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Flow characteristics in the Rayleigh step slider bearing with infinite width have been studied using both analytical and numerical methods. The conservation equations of mass and momentum were solved utilizing a finite volume approach and the whole flow field was simulated. More detailed information about the flow patterns and pressure distributions neglected by the Reynolds lubrication equation has been obtained, such as jumping phenomenon around a Rayleigh step, vortex structure, and shear stress distribution. The pressure distribution of the Rayleigh step bearing with optimum geometry has been numerically simulated and the results obtained agreed with the analytical solution of the classical Reynolds lubrication equation. The simulation results show that the maximum pressure of the flow field is at the step tip not on the lower surface and the increment of the strain rate from Navier-Stokes equation is approximately 49 percent greater than that from Reynolds theory at the step tip. It is also shown that the position of the maximum pressure of the lower surface is a little less than the length of the first region. These results neglected by the Reynolds lubrication equation are important for designing a bearing.

Rayleigh step bearing has been widely used in industry due to its highest load capacity among all other possible bearing geometries. Many researches on improving its load capacity were carried out using an analytical method by solving the classical Reynolds lubrication equation, assuming that the bearing length should be at least 100 times of the film thickness. In 1918, the theory of a step bearing was firstly discussed by Lord Rayleigh [

Because of its high load capacity and cheap manufacturing, the Rayleigh step bearing has been widely used in industry, such as thrust and pad bearings [

Through the numerical solutions of the conservation equations of mass and momentum, they found some special jumping phenomena around a Rayleigh step, which are important for designing a bearing and the study of wear characteristics. Zhu [

In previous literature, most researchers used an analytical method by solving the Reynolds lubrication equation, which is derived from fundamental equations governing fluid flows and based on several simplifying assumptions. Thus, the solutions of Reynolds lubrication equation for the pressure distribution, load capacity, and frictional coefficient were obtained through a relatively simple calculation and some detailed flow information on Rayleigh step bearing has been neglected [

In this study, the flow characteristics in the infinitely wide Rayleigh step bearing have been investigated by solving Navier-Stokes equations and comparing the numerical results with those results obtained by analytical solutions deduced from the Reynolds equation. From the comparison, more detailed flow characteristics and pressure and shear stress distributions have been obtained, and there are some new differences between the results obtained by two different methods. Moreover, the effects of step geometry and velocity of the lower surface on the recirculation flow and pressure and shear stress distributions in the whole flow field are investigated for better control of these bearings.

The geometry of a two-dimensional infinite Rayleigh step slider bearing is shown in Figure

The Rayleigh step bearing.

With reference to Figure

The dimensionless scales of the Rayleigh step bearing can be defined as

The boundary conditions for the velocity and the pressure are

Employing film thickness (

In addition, substituting (

Using pressure distribution (

When there is no pressure difference (

The derivation of (

So the maximum pressure

For steady state incompressible flow, the momentum and continuity equations of film region are^{5}Pa) and zero velocity gradient in the direction normal to sliding was assumed. This defines a fully developed flow approximation through these boundaries and it is important to set the boundaries relatively far from the region of interest so that they do not influence the numerical solution. The entire domain was fully flooded. At the solid walls, the “no-slip” boundary condition was assumed for the momentum equations. The film region was considered to be a fully developed thin film flow. The bearing working condition is specified by the velocity of the moving wall (^{3}. This approach was adopted because the present work is the first part of a study aimed at enlightening the flow characteristics in the Rayleigh step bearing. Meshes with 210295, 334793, and 5604066 cells were tested. The final refined mesh of the Rayleigh step bearing was built using 334793 cells and the solutions are independent of the refined mesh.

To solve the above equations, a CFD software package CFD-ACE+ (v2010, ESI Corporation) used for multiphysics computational analysis was employed. A finite volume method is utilized to turn the governing partial differential equations (PDE) into a system of algebraic equations and numerically integrated over each of the computational cells using a collocated cell-centered variable arrangement [

The dynamics of flow through a Rayleigh step was investigated. The simulations visualized both the velocity and pressure profiles, while varying system parameters including the depth and length of the step, inlet conditions (pressure and velocity), and shear speed.

The geometry and operation conditions are

Pressure distributions on lower (a) and upper surfaces (b).

The

The effects of height ratios on the maximum pressure were simulated. The Rayleigh step parameters are

Pressure contours around the steps with different height ratios (

Pressure contours (Pa) of

The flow characteristics of Rayleigh step bearing with different geometric structures were studied. Some new phenomena like the vortex and reversed flow zone which are not predicted by the Reynolds lubrication equation were investigated. Figure

Streamlines of different geometric structures.

The simulation results show that the pressure and shear stress distributions of the Rayleigh step bearing are significantly influenced by the vortex and velocity field. In Figure

Strain rates of fluid near the upper (a) and lower (b) surfaces.

The velocity of the lower surface

Lower surface pressures of different

The

The current work presents a parametric study of the Rayleigh step bearing with infinite width using two different approaches: (a) analytical solution of the Reynolds equation; (b) the numerical solution of the Navier-Stokes equations by CFD method. The numerical results agree well with the analytical results with parameters of

Coordinate of bearing leading end (mm)

Link coefficients

Coordinate of region I leading end (mm)

Coordinate of bearing trailing end (mm)

Integration constant

Integration constant

Height of region I (

Height of region II (

Length of the bearing (mm)

Length of region I (mm)

Length of region II (mm)

Pressure of fluid (Pa)

Pressure at bearing leading end (Pa)

Pressure at bearing trailing end (Pa)

The maximum pressure of the lower surface at

Pressure distribution along the bearing lower surface (Pa)

Velocity of the lower surface (m/s)

Velocity in the

Velocity in the

Load capacity of the bearing (N)

Viscosity (Pa.s)

Fluid density (Kg/m^{3})

Dimensionless length (

Dimensionless height (

All data included in this study are available upon request by contact with the corresponding author.

The authors declare that they have no conflicts of interest regarding the publication of the paper.

This work was supported by the National Natural Science Foundation of China (11572013), Beijing Municipal Natural Science Foundation (7152012), and Development Project of Beijing Municipal Education Commission (KM201610005002).

Description of each file of Supplementary Materials: Verification of the results compared with the ANSYS-FLUENT simulations. The results of pressure distributions in Figures