Reynolds Boundary Conditions Research Articles

Influence of mass conservation cavitation boundary on transient performance of water-lubricated bearings

Given the current research gap regarding cavitation phenomena and startup conditions in water-lubricated bearings (WLBs), this study offers an innovative approach by integrating the mass conservation boundary condition proposed by Jakobsson, Floberg, and Olsson (known as the JFO boundary condition) with a transient mixed lubrication model of WLBs. It also considers the impact of elastic deformation and surface roughness peaks, thus establishing a transient startup model of WLBs that incorporates the cavitation effect. Furthermore, the dynamic behaviors of this boundary condition are contrasted with the commonly used Reynolds boundary condition during the startup process of the bearings, as well as under eccentric and step load impacts. The findings demonstrate that the JFO boundary condition, accounting for cavitation effects, significantly influences the transient performance of WLBs. The presence of a cavitation region diminishes the damping of the bearing system following step load impacts, leading to increased overshoot and adjustment time while weakening the system's self-adjusting ability against step loads. This study offers valuable insights for theoretical analysis of ship propulsion systems operating under complex conditions.

Series solution of the Reynolds equation in finite-length tilting-pad journal bearing including pad deformation

This paper applies the variable separation method to the hydrodynamic pressure calculation of finite-length tilting pad journal bearings under Reynolds boundary conditions, and a series solution of the Reynolds equation is obtained based on the Sturm-Liouville theory, which considers pad deformation. The thermal effect is also taken into account. A comparison of the present solutions is made with the results obtained by other applied methods, as well as past and recent published data, and good agreement is noted. The steady-state and dynamic performance of a four-pad tilting pad journal bearing are investigated by the load capacity, pressure distribution, friction force, lubricant effective viscosity, journal center velocity, journal center orbits, and phase portrait under the proposed series solution of the Reynolds equation. This study contributes to quickly evaluating the nonlinear dynamic performance of a tilting pad journal bearing rotor system and supporting the nonlinear dynamic design of the system.

A New Geometrical Design to Overcome the Asymmetric Pressure Problem and the Resulting Response of Rotor-Bearing System to Unbalance Excitation

Improving the bearing design helps in reducing the negative consequences related to errors in installation, manufacturing, deflections under severe loading conditions, progressive wear of machine elements, and many other aspects. One of the methods of such a design improvement effort is changing the bearing profile along the bearing width to compensate for the reduction in the geometrical gap between the shaft and the bearing inner surface due to the aforementioned causes. Since in all rotating machinery, unbalance usually exists at some level, this paper deals with the response of this modified bearing to unbalanced excitation to evaluate the effectiveness of such geometrical design on the dynamic characteristics of the rotor-bearing system. The numerical solution is performed using the finite difference method by assuming Reynolds boundary conditions to determine the cavitation limits, and the 4th-order Range-Kutta method is used to determine the time responses resulting from the unbalance excitation. The time responses to this type of excitation show that the rotor-bearing with the improved geometrical design is more stable, particularly at high speeds. In addition, this modification leads to an improvement in the lubricant layer thickness and the reduction in the levels of the generated pressure between the surfaces despite the presence of large deviations from the perfectly aligned bearing system. Furthermore, the suggested geometrical design overcomes the problem of asymmetricity in the pressure field resulting from the shaft deviation to a large extent. The results of this work (the enhancement in the level of the film thickness and the improvement in the dynamic response of the system as well as the reduction of the maximum pressure value) extend the range of misalignment in which the rotor bearing systems can operate safely which represents a significant step in designing the rotor-bearing system.

Analysis of Modified Finite Length Journal Bearing under Position Perturbation

The performance of journal bearings is significantly affected by the presence of misalignment, which is usually an accompanying problem for this type of bearing. This includes exceeding the design limits for the maximum pressure and the minimum film thickness levels, which affect, in other words, the load-carrying capacity of the system. In addition, it raises the possibility of increasing the wear rate at the bearing edges and increases the friction coefficient at high levels of misalignment. This paper deals with the problem of finite-length misaligned journal bearings, considering a novel comparison between two cases of misalignments: the general 3D misalignment and the vertical misalignment problems for modified bearings. The effect of introducing a variable axial bearing profile on the bearing characteristics and the time responses of the rotor bearing system under position perturbation has also been investigated. The numerical solution of this hydrodynamic problem is based on the finite difference method using Reynolds boundary conditions method. Results show that using a variable bearing profile improves bearing characteristics, such as increasing the minimum film thickness significantly and reducing the pressure levels in addition to reducing the friction coefficient. Furthermore, the modification enhances the rotor-bearing stability under position perturbation, extending the speed range for a safe operation.

