Finite Elements Integration of the Bulk-Flow Equations

1992 ◽  
pp. 382-388
Author(s):  
B. Pizzigoni ◽  
E. Tanzi
Keyword(s):  
Finite Elements ◽  
Bulk Flow ◽  
Flow Equations

A Bulk-Flow Model of Angled Injection Lomakin Bearings

2002 ◽  
Author(s):  
Luis San Andre´s ◽  
Thomas Soulas ◽  
Florence Challier ◽  
Patrice Fayolle
Keyword(s):  
Flow Model ◽  
Control Volume ◽  
Dynamic Performance ◽  
Bulk Flow ◽  
Computational Method ◽  
Design Parameters ◽  
Dynamic Force ◽  
Injection Angle ◽  
Flow Equations ◽  
Force Coefficients

The paper introduces a bulk-flow model for prediction of the static and dynamic force coefficients of angled injection Lomakin bearings. The analysis accounts for the flow interaction between the injection orifices, the supply circumferential groove, and the thin film lands. A one control-volume model in the groove is coupled to a bulk-flow model within the film lands of the bearing. Bernoulli-type relationships provide closure at the flow interfaces. Flow turbulence is accounted for with shear stress parameters and Moody’s friction factors. The flow equations are solved numerically using a robust computational method. Comparisons between predictions and experimental results for a tangential-against-rotation injection water Lomakin bearing show the novel model predicts well the leakage and direct stiffness and damping coefficients. Computed cross-coupled stiffness coefficients follow the experimental trends for increasing rotor speeds and supply pressures, but quantitative agreement remains poor. A parameter investigation evidences the effects of the groove and land geometries on the Lomakin bearing flowrate and force coefficients. The orifice injection angle does not influence the bearing static performance, although it largely affects its stability characteristics through the evolution of the cross-coupled stiffnesses. The predictions confirm the promising stabilizing effect of the tangential-against-rotation injection configuration. Two design parameters, comprising the feed orifices area and groove geometry, define the static and dynamic performance of Lomakin bearing. The analysis also shows that the film land clearance and length have a larger impact on the Lomakin bearing rotordynamic behavior than its groove depth and length.

Comparison Between Numerical and Experimental Dynamic Coefficients of a Hybrid Aerostatic Bearing

2012 ◽  
Vol 134 (12) ◽  
Author(s):  
Mohamed Amine Hassini ◽  
Mihai Arghir ◽  
Manuel Frocot
Keyword(s):  
Experimental Data ◽  
Theoretical Models ◽  
Bulk Flow ◽  
Major Influence ◽  
Aerostatic Bearing ◽  
Complex Flow ◽  
Flow Equations ◽  
Dynamic Coefficients ◽  
Equivalent Area ◽  
Feeding Pressure

Hybrid journal bearings have been considered for many years as a possible replacement for ball bearings in turbopumps used by the aerospace industry. Due to flow regimes dominated by inertia and due to the nature of the lubricant (cryogenic fluids), the prediction of the linearized dynamic coefficients in these bearings must be based on the compressible bulk-flow equations. Theoretical models based on these equations were validated for hybrid bearings working with water or for liquid or gas annular seals. Validations for hybrid compressible bearings are missing. Experimental data obtained for an air lubricated hybrid aerostatic bearing designed with shallow pockets were recently presented; the data consist of linearized dynamic coefficients obtained for rotation speeds up to 50 krpm and up to 7 bars feeding pressure. The present work introduces a consolidated numerical approach for predicting static and linearized dynamic characteristics. Theoretical predictions are based on bulk flow equations in conjunction with CFD analysis. It was found that, for a given feeding pressure, the value of the pressure downstream the orifice has a major influence on all results. Special care was then taken to describe the complex flow in the feeding system and the orifice. Three dimensional CFD was employed because the bulk-flow equations are inappropriate in this part of the bearing. The pressure downstream the orifice stemming from CFD results and the feeding pressure were next imposed in the bulk flow model and the equivalent area of the orifice was obtained from the numerical solution of the steady flow in the bearing. Since the pockets of the hybrid bearing are shallow, this equivalent area is considered as being the harmonic average of the orifice cross section area and of the cylindrical curtain area located between the orifice and the rotor. The comparisons between theoretical dynamic coefficients and experimental data validated this approach of the equivalent area of the orifice.

