Pressure Distribution In Fluid Film Research Articles

Model and Algorithm for a Rotor-Bearing System Considering Journal Misalignment

Disturbances caused as a result of the misalignment and axial motion of the journal affect the characteristics of the rotor-bearing system. This paper aims to propose an algorithm for the theoretical analysis of a rotor-bearing system that considers these disturbances. A theoretical model for a journal bearing considering disturbances is given. The dynamic equations for a rigid rotor-bearing system are introduced. A detailed algorithm that can simultaneously solve the rotor-dynamic equations and the Reynolds equation is proposed. The static performance, such as the bearing attitude angle and the fluid film pressure, are given, and dynamic characteristics such as the nonlinear dynamic responses and the axial orbits of a rigid rotor-bearing system are presented. The hydrodynamic effect of the bearing is enhanced by the axial disturbance. Disturbances in the circumferential and radial directions lead to variations in the fluid film thickness distribution in the axial direction and the offset of the fluid film pressure distribution in the axial direction. When these disturbances work together, the variation trend is more obvious and affects the capacity and dynamic characteristics of the bearing. When the L/D value of the bearing increases, the clearance between the journal and the bearing decreases rapidly. When the value reaches a certain limit, contact and collision might occur. The theoretical analysis method and the algorithm proposed for a rotor-bearing system considering several disturbances could enhance the design level for a bearing and rotor-bearing system.

An analytical study on nonlinear viscoelastic lubrication in journal bearings

This paper presents a novel analytical solution for journal-bearing viscoelastic lubrication using the perturbation method. The nonlinear Giesekus model was used for the constitutive equations to study the effects of fluid elasticity, shear-thinning viscometric functions, and strain-hardening elongational viscosity of viscoelastic lubrication. The investigation focuses on the impact of characteristic parameters such as mobility factor, eccentricity ratio, and Weissenberg number on the fluid film pressure distribution, load capacity, and shear stress. Although distinguishing between the normal stress differences and extensional viscosity in mixed viscoelastic flows is complicated, we investigated the role and contribution of these two factors. By increasing the elasticity of the fluid, the portion of both mentioned parameters increases consequently. Furthermore, analyses and comparisons show the contributions of the first normal stress and elongational viscosity to the load capacity of the bearing through the stress ratio and flow type parameter for the first time. The research findings indicate that fluid elasticity enhances the load capacity of the bearing compared to a Newtonian lubricant with the same effective viscosity. Moreover, the bearing load capacity is divided into two regions. In the linear region, the mobility factor and Weissenberg numbers have minimal effects leading to a linear increase in the load distribution, and in the exponential region, the load capacity changes are considerable. This research provides valuable insights into the behavior of viscoelastic lubrication in journal-bearing systems.

Mathematical Model to Study the Squeeze Film Characteristics of Diseased Human Synovial Knee Joint

Synovial fluid functions as a biological lubricant and lubricates articular cartilage to reduce friction and wear. Lubricin and hyaluronic acid are the primary components of synovial fluid responsible for its lubricating properties. The loss of properties in synovial fluid and articular cartilage due to aging and activities can restrict mobility in synovial joints, resulting in the degradation of articular cartilage and ultimately leading to pathological synovial joints, which is a major cause of disability. Thus, research on synovial joints remains crucial, and despite previous investigations on synovial joint lubrication, there are several issues related to squeeze film lubrication that require further attention. The Newtonian model of squeeze film lubrication in synovial joints needs to be extensively studied. In this study, lubrication and other related properties of synovial fluid are studied theoretically. In this paper, we have studied the flow of synovial fluid in the diseased synovial joint. Further, we have studied the effect of the viscosity of the synovial fluid, the permeability of articular cartilage, thickness of articular cartilage, and fluid film thickness on the characteristic of the squeeze film formed between the articular cartilages of the diseased human knee joint. The flow of synovial fluid is modeled by considering it as a viscous, incompressible and Newtonian fluid. We have derived the modified Reynolds equation using the principle of hydrodynamic lubrication and continuum mechanics theory and solved it by applying the suitable boundary conditions according to the physical considerations. Subsequently, we obtained the expression for pressure distribution in fluid film, load-bearing capacity, and squeeze time and have done the theoretical analysis on these properties for different parameters. Pressure increases with squeeze velocity and viscosity of the synovial fluid and decreases with permeability and fluid film thickness. Human knee joint becomes diseased due to excessive pressure, and the mobility of the knee joint decreases. The load capacity increases with viscosity and squeeze velocity and decreases with permeability resulting in the reduction of the load-carrying capacity of the knee joint in diseased conditions. Moreover, the squeeze time also increased in the diseased state of the joint.

