Elrod Algorithm Research Articles

Stabilized finite elements for the solution of the Reynolds equation considering cavitation

The Reynolds equation, combined with the Elrod algorithm for including the effect of cavitation, resembles a nonlinear convection–diffusion–reaction (CDR) equation. Its solution by finite elements is prone to oscillations in convection-dominated regions, which are present whenever cavitation occurs. We propose a stabilized finite-element method that is based on the variational multiscale method and exploits the concept of orthogonal subgrid scales. We demonstrate that this approach only requires one additional term in the weak form to obtain a stable method that converges optimally when performing mesh refinement.

A decoupled approach to enhance the Elrod algorithm

The Elrod algorithm is widely used for the study of bearing cavitation problems. However, the potential instability of the algorithm makes it difficult to use in practical applications. A simplified Elrod algorithm based on a decoupled technique was developed. The new algorithm is well-regularized because the tough switch function used for the shear flow term is removed. The most significant advantage of the algorithm is the stability, unlike the traditional algorithm where the accuracy of results irregularly depends on the choice of grids. The numerical experiments, including a journal bearing and a surface-dimpled thrust bearing, show that the simplified Elrod algorithm converges consistently for all cases studied, while the traditional Elrod algorithm fails to converge for cases with a special number of grids. The simplified Elrod algorithm is considered to be a robust alternative to the traditional Elrod algorithm.

Influence of a transient bubble dynamics cavitation model for squeeze film dampers on the run‐up behaviour of a turbocharger rotor

AbstractThe force response of squeeze film dampers and hydrodynamic bearings is significantly influenced by the occurence of cavitation inside the lubricant films. The viscosity of the occuring mixture of oil and air bubbles is vastly different than that of pure lubricant and therefore changes the pressure build‐up and thereby the stiffness and damping properties of the lubrication films. To accurately predict the force response of a squeeze film damper, the cavitation and its effect on the pressure build‐up therefore has to be taken into account. Well known cavitation algorithms for lubrication problems, like the Elrod algorithm or the two‐phase‐model, neglect the transient nature of outgassing and assume instantaneous bubble formation and collapse. In the presented paper, a transient bubble dynamics model based on the Rayleigh‐Plesset‐Scriven equation is utilised instead and its effect on the rotordynamic behaviour of a turbocharger rotor is examined based on fully transient multi body dynamics simulations.

Semi-analytical solution of the Reynolds equation considering cavitation

The numerical effort of rotordynamic simulations with hydrodynamic bearings is often dominated by the solution of the Reynolds equation. Because of the nonlinear fluid–structure interaction, this equation needs to be solved in every time step. Although computationally efficient analytical approximations and look-up table techniques exist, these approaches often do not provide the necessary modeling depth and accuracy. Thus, in many cases, numerical methods are preferred despite their computational cost.In recent studies, a semi-analytical solution of the Reynolds equation has been developed based on the scaled boundary finite element method (SBFEM) with the objective of improving the numerical efficiency. The developed approach assumed Gümbel conditions, which means that cavitation was handled in a highly simplified manner. In this work, the SBFEM solution is for the first time combined with a more complex, nonlinear cavitation model based on the Elrod algorithm. Because of the semi-analytical technique, this requires a simplification of the switch function that defines the locations of the pressure and cavitation zones. However, as demonstrated in this paper, the resulting bearing forces are still in good agreement with a standard numerical reference solution based on the finite volume method (FVM). Since the SBFEM model uses only a one-dimensional discretization, its high-order formulation is very straightforward, even in case of a non-equidistant grid with varying shape functions that take the smooth and non-smooth regions of the solution into account. As a result, the numerical effort is reduced significantly in comparison to the standard FVM.

Implementation of the Elrod Algorithm in Practical Design of an Axial Lubricant Supply Groove in Journal Bearings

Abstract The Elrod algorithm was implemented to study the effects of axial lubricant supply groove parameters, i.e., groove length, width, and location, on the performance of a one-axial groove journal bearing, taking into consideration thermal effects. The results and presented recommendations will assist bearing designers in determining the groove’s preferred angular location and main dimensions to achieve a solution in terms of low total power loss and high minimum film thickness. It is suggested that with regard to low total power loss and high minimum film thickness, for a lighter loaded bearing, the axial groove should be located in the angular range of 100–140 deg to the load line with a groove length ratio Lg/L over 0.6 or the angular range of 300–350 deg with an Lg/L over 0.75. For a heavier loaded bearing, it is suggested that the groove be placed at the angular range of 80–115 deg with an Lg/L over 0.6.

