Hydrodynamic Lubrication

Steady-State Hydrodynamic Lubrication Modeled With the Payvar-Salant Mass Conservation Model

[+] Author and Article Information
Shangwu Xiong1

Department of Mechanical Engineering,  Northwestern University, 2145 Sheridan Road, Evanston, IL 60208s-xiong2@northwestern.edu

Q. Jane Wang

Department of Mechanical Engineering,  Northwestern University, 2145 Sheridan Road, Evanston, IL 60208qwang@northwestern.edu


Corresponding author.

J. Tribol 134(3), 031703 (Jun 18, 2012) (16 pages) doi:10.1115/1.4006615 History: Received February 22, 2012; Revised April 12, 2012; Published June 18, 2012; Online June 18, 2012

Steady-state smooth surface hydrodynamic lubrications of a pocketed pad bearing, an angularly grooved thrust bearing, and a plain journal bearing are simulated with the mass-conservation model proposed by Payvar and Salant. Three different finite difference schemes, i.e., the harmonic mean scheme, arithmetic mean scheme, and middle point scheme, of the interfacial diffusion coefficients for the Poiseuille terms are investigated by using a uniform and nonuniform set of meshes. The research suggests that for the problems with continuous film thickness and pressure distributions, the results obtained with these numerical schemes generally well agree with those found in the literatures. However, if the film thickness is discontinuous while the pressure is continuous, there may be an obvious deviation. Compared with both the analytical solution and other two schemes, the harmonic mean scheme may overestimate or underestimate the pressure. In order to overcome this problem artificial nodes should be inserted along the wall of the bearings where discontinuous film thickness appears. Moreover, the computation efficiency of the three solvers, i.e., the direct solver, the line-by-line the tridiagonal matrix algorithm (TDMA) solver, and the global successive over-relaxation (SOR) solver, are investigated. The results indicate that the direct solver has the best computational efficiency for a small-scale lubrication problem (around 40 thousand nodes). TDMA solver is more robust and requires the least storage, but the SOR solver may work faster than TDMA solver for thrust bearing lubrication problems. Numerical simulations of a group of grooved thrust bearings were conducted for the cases of different outer and inner radii, groove depth and width, velocity, viscosity, and reference film thickness. A curve fitting formula has been obtained from the numerical results to express the correlation of load, maximum pressure, and friction of an angularly grooved thrust bearing in lubrication.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Illustration of relative location of nodes (P, E, W, S, and N) and a cell with four control surfaces (i.e., w, s, e, and n “walls”), where E and e, W and w, S and s, and N and n are for east, west, south, and north to illustrate the control volume and flows

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Figure 2

Comparison of the pressure distibution by using the numerical algorithms with the analytical solution (C-W solution) for the 1-D plain journal bearing problem. (a) Solution with the three schemes (using the direct solver). (b) Solution with the harmonic scheme (using three solvers).

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Figure 3

Numerical results for the 2-D steady-state plain journal bearing. (a) Pressure distribution at ɛ= 0.9. (b), (c), and (d) Estimated load and maximum pressure by the three schemes using the TDMA solver when ɛ= 0.9, ɛ= 0.99, and ɛ= 0.995, respectively (mesh index: 1 for 57 × 41; 2 for 113 × 41; 3 for 169 × 41; 4 for 449 × 41; 5 for 889 × 41).

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Figure 4

Illustration of the pocketed bearing with an infinite width and the results obtained by different approximation schemes using the TMDA solver. (a) Illustration. (b) and (c) Pressure versus distance when hp /h0  = 2 and hp /h0  = 4, respectively (1601 × 41 nodes). (d) and (e) Effect of mesh density on maximum pressure when hp /h0  = 2 and hp /h0  = 4, respectively (mesh index: 1 for 51 × 41; 2 for 101 × 41; 3 for 201 × 41; 4 for 401 × 41; 5 for 1601 × 41).

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Figure 5

Pressure by different approximation schemes: (a) hp /h0  = 10, (b) hp /h0  = 21

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Figure 6

Pressure distribution, maximum pressure, and the computational efficiency of different solvers with the arithmetic mean scheme when hp /h0  = 21 and hp /h0  = 4. (a) and (d) Pressure versus distance using 1601 × 41 nodes. (b) and (e) Effect of mesh density on maximum pressure. (c) and (f) Effect of mesh density on accumulative CPU time. (mesh index: 1 for 51 × 41; 2 for 101 × 41; 3 for 201 × 41; 4 for 401 × 41; 5 for 1601 × 41).

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Figure 7

Illustration of the angular grooved thrust bearing and simulation conditions

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Figure 8

Results of pressure for an angularly grooved thrust bearing. (a) Pressure distribution obtained by the TMDA solver and the Yu-Sadeghi nodal iteration solver when Gd /h0  = 1 (with the harmonic scheme). (b) and (c) Pressure at the central section obtained by the TMDA solver using different schemes when Gd /h0  = 1 and when Gd /h0  = 4, respectively (101 × 101 uniform nodes).

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Figure 9

Mesh density effect on the results of maximum pressure, load, friction, and accumulative CPU time obtained by the three solvers using the arithmetic mean scheme. (a) and (c) Gd /h0  = 1. (b) and (d) Gd /h0  = 4. (mesh index: 1 for 26×26, 2 for 51×51, 3 for 101×101, 4 for 201×201, 5 for 301×301, 6 for 401×401, 7 for 801×801 and 8 for 1601×1601).

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Figure 10

Effects of rotating velocity ω and relative groove width θG/θ0 on maximum pressure, friction force, and load capacity (R1  = 42.25 mm, R2  = 65 mm). (a) ρc  = 18cp, Ng  = 5, h0  = Gd  = 32.5 μm, Gw  = 22.75 mm. (b) ρc  = 100 cp, Ng  = 10, h0  = Gd  = 16.25 μm, Gw  = 34.125 mm. (c), (d), and (e) C = h0  = 32.5 μm, η = 18 cp, Ng  = 5, ω = 2120.5 rpm.

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Figure 11

Effects of reference thickness, groove depth and the number of grooves at different θG/θ0 on maximum pressure, load and friction force (r1  = 42.25 mm, r2  = 65 mm). (a) and (b) η = 18 cp, Gd  = 32.5 μm, ω = 2120.5 rpm, θG/θ0 = 0.424242. (c) and (d) η = 18 cp, ω = 2120.5 rpm, Ng  = 5, h0  = 32.5 μm. (e), (f), and (g) ω = 2120.5 rpm, η = 100 cp, h0  = 16.25 μm and Gd  = 16.25 μm.

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Figure 12

Comparison of (a) load, (b) maximum pressure, and (c) friction force obtained from regressed formula with simulation data



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