Mihai Arghir

Theoretical Analysis of the Incompressible Laminar Flow in a Macro-Roughness Cell

The present work deals with the analysis of the incompressible laminar shear driven flow in a channel of which one of the walls carries a macro roughness pattern while the opposite one has a parallel velocity. The problem is discussed from the standpoint of lubrication theory and it is shown that the usual simplified models as the Reynolds or the Stokes equations are not applicable. Numerical results are presented for three types of two dimensional macro-roughness and two versions of a three dimensional one. It is shown that a pressure generation effect occurs with increasing the relative importance of convective inertia. Previous analyses found in the literature discussed only the increase of the shear stress due to the presence of the macro roughness but the lift effect due to the pressure generation has never been enlightened up to now. It is further discussed that, extrapolated to a very large number of macro roughness characterizing a textured surface, this new effect could be added to the other lift generating mechanisms of the lubrication theory. It could thus bring a different light on inertia effects stemming from the use of textured surfaces.

The finite volume solution of the Reynolds equation of lubrication with film discontinuities

The present work deals with the numerical solution of the Reynolds equation of lubrication in the specific case where the film thickness is discontinuous. The present approach is obtained within the framework of a finite volume discretization and enables concentrated inertia effects to be taken into account, as described by a generalized Bernoulli equation. The classic finite volume formulation is included as a special case when discontinuities are absent. Some numerical examples show that the conservative properties of the finite volume discretization are maintained even when the pressure field is discontinuous. A typical application shows that the film discontinuities should always be taken into account in a consistent physical manner in order to eliminate the awkward question of their impact on global results of technological interest.

Static and Dynamic Characterization of a Bump-Type Foil Bearing Structure

The performance of Gas Foil Bearings (GFBs) relies on a coupling between a thin gas film and an elastic structure with dissipative characteristics. Due to the mechanical complexity of the structure, the evaluation of its stiffness and damping is still largely inaccurate if not arbitrary. The goal of this paper is to improve the understanding of the behavior of the bump type FB structure under static and dynamic loads. The structure was modeled with finite elements by using a commercial code. The code employed the large displacements theory and took into account the friction between the bumps and the support and between the bumps and the deformable top foil. Static simulations enabled the estimation of the static stiffness of each bump of a strip. These simulations evidence a lack of reliable analytical models that can be easily implemented in a FB prediction code. The models found in the literature tend to over-estimate the foil flexibility because most of them do not consider the interactions between bumps that seem to be highly important. The transient simulations allowed the estimation of the dynamic stiffness and the damping of a single bump of the FB structure. The presence of stick-slip in the structure is evidenced and hysteretic plots are obtained. The energy dissipation due to Coulomb friction is quantified in function of materials, excitation amplitude and frequency. Some energetic considerations allow the calculation of the equivalent viscous damping coefficient and the results are related to experimental data found in literature. The influence of the number of bumps is also briefly addressed.Copyright

Nonlinear Numerical Prediction of Gas Foil Bearing Stability and Unbalanced Response

One of the main interests of gas foil bearings lies in their superior rotordynamic characteristics compared with conventional bearings. A numerical investigation on the stability limit and on the unbalanced response of foil bearings is presented in this paper. The main difficulty in modeling the dynamic behavior of such bearings comes from the dry friction that occurs within the foil structure. Indeed, dry friction is highly nonlinear and is strongly influenced by the dynamic amplitude of the pressure field. To deal with these nonlinearities, a structural dynamic model has been developed in a previous work. This model considers the entire corrugated foil and the interactions between the bumps by describing the foil bearing structure as a multiple degrees of freedom system. It allows the determination of the dynamic friction forces at the top and at the bottom of the bumps by simple integration of ordinary differential equations. The dynamic displacements of the entire corrugated sheet are then easily obtained at each time step. The coupling between this structural model and a gas bearing prediction code is presented in this paper and allows performing full nonlinear analyses of a complete foil bearing. The bearing stability is the first investigated problem. The results show that the structural deflection enhances the stability of compliant surface bearings compared with rigid ones. Moreover, when friction is introduced, a new level of stability is reached, revealing the importance of this dissipation mechanism. The second investigated problem is the unbalanced response of foil bearings. The shaft trajectories depict a nonlinear jump in the response of both rigid and foil bearings when the value of the unbalance increases. Again, it is evidenced that the foil bearing can support higher mass unbalance before this undesirable step occurs.