Modelling of Non-Newtonian Starved Thermal-elastohydrodynamic Lubrication of Heterogeneous Materials in Impact Motion

This study presents a numerical model for the thermal-elastohydrodynamic lubrication of heterogeneous materials in impact motion, in which a rigid ball bounces on a starved non-Newtonian oil-covered plane surface of an elastic semi-infinite heterogeneous solid with inhomogeneous inclusions. The impact–rebound process and the microscopic response of the subsurface inhomogeneous inclusions are investigated. The inclusions are homogenized according to Eshelby’s equivalent inclusion method. The Elrod algorithm is adopted to determine the lubrication starvation based on the solutions of pressure and film thickness, while the lubricant velocity and shear rate of the non-Newtonian lubricant are derived by using the separation flow method. The dynamic response of the cases subjected to constant impact mass, momentum, and energy is discussed to reveal the influence of the initial drop height on the impact–rebound process. The results imply that the inclusion disturbs the subsurface stress field and affects the dynamic response of the contact system when the surface pressure is high. The impact energy is the decisive factor for the stress peak, maximum hydrodynamic force, and restitution coefficient, while the dynamic response during the early approaching process is controlled by the drop height.

Experimental and numerical study on cavitation under elastohydrodynamic lubrication point contact

As a common phenomenon in elastohydrodynamic lubrication, cavitation has an effect on the completeness of the oil film in the contact area. Many studies have therefore been conducted on cavitation. Experimental researches on cavitation usually rely on optical interference observation, which offers a limited resolution and observation range. In this paper, an infrared thermal camera is used to observe the cavity bubbles on a ball-on-disc setup under sliding/rolling conditions. The results show that the cavity length increases with an increases of the entrainment speed and the viscosity of the lubricants. These observations are explained by a numerical model based on Elrod's algorithm. Effects of entrainment speed and lubricant viscosity on the breakup of cavitation bubbles and the cavitation states are investigated. Both the simulation and experimental results show that a negative pressure area is present behind the Hertzian contact area. The ambient pressure plays a role in maintaining cavitation state 1. The cavitation pressure is close to the vacuum pressure when the entrainment speed is low and to the ambient pressure instead when the entrainment speed is high.

A Finite Element Model for Non-Newtonian Starved Thermal-Elastohydrodynamic Lubrication of 3D Line Contact

A finite element model for the non-Newtonian starved thermal-elastohydrodynamic lubrication of 3D line contact is developed. The model integrates the mutually dependent solutions of the pressure, film thickness, and temperature in a coupled system. The finite difference Elrod algorithm is introduced in the constructed solution scheme to realize the starvation condition and determine the fractional film content. The lubricant velocity and shear rate of three typical non-Newtonian lubricants are derived by using the separation flow method. The interacted solutions are sequentially determined by their corresponding governing equations. After the model validation, the lubrication performances of the three non-Newtonian lubricants are studied at different levels of inlet lubricant supply. The results show that non-Newtonian rheological behaviors have substantial effects on the lubricant temperature, viscosity, shear rate, velocity, and shear stress, which further affect the accuracy of the predictions of friction and lubrication failure. The solutions of lubrication of machined rough surfaces are also presented to demonstrate the generality of the developed model.

Numerical analysis of textured piston compression ring conjunction using two-dimensional-computational fluid dynamics and Reynolds methods

The Reynolds equation, in which some items have been omitted, is a simplified form of the Navier–Stokes equations. When surface texturing exists, it may unreasonably reveal the tribological effects in some cases. In this paper, both the two-dimensional computational fluid dynamics method, which is based on the Navier–Stokes equations, and the corresponding one-dimensional Reynolds method are adopted to analyze the performance of the textured piston compression ring conjunction. To conduct a comparison between these two methods, the modified Elrod algorithm for Jakobsson–Floberg–Olsson cavitation model is chosen to solve the Reynolds equation. The results show that the Reynolds method is somewhat different from the computational fluid dynamics method in the minimum oil film thickness, pressure distribution, and cavitation at given operating conditions. Moreover, for a low ratio of texture depth to length, the Reynolds equation is still suitable to predict the overall effects of the designed groove textures. The simulation results also reveal that it is not always beneficial for the tribological performance and sometimes may increase the total friction force when the ring is textured.

The effect of lubricant inertia on fluid cavitation for high-speed squeeze film dampers

This work studies the effect of lubricant inertia on the fluid cavitation for partially sealed high-speed squeeze film dampers (SFDs) executing small amplitude circular-centered orbits (CCOs). The lubricant cavitation is modeled by both the Elrod algorithm and the Gumbel’s cavitation boundary condition to provide the comparison between the most common lubricant cavitation models. Additionally, the fluid inertia is integrated by adapting a finite-length SFD model for partially sealed dampers. The integrated SFD model is incorporated into a numerical simulation model and the results are validated by comparison with experimental data. The results of the analysis demonstrate that the fluid inertia effects significantly extend the cavitation region and influence the cavitation onset and the film reformation.