Complete Squeeze-Film Damper Analysis Based on the “Bulk Flow” Equations

2009 ◽  
Vol 53 (1) ◽  
pp. 84-96 ◽  
Author(s):  
Jérôme Gehannin ◽  
Mihai Arghir ◽  
Olivier Bonneau
Keyword(s):  
Bulk Flow ◽  
Squeeze Film ◽  
Squeeze Film Damper ◽  
Flow Equations

A Triangle Based Finite Volume Method for the Integration of Lubrication’s Incompressible Bulk Flow Equations

2000 ◽  
Vol 123 (1) ◽  
pp. 118-124 ◽  
Author(s):  
Mihai Arghir ◽  
Jean Fre^ne
Keyword(s):  
Flow System ◽  
A Priori ◽  
Simple Algorithm ◽  
Bulk Flow ◽  
System Of Equations ◽  
Flow Equations ◽  
Application Range ◽  
Lubrication Equation ◽  
Laminar And Turbulent Flow ◽  
Control Volumes

It is well known that for a reduced Reynolds number Re*=ρVH/μs˙H/L greater than unity, inertia forces have a dominant effect in the transport equations, thus rendering the classical lubrication equation inapplicable. The so called “bulk flow” system of equations is then the appropriate mathematical model for describing the flow in bearing and seals operating at Re*⩾1. The difficulty in integrating this system of equations is that one has to deal with coupled pressure and velocity fields. Analytic methods have a very narrow application range so a numerical method has been proposed by Launder and Leschziner in 1978. It represents a natural extrapolation of the successful SIMPLE algorithm applied to the bulk flow system of equations. The algorithm used rectangular, staggered control volumes and represented the state of the art at that moment. In the present work we introduced a method using triangular control volumes. The basic advantage of triangles versus rectangles is that non rectangular domains can be dealt without any a priori limitation. The present paper is focused on the description of the discretized equations and of the solution algorithm. Validations for bearings and seals operating in incompressible, laminar and turbulent flow regime are finally proving the accuracy of the method.

Comparison Between Numerical and Experimental Dynamic Coefficients of a Hybrid Aerostatic Bearing

2012 ◽  
Author(s):  
Mohamed Amine Hassini ◽  
Mihai Arghir ◽  
Manuel Frocot
Keyword(s):  
Experimental Data ◽  
Theoretical Models ◽  
Bulk Flow ◽  
Major Influence ◽  
Aerostatic Bearing ◽  
Complex Flow ◽  
Flow Equations ◽  
Dynamic Coefficients ◽  
Equivalent Area ◽  
Feeding Pressure

Hybrid journal bearings are considered since many years as a possible replacement for ball bearings in turbo-pumps used by the aerospace industry. Due to flow regimes dominated by inertia and due to the nature of the lubricant (cryogenic fluids), the prediction of the linearized dynamic coefficients in these bearings must be based on the compressible bulk-flow equations. Theoretical models based on these equations were validated for hybrid bearings working with water or for liquid or gas annular seals. Validations for hybrid compressible bearings are missing. Experimental data obtained for an air lubricated hybrid aerostatic bearing designed with shallow pockets were recently presented; the data consist of linearized dynamic coefficients obtained for rotation speeds up to 50 krpm and up to 7 bar feeding pressure. The present work introduces a consolidated numerical approach for predicting static and linearized dynamic characteristics. Theoretical predictions are based on bulk flow equations in conjunction with CFD analysis. It was found that for a given feeding pressure, the value of the pressure downstream the orifice has a major influence on all results. Special care was then taken for describing the complex flow in the feeding system and the orifice. Three dimensional CFD was employed because the bulk-flow equations are inappropriate in this part of the bearing. The pressure downstream the orifice stemming from CFD results and the feeding pressure were next imposed in the bulk flow model and the equivalent area of the orifice was obtained from the numerical solution of the steady flow in the bearing. Since the pockets of the hybrid bearing are shallow, this equivalent area is considered as being the harmonic average of the orifice cross section area and of the cylindrical curtain area located between the orifice and the rotor. The comparisons between theoretical dynamic coefficients and experimental data validated this approach of the equivalent area of the orifice.