Fluid Inertia Effect on Spiral-Grooved Mechanical Face Seals Considering Cavitation Effects

An extended Reynolds equation is presented to evaluate the effects of the fluid inertia and cavitation on the performance of spiral-grooved mechanical face seals under high-speed conditions. Segregated finite element procedures are employed to solve the fluid inertia terms and the fluid film pressure distribution with the artificial viscosity stabilization strategy and streamline-upwind/Petrov-Galerkin (SUPG) finite element method. The pressure jump is captured at the sudden variation in film thickness. The fluid inertia effect on the sealing performance is determined at different rotational speeds, groove depths, and spiral angles. The results show that the inertia effect due to the film thickness discontinuities improves the load-carrying capacity and increases the leakage rate. The convective inertia forces in the sealing gap flow promote fluid cavitation. The fluid inertia effect on the sealing performance is closely dependent on the spiral angles.

Analysis of nonlinear viscoelastic lubrication using Giesekus constitutive equation

Recent research has shown that adding polymeric materials to mineral oils and consequently changing the behavior of Newtonian lubricants into viscoelastic materials will enhance the lubrication performance. Therefore, in order to examine theoretically the actual behavior of such lubricants, a suitable viscoelastic model must be considered. Hence, in this paper, the solution of fluid film lubrication is presented analytically using the Giesekus viscoelastic model. This constitutive model is based on the concept of configuration-dependent molecular mobility and is suitable for predicting the nonlinear viscoelastic properties. Indeed, it can describe the power-law regions for viscosity, the normal-stress coefficients, the elongational viscosity, and also the complex viscosity. In order to linearize the momentum and constitutive equations and obtain the generalized Reynolds equation, the perturbation method is used and the mobility factor is considered as the perturbation parameter. Here, the effects of mobility factor, outlet-to-inlet height ratio, and Weissenberg number on fluid film pressure distribution, velocity profiles, load capacity, friction coefficient, and first normal stress difference are investigated in detail. Due to the normal stress difference in viscoelastic fluids, using a viscoelastic fluid in contrast a Newtonian fluid can significantly increase the load-carrying capacity of bearing. Another result is with increasing the value of mobility factor, the fluid viscosity decreases and consequently the pressure distribution decreases simultaneously while the lateral normal stress in the y-direction increases. The term pressure distribution is more negligible than the term lateral normal stress and as a result by increasing the mobility factor the load-carrying capacity increases too. It is also observed that, when the Weissenberg numbers tend to infinity regardless of the mobility factor, the friction coefficient tends towards a constant value and rubber-like elasticity is responses.

Simulation of the effects of non-Newtonian fluid on the behavior of a step hydraulic rod seal based on a power law fluid model

The rheological characteristics of the oil film on the rod-seal interface in the sealing zone have a major influence on the behavior of reciprocating seals. Because of the addition of polymers, the viscosity and temperature properties of hydraulic oil have improved and the fluid presents non-Newtonian characteristics. To investigate the influence of these characteristics on seal behavior, a soft elastohydrodynamic lubrication (EHL) numerical model is introduced to simulate a step seal under a mixed lubrication condition. A modified Reynolds equation is derived for calculating the fluid film pressure distribution more accurately. The equation is based on the power law fluid model and Jakobsson-Floberg-Olsson (JFO) cavitation theory. Results are presented to gain insight into the effect of non-Newtonian fluid characteristics on seal behavior, and the simulated results are compared to those of a Newtonian fluid to reveal the seal mechanism. The influence of operating parameters and the seal surface root mean square (RMS) roughness on sealing performance under different power law indexes is also investigated and discussed.