그루브 위치가 리니어 압축기용 피스톤과 실린더의 윤활특성에 미치는 영향

LG Electronics Inc.(Received December 10, 2015; Revised January 10, 2016; Accepted January 10, 2016)Abstract − In this paper hydrodynamic lubrication analysis is carried out to investigate the effects of groovelocation on the lubrication performance of a piston and cylinder system in a linear compressor. The rectangleshaped grooves having a constant groove depth and width are applied on the lubrication area of the piston. TheUniversal Reynolds equation is used to calculate the oil film pressure, and the Elrod algorithm with the finitedifferent method is used to solve the governing equation. The JFO boundary condition is applied to predict cav-itation regions. Transient analysis for different locations of the grooves on the piston is carried out using the typ-ical operating condition of the linear compressor in order to estimate the variations of frictional power losses andminimum film thicknesses. When the grooves are applied on the lubrication area, both the frictional power lossand the minimum film thickness decrease. The frictional power loss can be reduced effectively, while main-taining a minimum film thickness to enable the piston operation without direct contact with the cylinder surface,by means of choosing a proper location of the grooves. The optimum location of the grooves to improve a lubri-cation performance depends on the operation condition or the system requirements specification.Keywords − linear compressor(리니어 압축기), hydrodynamic lubrication(유체 윤활), piston(피스톤), cylinderliner(실린더 라이너), groove(그루브), groove location(그루브 위치)

Numerical analysis of spiral oil wedge sleeve bearing including cavitation and wall slip effects

AbstractThe modified Reynolds equation is established on the basis of critical shear stress model, in which the circumferential and axial wall slip of sleeve and journal surface is considered. Cavitation is treated using modified Elrod algorithm that simplifies the solution of modified Reynolds equation in the full‐film region. The modified Reynolds equations considering wall slip and cavitation effect for two‐dimensional sleeve bearing are established. The results show that wall slip decreases oil film pressure, carrying capacity, friction drag and temperature rise but increases end leakage and cavitation region. The obtained results using the mass‐conserving boundary condition are compared with the Reynolds boundary condition. Copyright © 2014 John Wiley & Sons, Ltd.

Spectral element analysis of herringbone-grooved journal bearings with groove-ridge discontinuity

Many researchers have investigated the performance of herringbone-grooved journal bearings (HGJBs). However, few have yet mentioned the issue of film thickness discontinuities in HGJBs with a finite number of grooves. Most studies have involved the application of a finite difference method for discretization. The present work utilizes the spectral element method to calculate the pressure distribution and dynamic coefficients of HGJBs, in which the thickness of the fluid film changes abruptly in the groove–ridge region. Conservation of mass is adopted to solve the problem. Additionally, the present method can be adopted for grooves with curvy geometry. The numerical results were compared with the analytical solution for a one-dimensional slider bearing and an HGJB. It also shows that for the case of HGJB, the numerical result by the present method is more accurate than the numerical results found in the literature (Trans. ASME J. Tribol. 2000; 122: 103–109, Int. J. Numer. Meth. Heat Fluid Flow 2002; 12: 518–540). Furthermore, employing the present method with the Elrod algorithm can improve the accuracy of deriving loads of HGJBs when cavitation occurs. In addition, the result displays the efficiency of the present method by observing the CPU time. Therefore, the approach can be employed to compute the critical mass of a HGJB. The influence of changing groove angle, groove depth, groove width, and the eccentricity on the critical mass are discussed. Observing the variations in critical mass shows that when the eccentricity is small, a larger groove angle, a lower groove depth, and smaller groove width correspond to a higher critical mass of the HGJB. Copyright © 2010 John Wiley & Sons, Ltd.

Estimation of Cavity Length in EHL Rolling Point Contact

Reciprocating lubricated contacts sometimes suffer from oil starvation due to cavitation at the reversal of motion. However, the behavior of cavities is not well understood such that starvation can be theoretically predicted. In this study, the length of cavity in a steady state elastohydrodynamic lubricated point contact was calculated. For numerical simulation, a modified Elrod algorithm was used. An equation was obtained for the cavity enclosed in the oil meniscus. The equation was constructed with Moes dimensionless parameters M and L, assumed pressure of cavity, and viscosity pressure index of the lubricant. The dimensionless cavity length (or the ratio of cavity length by Hertzian contact radius) is proportional to the product of M−a and Lb. Careful examination of the equation elucidated that the cavity length is dominated by the viscosity, sum velocity, cavity pressure, and geometry of the contact. Experimental measurements with a ball-on-disk apparatus have shown good agreement. The validity of the equation suggests that the algorithm is applicable for complete transient simulations. In practice, the estimated cavity length can be a parameter for starvation.