Numerical Study of the Pressure Pattern in a Two-Dimensional Hybrid Journal Bearing Recess, Laminar, and Turbulent Flow Results

The present work is a parametric study of the pressure pattern in a two-dimensional recess of a hybrid journal bearing (HJB). It is known that theoretical models of HJB are largely dependent on the recess pressure pattern especially for severe working conditions (high rotation speeds, shallow pockets, etc.). The difficulty is that the recess flow is dominated by the interaction of viscous and inertia forces and cannot be analyzed using a thin film model. The present analysis is based on the numerical resolution of the two-dimensional Navier-Stokes equations where only one recess is modeled (with the film lands and the supply region), the fluid being regarded as incompressible and isothermal. Both the laminar and the turbulent flow regimes are considered. The study is governed by two parameters, one related to the HJB operating conditions and the other related to the recess geometric characteristics. The first parameter is the ratio of the runner versus the supply Reynolds number, Re r /Re s ∈ {0, 1/ 4, 1/ 2, 1, 4, 8}. The supply Reynolds number is fixed at 100 for the laminar regime and at 5000 for the turbulent one. The second parameter is the ratio of the recess depth versus the film thickness. Six values of this ratio are considered, ranging from 4 (shallow recess) to 152 (deep recess). Detailed pressure patterns on the runner wall are presented in a systematic manner giving a clear insight of the flow effects intervening in the recess and of their mutual interaction. Some effects are explained by analyzing the recirculation zones inside the recess. It is also shown that for certain parameters turbulent flows have qualitatively similar effects as laminar ones but they can also have specific trends. In order to sustain this remark, the pressure variation at the recess downstream end is analyzed in the paper Finally, the present results and specially the turbulent ones are intended to contribute to the understanding of viscous and inertia effects interactions in a recess flow and to represent a database in view of HJB theoretical modeling.

Numerical Three-Dimensional Pressure Patterns in a Recess of a Turbulent and Compressible Hybrid Journal Bearing

The present work investigates the flow in the feeding recess of a hybrid journal bearing. Numerical integration of the complete Navier-Stokes equations was performed with an appropriate turbulence model. Of primary concern is the pressure field on the rotating journal surface that is commonly known as the recess pressure pattern. The goal of the work is to determine the influences of fluid compressibility, operating conditions and recess geometry. Reference parameters selected for this study comprise feeding Reynolds number Re a of 2.10 5 , sliding Reynolds number Re c of 5.10 3 and recess depth over film thickness ratio e/H of 2.2. Compressibility was considered first. Three values of the axial exit Mach number were selected for computation, namely 0.2, 0.45, and 0.7. As no significant variation was found, the Mach number was fixed at 0.45 in subsequent studies concerning other parameters: Feeding Reynolds number, Re a 2.10 4 ,2.10 5 ,4.10 5 Recess depth, e/H 0, 2.2, 8 Feedhole axis inclination 90°, 135°, 165° Feedhole location (Figs. 1(a) and 13) centered, downstream offset As each parameter is varied, wire mesh plot of pressure and its sectional profiles are examined and effects of varying various parameters are discussed in reference to flow processes as they may affect the support characteristics of the hybrid journal bearing.

A Dynamic Model for Dissipative Structures used in Bump-Type Foil Bearings

A gas bearing of bump-type foil comprises an underlying structure made of one or several strips of corrugated sheet metal covered by a top foil surface. The fluid film pressure thus needs to be coupled with the behavior of this structure for obtaining the equivalent bearing characteristics. The dynamic model of the bump-type foil structure is then of capital importance. The approach presented in this article is an extension of a previously presented static model considering the structure as a multidegree of freedom system. The continuous corrugated sheet is considered as a discrete one with two nodes and three degrees of freedom for each bump that are linked by linear springs. The hereby presented dynamic model is based on the time-domain integration of the elastic structure under a given load variation. Its efficiency is conditioned by the ability to deal with the discontinuities contained in the nonlinear Coulomb friction model. In order to avoid these numerical difficulties, the Coulomb friction forces are regularized by using Petrovs model (Petrov and Ewins (1)). The present work introduces the results obtained for a single bump and for strips made of several bumps. Stick-slip and hysteresis curves for a single bump under an oscillatory load are first presented. The dissipated energy per cycle and the load-deflection ratio variations in the function of the pressure excitation amplitude and of the friction coefficient are compared to the previous finite element results and exhibit good correlation. Then the case of a strip constituted of six bumps is analyzed and the resulting hysteresis loop is compared to the experimental results published by Ku (2). The two curves correlate well and validate the dynamic structural model.