Numerical and Experimental Analyses of Dynamic Characteristics for Liquid Annular Seals With Helical Grooves in Seal Stator

Numerical and experimental analyses were carried out to investigate the dynamic characteristics of liquid annular seals with helical grooves in the seal stator. In the numerical analysis, the governing equations were the momentum equations with turbulent coefficients and the continuity equation, all averaged across the film thickness and expressed using an oblique coordinate system in which the directions of coordinate axes coincided with the circumferential direction and the direction along the helical grooves. These governing equations were solved numerically to obtain the dynamic characteristics, such as the dynamic fluid-film forces, dynamic coefficients, and whirl-frequency ratio (WFR). The numerical analysis included the effect of both fluid inertia and energy loss at the steps between the helical groove and the land sections. In the experiments, the dynamic fluid-film pressure distributions, which were induced by a small whirling motion of the rotor about the seal center, were measured to obtain the dynamic characteristics. The equivalent numerical results reasonably agree with the experimental results, demonstrating the validity of the numerical analysis. The value of the tangential dynamic fluid force induced by the rotor whirling motion decreased with increasing the helix angle γ. Consequently, the values of the cross-coupled stiffness coefficient and WFR decreased with increasing γ and became negative for large γ. In general, pump rotors rotate with a forward whirling motion under normal operating conditions. Hence, the negative value of WFR for helically grooved seals contributes to rotor stability by suppressing the forward whirling motion of the rotor.

Numerical and Experimental Analyses of Static Characteristics for Liquid Annular Seals With Helical Grooves in Seal Stator

Numerical and experimental analyses were carried out to investigate the static characteristics of liquid annular seals with helical grooves in a seal stator. In the numerical analysis, the momentum equations with turbulent coefficients and the continuity equation, which were averaged across the film thickness, were numerically solved to obtain the leakage flow rate and the pressure distributions in the seal clearance. To accurately define the location of the step between the groove and the land regions in the calculation domain, these governing equations were expressed using an oblique coordinate system in which the directions of coordinate axes coincided with the circumferential direction and the direction along the helical grooves. The numerical analysis included the effects of both fluid inertia and energy loss due to expansion during the passage of fluid from the land region to the helical groove region and that due to contraction from the groove region to the land region. In the experimental analysis, the leakage flow rate and the fluid-film pressure distributions in the seal clearance were measured for the helically grooved seals with different helix angles of the helical groove. The numerical results of leakage flow rate and pressure distributions agree reasonably with the experimental results, which demonstrates the validity of the numerical analysis. The leakage flow rate of the helically grooved seals was influenced by two factors: fluid energy loss during passage through the step between the groove and the land, and the pumping effect by which the spinning motion of the rotor pushes the flow back upstream along the helical grooves. Under a low range of rotor spinning velocity, the leakage flow rate decreased with helix angle because the effect of fluid energy loss in the steps was significant. By contrast, under a high range of spinning velocity, the quantitative difference in the leakage flow rate due to the helix angle decreased compared to that under a low range because the reduction in the leakage flow rate due to the pumping effect was pronounced for a larger helix angle. The effects of helix angle and rotor spinning velocity on the leakage flow rate are explained qualitatively using a simplified model.

Finite element analysis of tilted thrust pad bearings of various recesses shapes considering thrust pad flexibility

The performance characteristics of capillary compensated tilted pad hydrostatic bearing operating with Newtonian lubricant and cubic law lubricant are studied numerically considering the flexibility of pad. The two-dimensional Reynolds equation coupled with three-dimensional equilibrium equations governing the deformation of thrust pad has been solved by using finite element method. The results indicate the influence of tilt parameter and bearing deformation on bearing performance characteristics parameters. In the analysis, four different types of recess shapes such as rectangular, circular, elliptical, and square have been chosen for the analysis. The results show that the consideration of bearing pad deformation significantly increases the value of fluid-film stiffness coefficient and decreases the value of fluid-film damping coefficient. The fluid-film pressure distributions also compared for circular, elliptical, square, and rectangular bearing.

Linear Stability Analysis of Hydrodynamic Journal Bearings with a Flexible Liner and Micropolar Lubrication

A linear stability analysis of hydrodynamic journal bearings is presented, including the effects of elastic distortion of the liner and micropolar lubrication. Hydrodynamic equations of the lubricant and equations of motion of the journal are solved simultaneously with the deformation equations for the bearing surface to predict the fluid film pressure distributions theoretically. The components of stiffness and damping coefficients, critical mass parameter, and whirl ratio, which reflect the dynamic characteristic of the journal bearing, are calculated for varying eccentricity ratio taking into account the flexibility of the liner and the micropolar properties of the lubricant. The results presented show that stability decreases with an increase in the value of the elasticity parameter of the bearing liner and micropolar fluids exhibit better stability in comparison to Newtonian fluids.

