Nonlinear Analysis of Rotordynamic Fluid Forces in the Annular Plain Seal by Using Extended Bulk-Flow Analysis: Influence of Static Eccentricity and Whirling Amplitude

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
Koya Yamada ◽  
Atsushi Ikemoto ◽  
Tsuyoshi Inoue ◽  
Masaharu Uchiumi
Keyword(s):  
Perturbation Analysis ◽  
Flow Analysis ◽  
Flow Theory ◽  
Bulk Flow ◽  
Design Stage ◽  
Fluid Force ◽  
Force Coefficients ◽  
Fluid Forces ◽  
Static Eccentricity ◽  
Harmonic Components

Rotor-dynamic fluid force (RD fluid force) of turbomachinery is one of the causes of the shaft vibration problem. Bulk flow theory is the method for analyzing this RD fluid force, and it has been widely used in the design stage of machine. The conventional bulk flow theory has been carried out under the assumption of concentric circular shaft's orbit with a small amplitude. However, actual rotating machinery's operating condition often does not hold this assumption, for example, existence of static load on the machinery causes static eccentricity. In particular, when such a static eccentricity is significant, the nonlinearity of RD fluid force may increase and become non-negligible. Therefore, conventional bulk flow theory is not applicable for the analysis of the RD fluid force in such a situation. In this paper, the RD fluid force of the annular plain seal in the case of circular whirling orbit with static eccentricity is investigated. The case with both the significant static eccentricity and the moderate whirling amplitude is considered, and the perturbation analysis of the bulk-flow theory is extended to investigate the RD fluid force in such cases. In this analysis, the assumption of the perturbation solution is extended to both static terms and whirling terms up to the third order. Then, the additional terms are caused by the coupling of these terms through nonlinearity, and these three kinds of terms are considered in the extended perturbation analysis of the bulk flow theory. As a result, a set of nonlinear analytical equations of the extended perturbation analysis of the bulk flow theory, for the case with both the significant static eccentricity and the moderate whirling amplitude, is deduced. The RD fluid force for such cases is analyzed, and the occurrence of constant component, backward synchronous component, and super-harmonic components in the RD fluid force is observed in addition to the forward synchronous component. The representation of RD fluid force coefficients (RD coefficients) are modified for the case with significant static eccentricity, and the variation of RD fluid force coefficients for the magnitude of static eccentricity is analyzed. These analytical results of RD fluid force and its RD coefficients are compared with the numerical results using finite difference analysis and experimental results. As a result, the validity of the extended perturbation analysis of the bulk-flow theory for the case with both the significant static eccentricity and the moderate whirling amplitude is confirmed.

Nonlinear Analysis of Rotordynamic Fluid Forces in the Annular Plain Seal by Using Extended Bulk-Flow Analysis: Influence of Static Eccentricity and Whirling Amplitude

2018 ◽  
Author(s):  
K. Yamada ◽  
A. Ikemoto ◽  
M. Uchiumi ◽  
T. Inoue
Keyword(s):  
Perturbation Analysis ◽  
Flow Analysis ◽  
Flow Theory ◽  
Bulk Flow ◽  
Design Stage ◽  
Fluid Force ◽  
Force Coefficients ◽  
Fluid Forces ◽  
Static Eccentricity ◽  
Harmonic Components