Thermal fluid lubrication analysis of hydrodynamic journal bearing based on mass conservation boundary conditions

The thermal-hydrodynamic lubrication analysis of bearing with different bearing speed and eccentricity was carried out under Reynolds boundary conditions and mass conservation boundary conditions, and the variation of performance parameters such as oil film carrying capacity with speed and eccentricity were analyzed in detail for the calculated results, and the calculation results under the two different boundary conditions were compared and analyzed. The results show that the boundary conditions have a greater influence on the lubrication flow calculation results, the Reynolds boundary conditions have a faster calculation process, and the mass conservation boundary conditions are recommended when focusing on the analysis of lubrication flow.

Minimizing Misalignment Effects in Finite Length Journal Bearings

This paper focuses on a method to reduce the detrimental effects that occur due to the misalignment in journal bearings by approaching it with the more complete model of a finite length bearing. Such a drawback is quite common in industrial applications, and it is generally accepted that misalignment causes a significant thinning in the film thickness in the area that is close to the bearing edges. Therefore, removing a certain volume of material from the inner surface of the bearing (bushing) over a distance that is at the bearing edges provides an additional clearance to compensate for the clearance reduction that is due to misalignment. A numerical solution that is used in this work is based on the finite difference method where the Reynolds boundary conditions are considered in the solution scheme, thereby, using an iterative procedure to identify the cavitation zone. A three-dimensional misalignment model is incorporated in the solution in order to provide a more realistic presentation of the deviations and errors that there are in comparison with the ideal aligned case. It has been found in the present work that the edge modification increases the thickness of the lubricant layer considerably and reduces the pressure spikes that are associated with the presence of misalignment. The suggested design also reduces the coefficient of friction in comparison with that of the misaligned case. Furthermore, this method helps in reducing the asymmetry of the hydrodynamic pressure field that results from the misalignment. This method enables the operation of journal bearings over a wider range of misalignment levels without sacrificing the load-carrying capacity of the bearing by maintaining a relatively thicker layer of lubricant at the critical positions that are not so due to the effects of misalignment.

Numerical Calculation of Fluid Film Force on Journal Bearings Based on a Biconjugate Gradient-Stabilized Algorithm

Abstract The aim of this work is to present a technical numerical method to improve the efficiency of solving the Reynolds equation for hydrodynamic bearings. Hydrodynamic bearings are largely employed in the shafting of high-speed rotating machinery to provide adequate support. The distribution of hydrodynamic pressure inside the bearing can be obtained by solving the Reynolds equation. Improving the efficiency of solving the Reynolds equation plays an essential role in the design and analysis of journal bearings. A numerical model has been developed to support the analysis and future design of hydrodynamic bearings. The primary objective of the model is to improve the efficiency of solving the Reynold equation during the steady-state and dynamic analysis. The developed method effectively combines the bi-conjugate gradient stabilized (Bi-CGSTAB) algorithm with the Reynolds boundary conditions, resulting in an effective methodology to characterize the pressure distribution within the lubricant for hydrodynamic bearings. A c++ program was implemented, and the difference between the BI-CGSTAB method and the successive over relaxation (SOR) method was evaluated against accuracy, convergence, and computational consumption. The Bi-CGSTAB algorithm has shown promising application in steady-state and dynamic analyses of hydrodynamic bearings. Validation of the results has been made with reference and analytical solutions.

