The subject of this monograph is the fluid dynamics of liquid turbomachines, particularly pumps. Rather than attempt a general treatise on turbomachines, we shall focus attention on those special problems and design issues associated with the flow of liquid through a rotating machine. There are two characteristics of a liquid that lead to these special problems, and cause a significantly different set of concerns than would occur in, say, a gas turbine. These are the potential for cavitation and the high density of liquids that enhances the possibility of damaging unsteady flows and forces.

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Hydrodynamics of Pumps

In this paper, we present a review of the experimental identification of dynamic parameters of bearings in a rotating machine. Major emphasis is given to vibration-based identification methods and the review encompasses descriptions of experimental measurement techniques, mathematical modeling, parameter extraction algorithms and uncertainty in the estimates applied to a variety of bearings. The parameter extraction algorithms include the descriptions of governing equations of the rotor-bearing system and identification methods in both time and frequency domains. The identification techniques have been classified based on methods used to excite the system. The review includes a variety of bearings and similar components, which play an active link between the rotating and stationary parts of a machine. Based on the state of the art in bearing identification, conclusions are made and future directions are suggested.

Identification of Dynamic Bearing Parameters: A Review

Labyrinth seals are utilized inside turbomachinery to provide noncontacting control of internal leakage. These seals can also play an important role in determining the rotordynamic stability of the machine. Traditional labyrinth seal models are based on bulk-flow assumptions where the fluid is assumed to behave as a rigid body affected by shear stress at the interfaces. To model the labyrinth seal cavity, a single, driven vortex is assumed and relationships for the shear stress and divergence angle of the through flow jet are developed. These models, while efficient to compute, typically show poor prediction for seals with small clearances, high running speed, and high pressure.* In an effort to improve the prediction of these components, this work utilizes three-dimensional computational fluid dynamics (CFD) to model the labyrinth seal flow path by solving the Reynolds Averaged Navier Stokes equations. Unlike bulk-flow techniques, CFD makes no fundamental assumptions on geometry, shear stress at the walls, as well as internal flow structure. The method allows modeling of any arbitrarily shaped domain including stepped and interlocking labyrinths with straight or angled teeth. When only leakage prediction is required, an axisymmetric model is created. To calculate rotordynamic forces, a full 3D, eccentric model is solved. The results demonstrate improved leakage anti rotordynamic prediction over bulk-flow approaches compared to experimental measurements.

Three-Dimensional CFD Rotordynamic Analysis of Gas Labyrinth Seals

Labyrinth seals are used in various kinds of turbo machines to reduce internal leakage flow. The working fluid, or the gas passing through the rotor shaft labyrinth seals, often generates driving force components that may increase the unstable vibration of the rotor. It is important to know the accurate rotordynamic force components for predicting the instability of the rotor-bearing-seal system. The major goals of this research were to calculate the rotordynamic force of a labyrinth seals utilizing a commercial CFD program and to further compare those results to an existing bulk flow computer program currently used by major US machinery manufacturers. The labyrinth seals of a steam turbine and a compressor eye seal are taken as objects of analysis. For each case, a 3D model with eccentric rotor was solved to obtain the rotordynamic force components. The leakage flow and rotor dynamics force predicted by CFX TASCFlow are compared with the results of the existing bulk flow analysis program DYNLAB. The results show that the bulk flow program gives a pessimistic prediction of the destabilizing forces for the conditions under investigation. Further research work will be required to fully understand the complex leakage flows in turbo machinery.

Application of Computational Fluid Dynamics Analysis for Rotating Machinery—Part II: Labyrinth Seal Analysis

A two-control-volume model is employed for honeycomb-stator/smooth-rotor seals, with a conventional control-volume used for the throughflow and a capacitance-accumulator model for the honeycomb cells. The control volume for the honeycomb cells is shown to cause a dramatic reduction in the effective acoustic velocity of the main flow, dropping the lowest acoustic frequency into the frequency range of interest for rotordynamics. In these circumstances, the impedance functions for the seals cannot be modeled with conventional (frequency-independent) stiffness, damping, and mass coefficients. More general transform functions are required to account for the reaction forces, and the transfer functions calculated here are a lead-lag term for the direct force function and a lag term for the cross-coupled function. Experimental measurements verify the magnitude and phase trends of the proposed transfer functions. These first-order functions are simple, compared to transfer functions for magnetic bearings or foundations. For synchronous response due to imbalance, they can be approximated by running-speed-dependent stiffness and damping coefficients in conventional rotordynamics codes. Correct predictions for stability and transient response will require more general algorithms, presumably using a state-space format.