Foil Bearing Design Guidelines for Improved Stability

Experimental evidence in the literature suggests that foil bearing-supported rotors can suffer from subsynchronous vibration. While dry friction between top foil and bump foil is thought to provide structural damping, subsynchronous vibration is still an unresolved issue. The current paper aims to shed new light onto this matter and discusses the impact of various design variables on stable foil bearing-supported rotor operation. It is shown that, while a time domain integration of the equations of motion of the rotor coupled with the Reynolds equation for the fluid film is necessary to quantify the evolution of the rotor orbit, the underlying mechanism and the onset speed of instability can be predicted by coupling a reduced order foil bearing model with a rigid-body, linear, rotordynamic model. A sensitivity analysis suggests that structural damping has limited effect on stability. Further, it is shown that the location of the axial feed line of the top foil significantly influences the bearing load capacity and stability. The analysis indicates that the static fluid film pressure distribution governs rotordynamic stability. Therefore, selective shimming is introduced to tailor the unperturbed pressure distribution for improved stability. The required pattern is found via multiobjective optimization using the foil bearing-supported rotor model. A critical mass parameter is introduced as a measure for stability, and a criterion for whirl instability onset is proposed. It is shown that, with an optimally shimmed foil bearing, the critical mass parameter can be improved by more than two orders of magnitude. The optimum shim patterns are summarized for a variety of foil bearing geometries with different L/D ratios and different degrees of foil compliance in a first attempt to establish more general guidelines for stable foil bearing design. At low compressibility (Λ < 2), the optimum shim patterns vary little with bearing geometry; thus, a generalized shim pattern is proposed for low compressibility numbers.

THERMOHYDRODYNAMIC PERFORMANCE OF A JOURNAL BEARING WITH 3D-SURFACE ROUGHNESS AND FLUID INERTIA EFFECTS

This theoretical work describe the combined influence of surface roughness, thermal and fluid-inertia effects on performance characteristics of hydrodynamic journal bearing. The average Reynolds equation that modified to include the surface roughness, viscosity variation due to temperature rise in lubricant fluid-film and fluid-inertia is used to obtain pressure field in the fluid-film. The matched solutions of modified average Reynolds, energy and conduction equations are obtained using finite element method and appropriate iterative schemes. The effects of surface roughness parameter, roughness orientation, and roughness characteristics of opposing surfaces on circumferential fluid-film pressure distribution, load carrying capacity and stability threshold speed of the bearing are studied by considering thermal and fluid-inertia effects.

A new fluid film force model of elliptical bearing: Modelling and case studies

The fluid film force model plays a very important role in the non-linear dynamic analysis of a rotor-bearing system. Based on the assumption of the short journal bearing, the fluid film pressure distribution in an elliptical bearing is obtained in terms of the Reynolds equation and the dynamic Gümbel boundary conditions. According to the structural characteristic of elliptical bearing and the integral of fluid film pressure distribution, the fluid force model is formulated. A numerical experiment on the Jeffcott flexible-rotor model is given to show that the model is effective when length-diameter ratio is smaller than 0.6, and the computing cost for dynamical response of rotor-bearing is reduced. Finally, the dynamical behaviour and bifurcation of a 200 MW turbine set are numerically obtained applying the new fluid film force model.

Quadratic programming algorithm for wall slip and free boundary pressure condition

AbstractWall slip is often observed in a highly sheared fluid film in a solid gap. This makes a difficulty in mathematical analysis for the hydrodynamic effect because fluid velocity at the liquid–solid interfaces is not known a priori. If the gap has a convergent–divergent wedge, a free boundary pressure condition, i.e. Reynolds pressure boundary condition, is usually used in the outlet zone in numerical solution. This paper, based on finite element method and parametric quadratic programming technique, gives a numerical solution technique for a coupled boundary non‐linearity of wall slip and free boundary pressure condition. It is found that the numerical error decreases with the number of elements in a negative power law having an index larger than 2. Our method does not need an iterative process and can simultaneously gives rise to fluid film pressure distribution, wall slip velocity and surface shear stress. Wall slip always decreases the hydrodynamic pressure. Large wall slip even causes a null hydrodynamic pressure in a pure sliding solid gap. Copyright © 2005 John Wiley & Sons, Ltd.