Rotor-dynamic fluid force (RD fluid force) of turbo-machinery is one of the causes of the shaft vibration problem. Bulk flow theory is the method for analyzing this RD fluid force, and it has been widely used in the design stage of machine. Conventional bulk flow theory has been carried out under the assumption of concentric circular shaft’s orbit with small amplitude. However, actual rotating machinery’s operating condition often does not hold this assumption, for example, existence of static load on the machinery causes static eccentricity. In particular, when such a static eccentricity is significant, the nonlinearity of RD fluid force may increase and become non-negligible. Therefore, conventional bulk flow theory is not applicable for the analysis of RD fluid force in such situation. In this paper, RD fluid force of the annular plain seal in the case of circular whirling orbit with static eccentricity is investigated. The case with both the significant static eccentricity and the moderate whirling amplitude is considered, and the perturbation analysis of the bulk-flow theory is extended to investigate RD fluid force in such cases. In this analysis, the assumption of the perturbation solution is extended to both static terms and whirling terms up to the third order. Then, the additional terms are caused by the coupling of these terms through nonlinearity, and these three kinds of terms are considered in the extended perturbation analysis of the bulk flow theory. As a result, a set of nonlinear analytical equations of the extended perturbation analysis of the bulk flow theory, for the case with both the significant static eccentricity and the moderate whirling amplitude, is deduced. RD fluid force for such cases are analyzed, and the occurrence of constant component, backward synchronous component and super-harmonic components in RD fluid force is observed in addition to the forward synchronous component. The representation of RD fluid force coefficients (RD coefficients) are modified for the case with significant static eccentricity, and the variation of RD fluid force coefficients for the magnitude of static eccentricity is analyzed. These analytical results of RD fluid force and its RD coefficients are compared with the numerical results using finite difference analysis and experimental results. As a result, the validity of the extended perturbation analysis of the bulk-flow theory for the case with both the significant static eccentricity and the moderate whirling amplitude is confirmed.

Nonlinear Analysis of Rotordynamic Fluid Forces in the Annular Plain Seal by Using Extended Perturbation Analysis of the Bulk-Flow Theory (Influence of Whirling Amplitude in the Case With Concentric Circular Whirl)

2018 ◽  
Vol 140 (4) ◽  
Author(s):  
Atsushi Ikemoto ◽  
Tsuyoshi Inoue ◽  
Kazukiyo Sakamoto ◽  
Masaharu Uchiumi
Keyword(s):  
Perturbation Analysis ◽  
Large Amplitude ◽  
Flow Analysis ◽  
Flow Theory ◽  
Bulk Flow ◽  
Rotating Shaft ◽  
Fluid Force ◽  
Fluid Forces ◽  
Shaft System ◽  
Large Amplitude Vibration

The bulk-flow theory for the rotordynamic (RD) fluid force has been investigated for many years. These conventional bulk-flow analyses were performed under the assumption and restriction that the whirl amplitude was very small compared to the seal clearance while actual turbomachinery often causes the large amplitude vibration, and these conventional analyses may not estimate its RD fluid force accurately. In this paper, the perturbation analysis of the bulk-flow theory is extended to investigate the RD fluid force in the case of concentric circular whirl with relatively large amplitude. A set of perturbation solutions through third-order perturbations are derived explicitly. It relaxes the restriction of conventional bulk flow analysis, and it enables to investigate the RD fluid force for the whirl amplitude up to about a half of the clearance. Using derived equations, the nonlinear analytical solutions of the flow rates and pressure are deduced, and the characteristics of the RD fluid force are investigated in both radial and tangential directions. The influence of the whirl amplitude on the RD fluid force is explained and validated by comparing with computational fluid dynamics (CFD) analysis. These results are useful for the analysis and prediction of frequency response of the vibration of the rotating shaft system considering the RD fluid forces.

scholarly journals Flow-Induced Vibration of Tubes in Cross-Flow

1996 ◽  
Vol 118 (4) ◽  
pp. 253-258 ◽  
Author(s):  
S. S. Chen ◽  
Y. Cai ◽  
S. Zhu
Keyword(s):  
Unsteady Flow ◽  
Mathematical Models ◽  
Cross Flow ◽  
Flow Theory ◽  
General Description ◽  
Flow Induced Vibration ◽  
Fluid Force ◽  
Force Coefficients ◽  
Fluid Forces

This paper presents an unsteady-flow theory for flow-induced vibration of tubes in cross-flow. It includes a general description of motion-dependent fluid forces, characteristics of fluid-force coefficients, and mathematical models. Detailed results are presented for the constrained mode in the lift direction for various tube arrangements.