The Acoustic Influence of Cell Depth on the Rotordynamic Characteristics of Smooth-Rotor/Honeycomb-Stator Annular Gas Seals

The basic equations are derived for compressible flow in a labyrinth seal. The flow is assumed to be completely turbulent in the circumferential direction where the friction factor is determined by the Blasius relation. Linearized zeroth and first-order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth-order pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equation. The first-order equations are solved by a separation of variables solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients. The results of this analysis are compared to published test results.

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An Iwatsubo-Based Solution for Labyrinth Seals: Comparison to Experimental Results

The basic equations are derived for a two-control-volume model for compressible flow in a labyrinth seal. The recirculation velocity in the cavity is incorporated into the model for the first time. The flow is assumed to be completely turbulent and isoenergetic. The wall friction factors are determined using the Blasius formula. Jet flow theory is used for the calculation of the recirculation velocity in the cavity. Linearized zeroth- and first-order perturbation equations are developed for small motion about a centered position by an expansion in the eccentricity ratio. The zeroth-order pressure distribution is found by satisfying the leakage equation while the circumferential velocity distribution is determined by satisfying the momentum equations. The first-order equations are solved by a separation of variable solution. Integration of the resultant pressure distribution along and around the seal defines the reaction force developed by the seal and the corresponding dynamic coefficients.

Theory Versus Experiment for the Rotordynamic Coefficients of Labyrinth Gas Seals: Part I—A Two Control Volume Model

An experimental test facility is used to measure the leakage and rotordynamic coefficients of teeth-on-rotor and teeth-on-stator labyrinth gas seals. The test results are presented along with the theoretically predicted values for the two seal configurations at three different radial clearances and shaft speeds to 16,000 cpm. The test results show that the theory accurately predicts the cross-coupled stiffness for both seal configurations and shows improvement in the prediction of the direct damping for the teeth-on-rotor seal. The theory fails to predict a decrease in the direct damping coefficient for an increase in the radial clearance for the teeth-on-stator seal.

Theory Versus Experiment for the Rotordynamic Coefficient of Labyrinth Gas Seals: Part II—A Comparison to Experiment

An experimental test facility is used to measure the rotordynamic coefficients of teeth-on-rotor and teeth-on-stator labyrinth gas seals Direct damping coefficients are presented for these seals for the first time The results are presented for the two seal configurations at identical operating conditions, and show that, in a rotordynamic sense, the teeth-on-stator seal is more stable than the teeth-on-rotor seal, for inlet tangential velocity in the direction of rotation

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Experimental rotordynamic coefficient results for teeth-on-rotor and teeth-on-stator labyrinth gas seals

A facility and apparatus are described for determining the rotordynamic coefficients and leakage characteristics of annular gas seals. The apparatus has a current top speed of 8000 cpm with a nominal seal diameter of 15.24 cmn (6 in). The air supply unit yields a seal pressure ratio of approximately 7. An external shaker is used to excite the test rotor. The capability to independently calculate all rotordynamic coefficients at a given operating condition with one excitation frequency are discussed.

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J. K. Scharrer

Papers

An Iwatsubo-Based Solution for Labyrinth Seals: Comparison to Experimental Results

Theory Versus Experiment for the Rotordynamic Coefficients of Labyrinth Gas Seals: Part I—A Two Control Volume Model

Theory Versus Experiment for the Rotordynamic Coefficient of Labyrinth Gas Seals: Part II—A Comparison to Experiment

Experimental rotordynamic coefficient results for teeth-on-rotor and teeth-on-stator labyrinth gas seals

Theory versus experiment for the rotordynamic coefficients of annular gas seals. Part 1: Test facility and apparatus