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Research Papers: Hydrodynamic Lubrication

# Numerical Approach for Solving Reynolds Equation With JFO Boundary Conditions Incorporating ALE Techniques

[+] Author and Article Information
Bernhard Schweizer

Department of Mechanical Engineering, Multibody Systems, University of Kassel, Mönchebergstrasse 7, 34109 Kassel, Germanyschweizer@mks.uni-kassel.de

J. Tribol 131(1), 011702 (Dec 02, 2008) (14 pages) doi:10.1115/1.2991170 History: Received January 09, 2008; Revised August 18, 2008; Published December 02, 2008

## Abstract

Calculating the fluid flow and pressure field in thin fluid films, lubrication theory can be applied, and Reynolds fluid film equation has to be solved. Therefore, boundary conditions have to be formulated. Well-established mass-conserving boundary conditions are the Jakobsson–Floberg–Olsson (JFO) boundary conditions. A number of numerical techniques, which have certain advantages and certain disadvantages, have been developed to solve the Reynolds equation in combination with JFO boundary conditions. In the current paper, a further method is outlined, which may be a useful alternative to well-known techniques. The main idea is to rewrite the boundary value problem consisting of the Reynolds equation and the JFO boundary conditions as an arbitrary Lagrangian–Eulerian (ALE) problem. In the following, an ALE formulation of the Reynolds equation with JFO boundary conditions is derived. Based on a finite element implementation of the governing boundary value problem, numerical examples are presented, and pressure fields are calculated for a plain hydrodynamic journal bearing with an axial oil groove.

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## Figures

Figure 16

Pressure field p(x,y) calculated with the mass-conserving biphase model for ε=0.5, D/B=1, pS=0.4, and pamb=0.1: 3D view (a); cross-section plot p(x,y=0) (b)

Figure 17

Mesh with quadrilateral elements of order 2

Figure 18

Convergence study for the second example (case 2) of Sec. 4

Figure 19

Convergence order for the second example (case 2) of Sec. 4

Figure 20

Cross-section plot p(ξ,η=0) and mean local relative error p̂(hi) for discretizations hi (i=1,…,5)

Figure 1

Hydrodynamic journal bearing with rotating shaft (left); developed journal surface with oil groove (right)

Figure 2

JFO boundary conditions at the developed fluid film gap in the Euler domain (I, pressure zone; II, cavitation zone)

Figure 3

Mapping from Euler domain (right) to Lagrange domain (left)

Figure 4

Boundary conditions for the Reynolds equation in Lagrange coordinates

Figure 5

Boundary conditions for the Laplace equation (mesh displacement) in Lagrange coordinates

Figure 6

Undeformed mesh in the initial configuration

Figure 7

Undeformed mesh with initially curved boundaries

Figure 8

Pressure field p(x,y) for ε=0.9, D/B=1, and pS=5 plotted on the deformed mesh: 3D view (a); top view (b)

Figure 9

Cross-section plot p(ξ,η=0) (a) and p,ξ(ξ,η=0) (b) for ε=0.9, D/B=1, and pS=5 displayed on the undeformed mesh

Figure 10

Mesh displacement field u(x,y) for ε=0.9, D/B=1, and pS=5 and an initial mesh size of 1≤ξ≤3.8,−1≤η≤1: 3D view (a); top view (b)

Figure 11

Pressure field p(x,y) for ε=0.9, D/B=1, and pS=1 plotted on the deformed mesh: 3D view (a); top view (b)

Figure 12

Cross-section plot p(ξ,η=0) (a) and p,ξ(ξ,η=0) (b) for ε=0.9, D/B=1, and pS=1 displayed on the undeformed mesh

Figure 13

Mesh displacement u(x,y) for ε=0.9, D/B=1, and pS=1 and an initial mesh size of 1≤ξ≤3.8,−1≤η≤1: 3D view (a); top view (b)

Figure 14

Pressure field p(x,y) calculated with the ALE approach for ε=0.5, D/B=1, pS=0.4, and pamb=0.1 plotted on the deformed mesh: 3D view (a); cross-section plot p(ξ,η=0) displayed on the undeformed mesh (b)

Figure 15

Pressure field p(x,y) calculated with the non-mass-conserving algorithm for ε=0.5, D/B=1, pS=0.4, and pamb=0.1: 3D view (a); cross-section plot p(x,y=0) (b)

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