Motion-Dependent Fluid Force Coefficients for Tube Arrays in Crossflow

2001 ◽  
Vol 123 (4) ◽  
pp. 429-436 ◽  
Author(s):  
S. S. Chen ◽  
G. S. Srikantiah
Keyword(s):  
Excitation Amplitude ◽  
Steam Generators ◽  
Fluid Force ◽  
Force Coefficients ◽  
Fluid Forces ◽  
Stiffness Coefficients ◽  
Tube Arrays ◽  
Fluidelastic Instability ◽  
Fluid Damping ◽  
Series Of Experiments

Fluidelastic instability of tube arrays in crossflow is interesting academically and important in steam generators and heat exchangers. The key elements necessary to accurately predict fluidelastic instability of tube arrays in crossflow are motion-dependent fluid force coefficients. This paper presents several series of experiments that measure motion-dependent fluid forces for various tube arrays. Fluid damping and stiffness coefficients based on the unsteady flow theory were obtained as a function of reduced flow velocity, excitation amplitude, and Reynolds number, and the characteristics of motion-dependent fluid force coefficients were applied to provide some additional insights into fluidelastic instability.

Analytical Investigation on the Rotational Speed Dependence of the Rotordynamic Fluid Force: The Case of Concentric Circular Whirl in the Annular Plain Seal

2016 ◽  
Author(s):  
Atsushi Ikemoto ◽  
Kazukiyo Sakamoto ◽  
Tsuyoshi Inoue ◽  
Masaharu Uchiumi
Keyword(s):  
Rotational Speed ◽  
High Speed ◽  
Quadratic Function ◽  
Analytical Investigation ◽  
Experimental Unit ◽  
Rotating Shaft ◽  
Fluid Force ◽  
Force Coefficients ◽  
Fluid Forces ◽  
Speed Dependence

Rotordynamic (RD) fluid forces of various kinds of seals has been investigated and reported by Childs [1], Iwatsusbo [2][3] and so on, because it has significant influence on the stability of rotating machinery. Those studies were carried out at lower speeds than the actual machines because of various restrictions such as the limitations of the experimental unit. Then, extrapolation approximations using the obtained results were used to predict the RD fluid force of the actual machines. However, when the rotor vibration is analyzed for the high speed rotating shaft such as a rocket turbopump, a more accurate evaluation of the rotational speed dependence of the derived RD fluid force is desired. In this study, the rotational speed dependence of RD fluid forces in the case of the concentric circular whirl in the annular plain seal is investigated. As a result, the characteristics of these fluid forces vary with the rotational speed significantly. In addition, the strong dependencies of RD fluid force coefficients calculated from these fluid forces on the rotational speed are observed. It is revealed that the changes of the RD fluid force coefficients to rotational speed were modeled by using the quadratic function.

A Computational Fluid Dynamics Modified Bulk Flow Analysis for Circumferentially Shallow Grooved Liquid Seals

2017 ◽  
Author(s):  
Luis San Andrés ◽  
Tingcheng Wu ◽  
Hideaki Maeda ◽  
Ono Tomoki
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Friction Factor ◽  
Flow Model ◽  
Flow Analysis ◽  
Bulk Flow ◽  
Rotor Speed ◽  
Force Coefficients ◽  
Annular Seal ◽  
Direct Stiffness

In straight-through centrifugal pumps, a grooved seal acts as a balance piston to equilibrate the full pressure rise across the pump. As the groove pattern breaks the development of fluid swirl, this seal type offers lesser leakage and lower cross-coupled stiffnesses than a similar size and clearance annular seal. Bulk-flow models predict expediently the static and dynamic force characteristics of annular seals; however they lack accuracy for grooved seals. Computational fluid dynamics (CFD) methods give more accurate results, but are not computationally efficient. This paper presents a modified bulk-flow model to predict the rotordynamic force coefficients of shallow depth circumferentially grooved liquid seals with an accuracy comparable to a CFD solution but with a simulation time of bulk-flow analyses. The procedure utilizes the results of CFD to evaluate the bulk flow velocity field and the friction factors for a 73 grooves annular seal (depth/clearance dg/ Cr = 0.98 and length/diameter L/D = 0.9) operating under various sets of axial pressure drop and rotor speed. In a groove, the flow divides into a jet through the film land and a strong recirculation zone. The penetration angle (α), specifying the streamline separation in the groove cavity, is a function of the operating conditions; an increase in rotor speed or a lower pressure difference increases α. This angle plays a prominent role to evaluate the stator friction factor and has a marked influence on the seal direct stiffness. In the bulk-flow code the friction factor model (f = nRem) is modified with the CFD extracted penetration angle (α) to account for the flow separation in the groove cavity. The flow rate predicted by the modified bulk-flow code shows good agreement with a measured result (6% difference). A perturbation of the flow field is performed on the bulk-flow equations to evaluate the reaction forces on the rotor surface. Compared to the rotordynamic force coefficients derived from the CFD results, the modified bulk-flow code predicts rotordynamic force coefficients within 10%, except that the cross-coupled damping coefficient is over-predicted up to 14%. An example test seal with a few grooves (L/D = 0.5, dg/Cr = 2.5) serves to further validate the predictions of the modified bulk-flow model. Compared to the original bulk-flow analysis, the current method shows a significant improvement in the predicted rotordynamic force coefficients, the direct stiffness and damping coefficients in particular.