Investigating the effects of chevron-textures on the performance of a porous journal bearing

The present article aimed for investigating the fluid-film pressures for chevron-textured porous journal bearing. The Reynolds equation for porous bearing has been adopted as governing equation, we took the Reynolds boundary conditions into account. The numerical computation has been performed with the help of finite difference methods. The numerical model is well validated with the previous work of researchers. The calculated results revealed that the incorporation of chevron-textures in the convergent zone significantly increase the film pressures as compared to smooth bearing case which ultimately lead to improve the tribological performance. Moreover, permeability play a detrimental role in enhancement of film pressure but incorporation of texture definitely compensate to these effects up to some extent.

Lubrication Characteristics of Water-Lubricated Rubber Bearings Based on an Improved Algorithm of the Mass Conservation Boundary Condition

Abstract Water-lubricated rubber bearings (WLRBs) are widely used in the propulsion systems of ships and military crafts. Based on the mass conservation boundary condition, the elastohydrodynamic lubrication model of WLRBs is established after considering the elastic deformation of the rubber liner and solved by the finite difference method. An improved algorithm is proposed to track the mass conservation boundary. For the first time, the algorithm allocates independent finite difference grids for the film pressure and fluid fraction to overcome the instability issues and to ensure the flow conservation of each grid cell. The accuracy and stability of the algorithm are verified by the experimental data in the literature, comparison of film pressures calculated by two different finite difference schemes, and flowrate calculations of two different bearings. The lubrication characteristics under the mass conversation boundary condition are compared with classical boundary conditions. The results indicate that the film pressure distribution, velocity distribution, and size and shape of the cavitation zone vary greatly under different boundary conditions in the clearance divergence region. The bearing capacity, attitude angle, and friction force under the mass conservation boundary condition are the largest; those under the double Reynolds boundary condition are smaller; and those under the Reynolds boundary condition are the smallest.

Mixed-lubrication analysis of misaligned journal bearing considering turbulence and cavitation

By integrating the average flow equation and Ng–Pan turbulence model using the boundary condition of mass conserving [Jakobsson-Floberg-Olsson (JFO)], a mixed lubrication model for misaligned bearing was established considering the effects of turbulence and cavitation. The numerical solution was obtained using a finite difference scheme. The results indicate that the frictional force and moment calculated using the JFO boundary condition are greater than those calculated using the Reynolds boundary condition under the constant external load. In high speed conditions, the turbulence begins to play a role, which makes the maximum oil film pressure and moment decrease and the minimum oil film thickness, cavitation zone, and friction coefficient increase. The boundary condition and turbulence have important effects on the equilibrium position of the axis. The misalignment leads to an obvious increase in the friction of the mixed lubrication area and a slight decrease in that of the hydrodynamic lubrication area.

Influence of surface textures on the dynamic stability and performance parameters of hydrodynamic two-lobe journal bearings

Surface texturing can improve the performance of journal bearing system. The present study theoretically investigates the impact of surface textures on the dynamic stability and performance parameters of two-lobe journal bearing system. Galerkin's finite element method is used to solve the Reynolds equation governing the flow of lubricant in the gap between the bearing and the journal. Reynolds boundary conditions are applied in the simulation study of plain, full-textured, partially textured-I and partially textured-II configurations of two-lobe journal bearing. The dynamic stability and performance parameters of textured two-lobe journal bearings are computed with the variation of eccentricity ratio and dimple depth and compared with circular bearing results. The results indicate that the existence of surface textures in the pressure build-up zone ranging from 126°–286° and at unity dimple aspect ratio can significantly improve the dynamic stability and performance parameters of two-lobe bearing system.

Issues related to the numerical simulation of herringbone grooved journal bearing including cavitation condition

This study aims to address the complexities involved in determining the numerical solution of herringbone grooved journal bearing considering cavitation. A modification is made to the Reynolds’ equation to include the effect of cavitation as given by Elrod's cavitation algorithm. Grid transformation is performed to consider the effect of the inclined grooves. The partial differential equation is discretised using finite-difference method. Then, the solution of the resulting set of equations is determined by the alternating-direction implicit method and the pressure, load capacity and attitude angle are obtained. Time step (Δ t) and Bulk modulus have a significant impact on the convergence of the numerical solution incorporating Elrod's cavitation algorithm. Use of alternating-direction implicit method over point by point method like Gauss–Seidel is essential to obtain convergence. Load capacity of the herringbone grooved journal bearing rises with the rise in eccentricity ratio. As compared to the Reynolds boundary conditions, Elrod's model results into lower attitude angle for herringbone grooved journal bearing. Cavitation distribution for herringbone grooved journal bearing is much lower than that of plain journal bearing. The effect of variation of groove angle on the herringbone grooved journal bearing's load capacity, side leakage and friction parameter is also determined. A detailed discussion on the various complexities such as treatment at groove ridge boundaries; numerical oscillations; choice of time step and bulk modulus; and influence of compressibility in the Couette term in full film region in the numerical analysis of herringbone grooved journal bearing specifically considering cavitation is given in this work. Multiple methods to deal with the aforementioned complexities are examined and appropriate solutions are obtained.

