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Research Papers: Hydrodynamic Lubrication

# A Robust Modification to the Universal Cavitation Algorithm in Journal Bearings

[+] Author and Article Information

Mechanical Engineering Department,
University of British Columbia,
2054-6250 Applied Science Lane,

Mechanical Engineering Department,
University of British Columbia,
2054-6250 Applied Science Lane,
e-mail: farhemmati@alumni.ubc.ca

Alireza Jalali

Mechanical Engineering Department,
University of British Columbia,
2054-6250 Applied Science Lane,
e-mail: arjalali@interchange.ubc.ca

Department of Mechanical
and Industrial Engineering,
College of Engineering,
Qatar University,
P.O. Box 2713,
Doha, Qatar
e-mail: myq@qu.edu.qa

Mechanical Engineering Department,
University of British Columbia,
2054-6250 Applied Science Lane,
Department of Mechanical Engineering,
Abu Dhabi University,
Abu Dhabi, UAE

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received March 9, 2016; final manuscript received June 14, 2016; published online November 9, 2016. Assoc. Editor: Jordan Liu.

J. Tribol 139(3), 031703 (Nov 09, 2016) (17 pages) Paper No: TRIB-16-1078; doi: 10.1115/1.4034244 History: Received March 09, 2016; Revised June 14, 2016

## Abstract

In the current study, a modified fast converging, mass-conserving, and robust algorithm is proposed for calculation of the pressure distribution of a cavitated axially grooved journal bearing based on the finite volume discretization of the Adams/Elrod cavitation model. The solution of cavitation problem is shown to strongly depend on the specific values chosen for the lubricant bulk modulus. It is shown how the new proposed scheme is capable of handling the stiff discrete numerical system for any chosen value of the lubricant bulk modulus ($β$) and hence a significant improvement in the robustness is achieved compared to traditionally implemented schemes in the literature. Enhanced robustness is shown not to affect the accuracy of the obtained results, and the convergence speed is also shown to be considerably faster than the widely used techniques in the literature. Effects of bulk modulus, static load, and mesh size are studied on numerical stability of the system by means of eigenvalue analysis of the coefficient matrix of the discrete numerical system. It is shown that the impact of static load and mesh size is negligible on numerical stability compared to dominant significance of varying bulk modulus values. Common softening techniques of artificial bulk modulus reduction is found to be tolerable with maximum two order of magnitudes reduction of $β$ to avoid large errors of more than 3–40% in calculated results.

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## Figures

Fig. 1

Structured 50×20 mesh (a) and the computational domain (b)

Fig. 2

Performance study of the softened system of equations at β=20 MPa for the AF solver (a) and the FLS solver (b)

Fig. 3

CPU computation time of AF and FLS schemes for a 50 × 20 and a 100 × 40 mesh, total iterations to converge (a), first 100 iterations for 50 × 20 mesh (b), and first 120 iterations for 100 × 40 mesh (c)

Fig. 4

Performance study (a) and CPU computation time (b) of the softened system of equations at β=20 MPa for AF and FLS schemes with binary switch function and fixed time-step

Fig. 5

Performance study of the stiff system of equations at β=2 GPa for the AF solver (a) and the full linear system solver (b)

Fig. 6

Error distribution dP (a) and L2 norms of the residual (R2) versus iterations for rigid and softened system based on AF and FLS schemes (b)

Fig. 7

Mesh-sensitivity analysis in Z (a) and X (b) directions

Fig. 8

Schematic of a convergent–divergent bearing under pure sliding motion as in Ref. [25]

Fig. 9

Comparison of the calculated film pressure based on the FLS in the current study versus the LCP based results of Giacopini et al. [25] for case 1-1, (a) and 1-2, (b)

Fig. 10

Pressure distribution of a nonpressured supply groove journal bearing of case 2-1 (a) and for a pressurized supply groove with  Pg=2 π of case 2-2 (b) both at ϵ=0.8 with a physical choice of bulk modulus β=2GPa

Fig. 11

Error in obtaining solution parameters versus choice of bulk modulus, β (a) and pressure variation at midspan with β (b)

Fig. 12

Pressure contours for a high eccentric journal bearing at ϵ=0.95 with pressurized groove at Pg=20 π  and physical bulk modulus of β=2 GPa  (a) and the corresponding convergence plots for a range of eccentricity ratios (b)

Fig. 13

Eigenvalues of the full linear system for a range of lubricant bulk modulus

Fig. 14

Numerical stiffness versus the lubricant bulk modulus (a), eccentricity ratios (b), and number of mesh elements in x (c) and z directions x (d)

Fig. 15

A 4×4  mesh

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