A Computational Fluid Dynamics Modified Bulk Flow Analysis for Circumferentially Shallow Grooved Liquid Seals

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Luis San Andrés ◽  
Tingcheng Wu ◽  
Hideaki Maeda ◽  
Ono Tomoki
Keyword(s):  
Fluid Dynamics ◽  
Computational Fluid Dynamics ◽  
Friction Factor ◽  
Flow Analysis ◽  
Operating Conditions ◽  
Bulk Flow ◽  
Rotor Speed ◽  
Force Coefficients ◽  
Annular Seal ◽  
Direct Stiffness

In straight-through centrifugal pumps, a grooved seal acts as a balance piston to equilibrate the full pressure rise across the pump. As the groove pattern breaks the development of fluid swirl, this seal type offers lesser leakage and lower cross-coupled stiffnesses than a similar size and clearance annular seal. Bulk-flow models (BFMs) predict expediently the static and dynamic force characteristics of annular seals; however they lack accuracy for grooved seals. Computational fluid dynamics (CFD) methods give more accurate results, but are not computationally efficient. This paper presents a modified BFM to predict the rotordynamic force coefficients of shallow depth, circumferentially grooved liquid seals with an accuracy comparable to a CFD solution but with a simulation time of bulk-flow analyses. The procedure utilizes the results of CFD to evaluate the bulk flow velocity field and the friction factors for a 73 grooves annular seal (depth/clearance dg/Cr = 0.98 and length/diameter L/D = 0.9) operating under various sets of axial pressure drop and rotor speed. In a groove, the flow divides into a jet through the film land and a strong recirculation zone. The penetration angle (α), specifying the streamline separation in the groove cavity, is a function of the operating conditions; an increase in rotor speed or a lower pressure difference increases α. This angle plays a prominent role to evaluate the stator friction factor and has a marked influence on the seal direct stiffness. In the bulk-flow code, the friction factor model (f = nRem) is modified with the CFD extracted penetration angle (α) to account for the flow separation in the groove cavity. The flow rate predicted by the modified bulk-flow code shows good agreement with the measured result (6% difference). A perturbation of the flow field is performed on the bulk-flow equations to evaluate the reaction forces on the rotor surface. Compared to the rotordynamic force coefficients derived from the CFD results, the modified bulk-flow code predicts rotordynamic force coefficients within 10%, except that the cross-coupled damping coefficient is over-predicted up to 14%. An example test seal with a few grooves (L/D = 0.5, dg/Cr = 2.5) serves to further validate the predictions of the modified BFM. Compared to the original bulk-flow analysis, the current method shows a significant improvement in the predicted rotordynamic force coefficients, the direct stiffness and damping coefficients, in particular.

Hybrid Analysis of Gas Annular Seals With Energy Equation

2013 ◽  
Author(s):  
Patrick J. Migliorini ◽  
Alexandrina Untaroiu ◽  
William C. Witt ◽  
Neal R. Morgan ◽  
Houston G. Wood
Keyword(s):  
Experimental Data ◽  
Hybrid Method ◽  
Energy Equation ◽  
Flow Analysis ◽  
Experimental Studies ◽  
Rotor System ◽  
Bulk Flow ◽  
Outlet Temperature ◽  
Cfd Analysis ◽  
Annular Seals