Reynolds boundary condition realization in journal bearings: Location of oil film rupture boundary with layering-sliding mesh method

A layering-sliding mesh method was developed to realize Reynolds boundary condition in three-dimensional Computational Fluid Dynamics (CFD) analysis of journal bearings. Hybrid dynamic mesh method and pseudo transient analysis were employed to locate the oil film rupture boundary. A numerical model of a full circular journal bearing was established to obtain the steady-state full-Sommerfeld solution. Using the solution as the initial condition, a pseudo transient analysis was performed. The axial boundary of maximum film thickness in the negative pressure region was specified as moving boundary. A dynamic layering method was employed to degenerate the mesh of the divergent clearance region along circumference, and a sliding mesh method was employed to locate the curvature of the moving boundary. The oil film rupture boundary was precisely captured through the criterion of local Reynolds boundary condition. The resulting oil film pressure was validated by comparison with the results of Reynolds equation and published works. The proposed method realized the Reynolds boundary condition in the Navier–Stokes approach. It is an alternative to other cavitation models.

Theoretical Study of influence of MHD in an infinitely long rough Porous Journal Bearing

Analytical study of porous rough journal bearing with magneto hydrodynamic fluid in the presence of longitudinal and transverse roughness is investigated. Surface roughness effect is studied by using Christensen stochastic theory. The generalized Reynolds equations for both types of roughness patterns are derived. Reynolds boundary conditions are used to study the characteristics of infinitely long journal bearing. Cavitation angle is obtained by using bisection method. The expressions for pressure, load carrying capacity and attitude angle are obtained. The results are shown in the graph for longitudinal roughness pattern and concluded that, load carrying capacity increases for the increasing values of Hartmann number.

Reynolds Boundary Condition Approximation in Journal Bearings Based on Dynamic Mesh Method

Reynolds Boundary Condition Approximation in Journal Bearings Based on Dynamic Mesh Method

Dynamics of on-board rotors on finite-length journal bearings subject to multi-axial and multi-frequency excitations: numerical and experimental investigations

On-board rotating machinery subject to multi-axial excitations is encountered in a wide variety of high-technology applications. Such excitations combined with mass unbalance forces play a considerable role in their integrity because they can cause parametric instability and rotor–stator interactions. Consequently, predicting the rotordynamics of such machines is crucial to avoid triggering undesirable phenomena or at least limiting their impacts. In this context, the present paper proposes an experimental validation of a numerical model of a rotor-shaft-hydrodynamic bearings system mounted on a moving base. The model is based on a finite element approach with Timoshenko beam elements having six degrees of freedom (DOF) per node to account for the bending, torsion and axial motions. Classical 2D rectangular finite elements are also employed to obtain the pressure field acting inside the hydrodynamic bearing. The finite element formulation is based on a variational inequality approach leading to the Reynolds boundary conditions. The experimental validation of the model is carried out with a rotor test rig, designed, built, instrumented and mounted on a 6-DOF hydraulic shaker. The rotor’s dynamic behavior in bending, torsion and axial motions is assessed with base motions consisting of mono- and multi-axial translations and rotations with harmonic, random and chirp sine profiles. The comparison of the predicted and measured results achieved in terms of shaft orbits, full spectrums, transient history responses and power spectral densities is very satisfactory, permitting the experimental validation of the model proposed.