A Bulk-Flow Analysis of Static and Dynamic Characteristics of Eccentric Circumferentially-Grooved Liquid Annular Seals

The bulk-flow equations used for inertia dominated thin-film flows is an attractive model for the analysis of circumferentially grooved annular seals because the solutions based on the numerical integration of the complete Navier-Stokes equations can be very time-consuming. By using three types of control volumes and some user-tuned constants, the bulk-flow model can be used for calculating the static and the dynamic characteristics. Until now, this has been carried out for centered seals where the flow is governed by ordinary differential equations but no solutions have yet been given for eccentric working conditions. In this latter case, the model is governed by partial differential equations of an elliptic type. The main problem is that for describing the groove effects, the pressure field must incorporate the concentrated drop or recovery effects that occur at the interface between the groove and the land zone. This means that the numerical procedure used for solving the elliptic equations should be able to handle a pressure field having discontinuous values and discontinuous first order derivatives. In the present work, the method used for integrating the system of bulk-flow equations is the SIMPLE algorithm. The algorithm is extended for handling pressure jumps by adding two pressure values on each side of the discontinuity. These values are then expressed in terms of cell centered pressures by imposing the mass conservation and the generalized Bernoulli equation at the discontinuity. This numerical solution is original and has never previously been presented in the finite volume related literature. Comparisons between the numerical predictions (leakage flow rate and rotordynamic coefficients) and experimental data taken from the literature Marquette and Childs (1997) are subsequently presented for an eccentric ten-groove annular seal.

Experimental Analyses of a First Generation Foil Bearing: Startup Torque and Dynamic Coefficients

This paper presents the results of the experimental analysis of static and dynamic characteristics of a generation 1 foil bearing of 38.1 mm diameter and L/D=1. The test rig is of floating bearing type, the rigid shaft being mounted on ceramic ball bearings and driven up to 40 krpm. Two different casings are used for startup and for measurement of dynamic coefficients. In its first configuration, the test rig is designed to measure the startup torque. The foil bearing casing is made of two rings separated by a needle bearing to enable an almost torque free rotation between the foil bearing and the static load. The basic results are the startup torque and the lift-off speed. In its second configuration, a different casing is used to measure the impedances of the foil bearing. Misalignment is a problem that is minimized by using three flexible stingers connecting the foil bearing casing to the base plate of the test rig. The test rig enables the application of a static load and of the dynamic excitation on the journal bearing casing and can measure displacements, forces, and accelerations. Working conditions consisted of static loads comprised between 10 N and 50 N and rotation frequencies ranging from 260 Hz to 590 Hz. Excitation frequencies comprised between 100 Hz and 600 Hz are applied by two orthogonally mounted shakers for each working condition. Stiffness and damping coefficients are identified from the complex impedances and enable the calculation of natural frequencies. The experimental results show that the dynamic characteristics of the tested bearing have a weak dependence on the rotation speed but vary with the excitation frequency.

Combined navier-stokes and bulk-flow analysis of hybrid bearings : Radial and angled injection

The present work deals with the Navier-Stokes and bulk-flow analysis of hybrid bearings intended for use in aerospace applications. These bearings are expected to work at high rotational speeds and high feeding pressures. In such a case, the pressure in the shallow pockets of the bearing is no longer constant and is influenced by hydrostatic and hydrodynamic effects. It has been shown in the literature that the recess pressure pattern can have an important influence on the dynamic characteristics of the bearing. The present work investigates the pressure field in the recess of centered hybrid bearings with radial and angled injection by using a numerical Navier-Stokes analysis. The recess pressure pattern is then subsequently characterized by combining these results with some parametric descriptions. For calculating the dynamic characteristics of the bearing, the parametric pressure pattern is then injected into a bulk-flow model. The proposed model is an alternative analysis to the one advanced by San Andres [ASME J. Tribole, 112, pp. 699-707; 119, 179-187] and in order to evaluate the validity of the bulk-flow code, the numerical predictions are compared with experimental data taken from the literature for radial and angled injection. The favourable effect of the counter-rotating angled injection is then explained by using the velocity field issued from the Navier-Stokes analysis and the pressure field given by the bulk-flow model.