Advances in Tribology

Volume 2018, Article ID 9480636, 8 pages

https://doi.org/10.1155/2018/9480636

## Recirculation Flow and Pressure Distributions in a Rayleigh Step Bearing

College of Mechanical Engineering and Applied Electronics Technology & Institute for Advanced Mechanics in Engineering, Beijing University of Technology, Beijing 100124, China

Correspondence should be addressed to Zhao-Miao Liu; nc.ude.tujb@mzl

Received 26 January 2018; Revised 16 April 2018; Accepted 26 April 2018; Published 21 June 2018

Academic Editor: Enrico Ciulli

Copyright © 2018 Feng Shen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1], determining the optimum geometry with maximum load capacity per unit width for a given film thickness and bearing length. This configuration is now referred to as Rayleigh step bearing.

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 [2–5]. In addition, a series of Rayleigh steps are used in journal bearings to form a grooved bearing with higher performance. Since then, researches into optimum design and fluid dynamic characteristics of this bearing have attracted much attention [6, 7]. Rahmani et al. [8] comprehensively studied the Rayleigh step slider bearing including the effect of variations of pressure at the boundaries on the optimum parameters. The bearing is also optimized considering the lubricant flow rate, friction force, and friction coefficient. Auloge et al. [9] studied the optimum design of Rayleigh step bearing and determined the relationships between step location and height along with non-Newtonian lubricants. Artiles et al. [10] described the analysis and design of a 50mm diameter floating-ring seals with the Rayleigh step lift pads, considering influences of the surface speed, gas pressure, and inertia. Hong et al. [11] investigated the flow of a Newtonian fluid and Bingham fluid in a Rayleigh step bearing.

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 [12] investigated both numerically and experimentally the response of a rotor supported on Rayleigh step gas bearing using Galerkin finite element method. Constantinescu et al. [13] analyzed the pressure variation due to fluid inertia effects in Rayleigh step bearings. Faria and San Andrés [14] calculated the bearing load capacity, static stiffness coefficients, and frequency-dependent force coefficients for the gas-lubricated plane and Rayleigh step slider bearings using both finite element and finite difference methods. Lee and Kim [15] calculated the air film temperature of Rayleigh step air foil thrust bearing by solving the Reynolds equation and 3D energy equation with thermohydrodynamic boundary conditions at the top foil, thrust disc, and cooling air plenum. By using the non-Newtonian fluid as a lubricant, the pressure distribution and load capacity for Rayleigh step bearing have also been investigated [11, 16–19].

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 [20]. Only a few investigations used the calculation of the Navier-Stokes equations with finite volume method [11] or infinite element method [12, 14]. In recent years, the full Navier-Stokes equations have been increasingly applied to solve some lubrication problems [21–23]. The trend is driven by a growing interest in describing lubricant flow behavior within a whole lubricated system. The Reynolds equation is confined to just the contact region, but the Navier-Stokes equations can be applied throughout the flow field [24, 25]. In order to optimize Rayleigh step bearing design scientifically, it is beneficial to obtain detailed flow information of the whole lubricant flow field [26].

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.

#### 2. Analytical Solution

The geometry of a two-dimensional infinite Rayleigh step slider bearing is shown in Figure 1. There are two opposed surfaces: the upper surface separates the bearing into two regions and the lower surface has a pure tangential sliding motion relative to the upper surface. The one-dimensional Reynolds equation for a steady flow of an incompressible fluid of constant viscosity can be expressed in the following form [27]:Here is the oil film pressure, is the oil film thickness, is the fluid dynamic viscosity, is the velocity of the lower surface, and is the Cartesian position variable.