Steady-State and Dynamic Characteristics of Water-lubricated Rubber Bearings under Two Sets of Reynolds Boundary Conditions

The lubrication characteristics of bearings are significantly influenced by boundary conditions. The double Reynolds boundary conditions (DRBCs) are introduced to analyze the hydrodynamic lubrication characteristics of water-lubricated rubber bearings for turbulent flows, considering the elasticity of rubber liner. Differences in its steady state and dynamic characteristics arising from the DRBCs and the Reynolds boundary conditions (RBCs) are discussed based on the finite difference method. The results show that the water-film reformation boundary is significantly different between the two sets of conditions, however, the load capacity of the bearing is only slightly different. The attitude angle and friction are greater for DRBCs than for RBCs. In the horizontal direction, direct stiffness and direct damping coefficients are larger for DRBCs than for RBCs, but in the vertical direction the opposite result holds.

Effects of double parabolic profiles with groove textures on the hydrodynamic lubrication performance of journal bearing under steady operating conditions

The textures on the bushing surface have important effects on the performance of journal bearing. In this study, the effects of double parabolic profiles with groove textures on the hydrodynamic lubrication performance of journal bearing under steady operating conditions are investigated theoretically. The journal misalignment, asperity contact and thermal effects are considered, while the profile modifications due to running-in are neglected. The Winkler/Column model is used to calculate the elastic deformation of bushing surface and the adiabatic flow hypothesis is adopted to obtain the effective temperature of lubricating oil. The numerical solution is established by using finite difference and overrelaxation iterative methods, and the rupture zone of oil film is determined by Reynolds boundary conditions. The numerical results reveal that the double parabolic profiles with groove textures with proper location and geometric sizes can increase load carrying capacity and reduce friction loss under steady operating conditions, which effectively overcome the drawbacks of double parabolic profiles. This novel bushing profile may help to reduce the bushing edge wear and enhance the lubrication performance of journal bearing.

Heat transfer and fluid flow of Biodiesel at a backward-Facing step

You should Three-dimensional simulation of a biodiesel fluid flow within a rectangular duct over a backward-facing step is investigated in the present paper. The fluid, which obeys to the Newtonian rheological behavior, is obtained by transformation of Algerian waste cooking oil into a biodiesel. Flow through a rectangular channel subjected to a constant wall temperature or constant heat flux as boundary conditions. The partial differential equations governing fluid flow and heat transfer are solved by the Fluent CFD computational code based on the Finite Volume Method. The numerical experiments are carried out to examine the effect of the Reynolds number by fluid inlet velocity variation for the two boundary conditions. The results are analyzed through the distribution of the temperature and the velocity contours. The variation of the Reynolds number and boundary conditions affects greatly the heat transfer and the fluid flow, in particular near the step region.

Three-Dimensional Thermo-Elasto-Hydrodynamic Computational Fluid Dynamics Model of a Tilting Pad Journal Bearing—Part I: Static Response

Abstract This paper presents the first simulation model of a tilting pad journal bearing (TPJB) using three-dimensional (3D) computational fluid dynamics (CFD), including multiphase flow, thermal-fluid, transitional turbulence, and thermal deformation of the shaft and pads employing two-way fluid–structure interaction (FSI). Part I presents a modeling method for the static performance. The model includes flow between pads BP, which eliminates the use of an uncertain, mixing coefficient (MC) in Reynold's equation approaches. The CFD model is benchmarked with Reynold's model with a 3D thermal-film, when the CFD model boundary conditions are consistent with the Reynolds boundary conditions. The Reynolds model employs an oversimplified MC representation of the three-dimensional mixing effect of the BP flow and heat transfer, and it also employs simplifying assumptions for the flow and heat transfer within the thin film between the journal and bearing. This manufactured comparison shows good agreement between the CFD and Reynold's equation models. The CFD model is generalized by removing these fictitious boundary conditions on pad inlets and outlets and instead models the flow and temperature between pads. The results show that Reynold's model MC approach can lead to significant differences with the CFD model including detailed flow and thermal modeling between pads. Thus, the CFD approach provides increased reliability of predictions. The paper provides an instructive methodology including detailed steps for properly applying CFD to tilt pad bearing modeling. Parts I and II focus on predicting static and dynamic response characteristic responses, respectively.