide Range ◽  
The Stability

An analytical model to describe the flutter onset of straight-through labyrinth seals has been numerically validated using a frequency domain linearized Navier-Stokes solver. A comprehensive set of simulations has been conducted to assess the stability criterion of the analytical model originally derived by Corral and Vega (2018), “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models - Part I: Theoretical Support,” ASME J. Turbomach., 140 (12), pp. 121006. The accuracy of the model has been assessed by using a simplified geometry consisting of a two-fin straight-through labyrinth seal with identical gaps. The effective gaps and the kinetic energy carried over are retained and their effects on stability are evaluated. It turns out that is important to inform the model with the correct values of both parameters to allow a proper comparison with the numerical simulations. Moreover, the non-isentropic perturbations included in the formulations are observed in the simulations at relatively low frequencies whose characteristic time is of the same order as the discharge time of the seal. This effect is responsible for the bending of the stability limit in the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"><mml:mn>0</mml:mn><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:math></inline-formula> ND stability map obtained both in the model and the simulations. It turns out that the analytical model can predict accurately the stability of the seal in a wide range of pressure ratios, vibration mode-shapes, and frequencies provided that this is informed with the fluid dynamic gaps and the energy carried over to the downstream fin from a steady RANS simulation. The numerical calculations show for the first time that the model can be used to predict accurately not only the trends of the work-per-cycle of the seal but also quantitative results.

EFFECTIVE CLEARANCE AND DIFFERENTIAL GAPPING IMPACT ON SEAL FLUTTER MODELLING AND VALIDATION

2021 ◽  
pp. 1-16
Author(s):  
Roque Corral ◽  
Michele Greco ◽  
Almudena Vega
Keyword(s):  
Labyrinth Seal ◽  
Analytical Models ◽  
Navier Stokes ◽  
Labyrinth Seals ◽  
Theoretical Support ◽  
Gap Ratio ◽  
Flutter Analysis ◽  
The Impact ◽  
Cv Model ◽  
Additional Loss

Abstract This paper presents an update of the model derived by Corral and Vega (2018, “Conceptual Flutter Analysis of Labyrinth Seal Using Analytical Models. Part I - Theoretical Support”, ASME J. of Turbomach., 140 (12), pp. 121006) for labyrinth seal flutter stability, providing a method of accounting for the effect of dissimilar gaps. The original CV model was intended as a conceptual model for understanding the effect of different parameters on the seal stability comprehensively, providing qualitative trends for seal flutter stability. However, the quantitative evaluation of seal flutter, and the comparison of the CV model with detailed unsteady numerical simulations or experimental data, require including additional physics. The kinetic energy generated in the inlet gap is not dissipated entirely in the inter-fin cavity of straight-through labyrinth seals, and part is recovered in the downstream knife. This mechanism needs to be retained in the model. It is concluded that when the theoretical gaps are identical, the impact of the recovery factor on the seal stability can be high. The sensitivity of the seal stability to large changes in the outlet to inlet gap ratio is high as well. It is concluded that fin variations due to rubbing or wearing inducing inlet gaps more open than the exit gaps lead to an additional loss of stability concerning the case of identical gaps. The agreement between the updated model and 3D linearized Navier-Stokes simulations is excellent when the model is informed with data coming from steady RANS simulations of the seal.

scholarly journals Reliability Estimation of Reinforced Slopes to Prioritize Maintenance Actions

2021 ◽  
Vol 18 (2) ◽  
pp. 373
Author(s):  
Farshad BahooToroody ◽  
Saeed Khalaj ◽  
Leonardo Leoni ◽  
Filippo De Carlo ◽  
Gianpaolo Di Bona ◽  
...  
Keyword(s):  
Urban Areas ◽  
Evaluation Method ◽  
Likelihood Function ◽  
Reliability Estimation ◽  
Analytical Models ◽  
Geotechnical Structures ◽  
Reinforced Slopes ◽  
The Stability ◽  
Methods Analytical ◽  
Asset Managers

Geosynthetics are extensively utilized to improve the stability of geotechnical structures and slopes in urban areas. Among all existing geosynthetics, geotextiles are widely used to reinforce unstable slopes due to their capabilities in facilitating reinforcement and drainage. To reduce settlement and increase the bearing capacity and slope stability, the classical use of geotextiles in embankments has been suggested. However, several catastrophic events have been reported, including failures in slopes in the absence of geotextiles. Many researchers have studied the stability of geotextile-reinforced slopes (GRSs) by employing different methods (analytical models, numerical simulation, etc.). The presence of source-to-source uncertainty in the gathered data increases the complexity of evaluating the failure risk in GRSs since the uncertainty varies among them. Consequently, developing a sound methodology is necessary to alleviate the risk complexity. Our study sought to develop an advanced risk-based maintenance (RBM) methodology for prioritizing maintenance operations by addressing fluctuations that accompany event data. For this purpose, a hierarchical Bayesian approach (HBA) was applied to estimate the failure probabilities of GRSs. Using Markov chain Monte Carlo simulations of likelihood function and prior distribution, the HBA can incorporate the aforementioned uncertainties. The proposed method can be exploited by urban designers, asset managers, and policymakers to predict the mean time to failures, thus directly avoiding unnecessary maintenance and safety consequences. To demonstrate the application of the proposed methodology, the performance of nine reinforced slopes was considered. The results indicate that the average failure probability of the system in an hour is 2.8×10−5 during its lifespan, which shows that the proposed evaluation method is more realistic than the traditional methods.

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