Numerical Solution of Lubrication’s Compressible Bulk Flow Equations: Applications to Annular Gas Seals Analysis

2001 ◽  
Author(s):  
Mihai Arghir ◽  
Jean Frene
Keyword(s):  
Compressible Flows ◽  
Reynolds Equation ◽  
Specific Treatment ◽  
Reynolds Numbers ◽  
Numerical Approach ◽  
Analytic Solutions ◽  
Bulk Flow ◽  
Flow Equations ◽  
The Family ◽  
Gas Seals

The appropriate theoretical model for lubrication flows characterized by reduced Reynolds numbers greater than unity (Re*=ρVH/μ·H/L≥l) is the so-called “bulk flow” system of equations. Its solution is more difficult than for the simple Reynolds equation because one has to deal with coupled pressure and velocity fields and the equations are of mixed elliptic-hyperbolic type. Adapting some methods borrowed from CFD can develop a suitable numerical approach. In the present work we introduce a pressure-based method belonging to the family of SIMPLE algorithms where compressibility is taken into account by coupling density corrections with pressure corrections. The method is capable of handling all compressible flows and doesn’t require any specific treatment of the incompressible flow regime. It is an adaptation for the bulk flow equations of Karki and Patankar’s (1989) work and is developed in conjunction with a triangular finite volume formulation. The present paper is focused on the description of the discretized equations, of the accompanying boundary conditions and of the solution algorithm. Comparisons with analytic solutions for subsonic and supersonic channel flows prove the ability of the method to handle all compressible flow regimes. Published results for gas (air) annular seals are further used to evaluate the proposed numerical model.

On the Dynamic Properties of Pump Liquid Seals

2013 ◽  
Vol 135 (5) ◽  
Author(s):  
Alexandrina Untaroiu ◽  
Vahe Hayrapetian ◽  
Costin D. Untaroiu ◽  
Houston G. Wood ◽  
Bruno Schiavello ◽  
...  
Keyword(s):  
Complex Geometry ◽  
Dynamic Properties ◽  
Bulk Flow ◽  
Damping Factor ◽  
Steam Turbines ◽  
Fluid Inertia ◽  
Flow Equations ◽  
Dynamic Coefficients ◽  
Aspect Ratios ◽  
Rotor Dynamic

Rotordynamic instability due to fluid flow in seals is a well known phenomenon that can occur in pumps as well as in steam turbines and air compressors. While analysis methods using bulk-flow equations are computationally efficient and can predict dynamic properties fairly well for short seals, they often lack accuracy in cases of seals with complex geometry or with large aspect ratios (L/D above 1.0). This paper presents the linearized rotordynamic coefficients for a liquid seal with large aspect ratio subjected to incompressible turbulent flow. The fluid-induced forces acting on the rotor are calculated by means of a three-dimensional computational fluid dynamics (3D-CFD) analysis, and are then expressed in terms of equivalent linearized stiffness, damping, and fluid inertia coefficients. For comparison, the seal dynamic coefficients were calculated using two other codes: one developed with the bulk flow method and one based on the finite difference method. The three sets of dynamic coefficients calculated in this study were used then to predict the rotor dynamic behavior of an industrial pump. These estimations were then compared to the vibration characteristic measured during the pump shop test, results indicating that the closest agreement was achieved utilizing the CFD generated coefficients. The results of rotor dynamic analysis using the coefficients derived from CFD approach, improved the prediction of both damped natural frequency and damping factor for the first mode, showing substantially smaller damping factor which is consistent with the experimentally observed instability of the rotor-bearing system. As result of continuously increasing computational power, it is believed that the CFD approach for calculating fluid excitation forces will become the standard in industry.

Hole-Pattern Seals: A Three Dimensional CFD Approach for Computing Rotordynamic Coefficient and Leakage Characteristics

2009 ◽  
Author(s):  
Alexandrina Untaroiu ◽  
Patrick Migliorini ◽  
Houston G. Wood ◽  
Paul E. Allaire ◽  
John A. Kocur
Keyword(s):  
Shear Stress ◽  
Stokes Equations ◽  
Complex Geometry ◽  
Fluid Velocity ◽  
Three Dimensional ◽  
Bulk Flow ◽  
Fluid Inertia ◽  
Flow Equations ◽  
Flow Models ◽  
Rotordynamic Coefficients