Dynamic analysis of a HDD spindle system with FDBs due to the bearing width and asymmetric grooves of journal bearing

This paper investigates the dynamic behavior of a HDD spindle system with fluid dynamic bearings (FDBs) by solving the Reynolds equation and the equations of a motion of a HDD spindle system in five degrees of freedom. FEM is used to solve the Reynolds equation in order to calculate the pressure distribution in fluid film. Reaction forces and friction torque are obtained by integrating the pressure and shear stress along the fluid film, respectively. Dynamic behaviors of a HDD spindle system, such as the whirling and the tilting motion, are determined by solving its nonlinear equations of motion with the Runge-Kutta method. This paper also proposes two design methods to improve the dynamic characteristics of a HDD spindle system without increasing friction torque, i.e., optimization of the width of the lower and the upper journal bearings and the journal bearings with asymmetric grooves.

A Numerical Model for Elastohydrodynamic Analysis of Plunger and Barrel Clearances in Fuel Injection Equipment

This paper describes a numerical model developed to predict the elastohydrodynamic (coupled solid-fluid) response of unit injector fuel systems. These systems consist of a concentric barrel and plunger with a small annular clearance. During operating (axial movement of the plunger), highly nonuniform pressure and clearance fields are developed which are strongly coupled with each other. The model simultaneously solves for the transient response of the fluid film pressure distribution and three different structural deformation components in a two-dimensional (axial-circumferential) domain. These structural components are the transverse bending of the plunger, radial expansion of the barrel, and radial growth of the plunger from a Poisson effect. The fluid film pressure distribution is governed by the transient Reynolds equation (i.e., lubrication theory) and the structural deformation components governed by linear elastic theory. Full account is taken of these hydrostatic, hydrodynamic, and squeeze-film forces generated in the fluid. The model has been applied to several injector designs. Results have been compared with known performance characteristics and have been found to be qualitatively accurate, in that locations of plunger/barrel contact, and potential for failure, have been accurately predicted.

Nonlinear Effects in a Plain Journal Bearing: Part 1—Analytical Study

Hydrodynamic/hydrostatic journal bearings have been widely used in various types of high speed rotating machinery. For space applications, the issue of using cryogenic fluids as working lubricants has steadily gained in significance. The primary goal of this paper is to model the nonlinearities that occur in a hydrodynamic journal bearing with both cryogenic and oil lubricants. Results will be examined through bearing fluid film pressure distribution and bearing linear and nonlinear stiffness characteristics. The numerical model that couples a variable property Reynolds equation with the dynamics of the rotor is solved by means of a finite difference solution technique. The procedure for the fluid film pressure solution involves an iterative scheme that solves the Reynolds equation coupled with the equations of state for liquid oxygen (LO2). The pressure curve is then integrated to calculate bearing supporting forces. A two-dimensional Newton-Raphson iteration method is used to locate the journal equilibrium position from which both linear and nonlinear bearing stiffness are evaluated by means of the small perturbation technique. The effects of load on the linear/nonlinear plain journal bearing characteristics are analyzed and presented in a parametric form. The relationship between the accuracy of the linear solution and the various orders (3rd, 5th, and 7th power for ΔX) of the nonlinear approximation are also discussed. The validity of both linear and nonlinear solutions at various distances from the journal equilibrium position is also examined. A complete parametric study on the effects of load, temperature, operating speed, and shaft misalignment will be given in Part 2 of this paper.

On The Role of a Non-Newtonian Fluid in Short Bearing Theory

The importance of rheological properties of lubricants has arisen from the realization that non-Newtonian fluid effects are manifested over a broad range of lubrication applications. In this paper a theoretical investigation of short journal bearings performance characteristics for non-Newtonian power-law lubricants is given. A modified form of the Reynolds’ equation for hydrodynamic lubrication is studied in the asymptotic limit of small slenderness ratio (i.e., bearing length to diameter, L/D = λ→0). Fluid film pressure distributions in short bearings of arbitrary azimuthal length are studied using matched asymptotic expansions in the slenderness ratio. The merit of the short bearing approach used in solving a modified Reynolds’ equation by the method of matched asymptotic expansions is emphasized. Fluid film pressure distributions are determined without recourse to numerical solutions to a modified Reynolds’ equation. Power-law rheological exponents less than and equal to one are considered; power-law fluids exhibit reduced load capacities relative to the Newtonian fluid. The cavitation boundary shape is determined from Reynolds’ free surface condition; and the boundary shape is shown to be independent of the bearing eccentricity ratio.