Annular seals are used in turbomachinery to reduce secondary flow between regions of high and low pressure. In a vibrating rotor system, the non-axisymmetric pressure field developed in the small clearance between the rotor and the seal generate reactionary forces that can affect the stability of the entire rotor system. Traditionally, two analyses have been used to study the fluid flow in seals, bulk-flow analysis and computational fluid dynamics (CFD). Bulk-flow methods are computational inexpensive, but solve simplified equations that rely on empirically derived coefficients and are moderately accurate. CFD analyses generally provide more accurate results than bulk-flow codes, but solution time can vary between days and weeks. For gas damper seals, these analyses have been developed with the assumption that the flow can be treated as isothermal. Some experimental studies show that the difference between the inlet and outlet temperature temperatures is less than 5% but initial CFD studies show that there can be a significant temperature change which can have an effect on the density field. Thus, a comprehensive analysis requires the solution of an energy equation. Recently, a new hybrid method that employs a CFD analysis for the base state, unperturbed flow and a bulk-flow analysis for the first order, perturbed flow has been developed. This method has shown to compare well with full CFD analysis and experimental data while being computationally efficient. In this study, the previously developed hybrid method is extended to include the effects of non-isothermal flow. The hybrid method with energy equation is then compared with the isothermal hybrid method and experimental data for several test cases of hole-pattern seals and the importance of the use of energy equation is studied.

A NONHOMOGENEOUS BULK FLOW MODEL FOR GAS IN LIQUID FLOW ANNULAR SEALS: AN EFFORT TO PRODUCE ENGINEERING RESULTS

2021 ◽  
pp. 1-31
Author(s):  
Xueliang Lu ◽  
Luis San Andres ◽  
Jing Yang
Keyword(s):  
Pressure Drop ◽  
Test Data ◽  
Liquid Flow ◽  
Flow Model ◽  
Bulk Flow ◽  
Experimental Results ◽  
Pure Liquid ◽  
Dynamic Force ◽  
Force Coefficients ◽  
Direct Stiffness

Abstract Seals in multiple phase rotordynamic pumps must operate without compromising system efficiency and stability. Both field operation and laboratory experiments show that seals supplied with a gas in liquid mixture (bubbly flow) can produce rotordynamic instability and excessive rotor vibrations. This paper advances a nonhomogeneous bulk flow model (NHBFM) for the prediction of the leakage and dynamic force coefficients of uniform clearance annular seals lubricated with gas in liquid mixtures. Compared to a homogeneous BFM (HBFM), the current model includes diffusion coefficients in the momentum transport equations and a field equation for the transport of the gas volume fraction (GVF). Published experimental leakage and dynamic force coefficients for two seals supplied with an air in oil mixture whose GVF varies from 0 (pure liquid) to 20% serve to validate the novel model as well as to benchmark it against predictions from a HBFM. The first seal withstands a large pressure drop (~ 38 bar) and the shaft speed equals 7.5 krpm. The second seal restricts a small pressure drop (1.6 bar) as the shaft turns at 3.5 krpm. The first seal is typical as a balance piston whereas the second seal is found as a neck-ring seal in an impeller. For the high pressure seal and inlet GVF = 0.1, the flow is mostly homogeneous as the maximum diffusion velocity at the seal exit plane is just ~0.1% of the liquid flow velocity. Thus, both the NHBFM and HBFM predict similar flow fields, leakage (mass flow rate) and drag torque. The difference between the predicted leakage and measurement is less than 5%. The NHBFM direct stiffness (K) agrees with the experimental results and reduces faster with inlet GVF than the HBFM K. Both direct damping (C) and cross-coupled stiffness (k) increase with inlet GVF < 0.1.Compared to the test data, the two models generally under predict C and k by ~ 25%. Both models deliver a whirl frequency ratio (fw) ~ 0.3 for the pure liquid seal, hence closely matching the test data. fw raises to ~0.35 as the GVF approaches 0.1. For the low pressure seal the flow is laminar, the experimental results and both NHBFM and HBFM predict a null direct stiffness (K). At an inlet GVF = 0.2, the NHBFM predicted added mass (M) is ~30 % below the experimental result while the HBFM predicts a null M. C and k predicted by both models are within the uncertainty of the experimental results. For operation with either a pure liquid or a mixture (GVF = 0.2), both models deliver fw = 0.5 and equal to the experimental finding. The comparisons of predictions against experimental data demonstrate the NHBFM offers a marked improvement, in particular for the direct stiffness (K). The predictions reveal the fluid flow maintains the homogeneous character known at the inlet condition.

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