Labyrinth and other annular seals are commonly used in the turbomachinery industry to limit the leakage between different pressure regions. The pressure driven flow these seals experience can produce significant forces on the rotor. These fluid-induced excitation forces can exert a strong influence on the dynamic characteristics of the machine. Such seal forces can cause the rotor to become unstable, or when properly designed, stabilize a troublesome machine. Thus, it is important to accurately quantify the fluid-induced forces exerted on the rotor to effectively predict the dynamic behavior. Traditional annular seal models are based on bulk flow theory. While these methods are computationally efficient, due to the assumptions made to simplify the flow equations, seal bulk flow models lack accuracy when dealing with more complex geometry seals, such as hole-pattern seals. Unlike the bulk flow model, computational fluid dynamics (CFD) makes no simplifying assumption on the seal geometry, shear stress at the wall, relationship between wall shear stress and mean fluid velocity, or characterization of interfaces between control volumes. This paper presents a method to calculate the linearized rotordynamic coefficients for a hole-pattern seal by means of a three dimensional CFD approach to estimate the fluid-induced forces acting on the rotor. The system is modeled as a rigid rotor, with rotational speed, ω, and whirl frequency, Ω, describing non-synchronous whirl orbits around a static operating point. The Reynolds-averaged Navier-Stokes equations for fluid flow are solved by dividing the volume of fluid into a discrete number of points at which unknown variables (velocity, pressure, etc.) are computed. As a result, all the details of the flow field, including the fluid forces with potential destabilizing effects, are calculated. A 2nd order regression method is then utilized to express the fluid induced forces in terms of equivalent linearized stiffness, damping, and fluid inertia coefficients.

Static and Dynamic Analysis of Annular Seals

2006 ◽  
Author(s):  
Mihai Arghir ◽  
Jean Frene
Keyword(s):  
Static Problem ◽  
Bulk Flow ◽  
Friction Coefficients ◽  
Flow Equations ◽  
Static And Dynamic Analysis ◽  
Dynamic Coefficients ◽  
Theoretical Approaches ◽  
Eccentric Rotor ◽  
Inertia Forces ◽  
Annular Seals

This work is an overview of theoretical approaches used for estimating the characteristics of straight or grooved annular seals. The flow in annular seals is dominated by inertia forces. The goal of the static analysis is to describe the relation between the pressure difference across the seal and the (mass) flow rate. The presentation introduces different approaches of the static problem (analytic, simplified–“bulk flow” and CFD) and underlines the main difficulties in analysing annular seals. The forces on an eccentric rotor are described as a superposition between three effects (Lomakin, viscous and Bernoulli forces). This approach is then used to describe the dynamic characteristics of the seal for a rotor whirling around its centred position. The specific aspects that compressibility adds to gas annular seals analysis are next discussed, with its most important consequence, the flow choking in the exit section. Finally, some recent findings concerning the analysis of textured stator annular seals are presented. The results show that the presence of textures engenders stator and rotor friction coefficients obeying different laws. The use of these new friction coefficients in the bulk-flow equations enables to match the values of the experimental dynamic coefficients. A discussion about the further needs (development and research) in annular seals analysis is carried out at the end of this work.

A Bulk-Flow Model of Angled Injection Lomakin Bearings

2002 ◽  
Vol 129 (1) ◽  
pp. 195-204
Author(s):  
Luis San Andrés ◽  
Thomas Soulas ◽  
Patrice Fayolle
Keyword(s):  
Flow Model ◽  
Control Volume ◽  
Dynamic Performance ◽  
Bulk Flow ◽  
Computational Method ◽  
Design Parameters ◽  
Dynamic Force ◽  
Injection Angle ◽  
Flow Equations ◽  
Force Coefficients

This paper introduces a bulk-flow model for prediction of the static and dynamic force coefficients of angled injection Lomakin bearings. The analysis accounts for the flow interaction between the injection orifices, the supply circumferential groove, and the thin film lands. A one control-volume model in the groove is coupled to a bulk-flow model within the film lands of the bearing. Bernoulli-type relationships provide closure at the flow interfaces. Flow turbulence is accounted for with shear stress parameters and Moody’s friction factors. The flow equations are solved numerically using a robust computational method. Comparisons between predictions and experimental results for a tangential-against-rotation injection water Lomakin bearing show that novel model well predicts the leakage and direct stiffness and damping coefficients. Computed cross-coupled stiffness coefficients follow the experimental trends for increasing rotor speeds and supply pressures, but quantitative agreement remains poor. A parameter investigation shows evidence of the effects of the groove and land geometries on the Lomakin bearing flowrate and force coefficients. The orifice injection angle does not influence the bearing static performance, although it largely affects its stability characteristics through the evolution of the cross-coupled stiffnesses. The predictions confirm the promising stabilizing effect of the tangential-against-rotation injection configuration. Two design parameters, comprised of the feed orifices area and groove geometry, define the static and dynamic performance of Lomakin bearing. The analysis also shows that the film land clearance and length have a larger impact on the Lomakin bearing rotordynamic behavior than its groove depth and length.

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