A Theoretical Model for Squeeze Film Dampers Operating with Air Entrainment

A continuum model of the evolution of air ingestion and entrainment for open-ended squeeze film dampers is proposed in this paper. Hydrodynamic lubrication theory is extended to lubrication with mixture of a Newtonian liquid and an ideal gas. The solution to the universal Reynolds equation is determined numerically using a control volume method (Elrod algorithm) and the forth-order Range-Kutta method. This method conserves mass throughout the computational domain including air ingestion and entrainment. Excellent agreement is found with the experimental works of Diaz and San Andrès for the squeeze film damper [1, 2].

HISTORY MOVING OF CAVITATION FOR DYNAMICALLY LOADED JOURNAL BEARING

On the base of the Elrod algorithm, a generalized equation is derived to describe the lubrication in full film region and in cavitation region of dynamically loaded journal bearing, which is nominated as the incompressible mass-conserving cavitation algorithm. A numerical technique with this algorithm has been solved. By the definition of a switch function g and a variable ξ, which have different meaning in different zone, a generalized equation is derived to integrate the Reynolds equation in full film region with the lubrication equation in cavitation region. This algorithm can predict the moving oil boundary automatically and make the mass conserving boundary condition (JFO theory) more easily and precisely to be applied in actual analysis. It is found that the analysis results with the incompressible mass-conserving cavitation algorithm have good agreement with the existing experiment data. The incompressible cavitation algorithm can predict the history moving of cavitation well. And it is also shown that the pressure value in the cavitation has great effect on the analysis results.

Dynamic Coefficient Prediction in Multi-lobe Journal Bearings Using a Mass Conservation Algorithm

A numerical procedure incorporating cavitation modeling in the predication of dynamic coefficients for four multi-lobe bearings is presented. A first order perturbation solution of the density (or fractional film content) is obtained by using the Elrod universal equation. The Elrod algorithm conserves mass throughout the bearing and automatically predicts the full film and cavitation regions using a switch function. In this work, the cavitation boundary predicted under steady state is considered for calculation of four unsteady pressure gradients. Results of stiffness and damping coefficients are presented for four multi-lobe bearing configurations, viz., two-axial groove, elliptical, three-lobe and offset cylindrical bearing for various L/D and eccentricity ratios. The results of bearing dynamic coefficients obtained using the cavitation algorithm agree well with the published work based on Reynolds boundary on the conditions as the film rupture boundary is predicted based on the continuity of fluid flow in the journal bearings. Presented at the 58th Annual Meeting in New York City April 28–May 1, 2003

Lubrication of journal bearings—influence of stress jump condition at the porous-media/fluid film interface

In this study, the microstructure of bearing surface is modeled as a thin porous film press-fitted on an impermeable surface. On the basis of the Brinkman-extended Darcy model with stress jump condition at the porous-media/fluid film interface, the effects of viscous shear stress and stress jump on the performance of finite journal bearings are examined. The Elrod algorithm as well as the successive over-relaxation method is utilized to solve the modified Reynolds equation and determine the accurate cavitation region. The results show that the increase in stress jump parameter ( β) provides a decrease in the load capacity, an increase in the attitude angle, and an increase in the coefficient of friction.

3012 コンロッド軸受供給油量の計算と測定

Oil flow rate to a con-rod big-end bearing was studied with theoretical and experimental approach. The Elrod's algorithm was applied to the mass conserving EHL theory with a simple model of oil flow in a passage. A hot film anemometer was calibrated and applied to the measurement of oil flow rate. The oil flow rate to con-rod big-end bearing in an experimental engine was calculated and measured under full load operations.

Surface Roughness Effects in Journal Bearings with Non-Newtonian Lubricants

The static performance of finite journal bearings lubricated with non-Newtonian power law fluids is analyzed by using a control volume method with an Elrod algorithm to solve the average Reynolds equation and determine the cavitation region accurately. The results show that the flow behavior index of power law fluids has an insignificant affect on the load ratios, side flow ratios and cavitation regions, while it significantly affects load capacities and side flow rates. Furthermore, the effects of film thickness ratios, pressure flow factors, shear flow factors, slenderness ratios, eccentricities and inlet pressures on the variations of cavitation regions are also discussed.