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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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## References

Temperley, H. , and Trevena, D. , 1977, “ Metastability of Phase Transitions and the Tensile Strength of Liquids,” Proc. R. Soc. London, Ser. A, 357(1690), pp. 395–402.
Braun, M. , and Hannon, W. , 2010, “ Cavitation Formation and Modelling for Fluid Film Bearings: A Review,” Proc. Inst. Mech. Eng., Part J, 224(9), pp. 839–863.
Brewe, D. E. , 1986, “ Theoretical Modeling of the Vapor Cavitation in Dynamically Loaded Journal Bearings,” ASME J. Tribol., 108(4), pp. 628–637.
Szeri, A. Z. , 2011, Fluid Film Lubrication, Cambridge University Press, Cambridge, UK.
Christopherson, D. G. , 1941, “ A New Mathematical Method for the Solution of Film Lubrication Problems,” Proc. Inst. Mech. Eng., 146(1), pp. 126–135.
Ausas, R. , Ragot, P. , Leiva, J. , Jai, M. , Bayada, G. , and Buscaglia, G. C. , 2007, “ The Impact of the Cavitation Model in the Analysis of Microtextured Lubricated Journal Bearings,” ASME J. Tribol., 129(4), pp. 868–875.
Ausas, R. F. , Jai, M. , and Buscaglia, G. C. , 2009, “ A Mass-Conserving Algorithm for Dynamical Lubrication Problems With Cavitation,” ASME J. Tribol., 131(3), p. 031702.
Jakobsson, B. , and Floberg, L. , 1957, The Finite Journal Bearing Considering Vaporization (Transactions of Chalmers University of Technology), Vol. 190, Guthenberg, Sweden, p. 308.
Olsson, K. O. , 1965, Cavitation in Dynamically Loaded Bearing (Transactions of Chalmers University of Technology), Vol. 308, Guthenberg, Sweden, p. 308.
Etsion, I. , and Ludwig, L. , 1982, “ Observation of Pressure Variation in the Cavitation Region of Submerged Journal Bearings,” ASME J. Lubr. Technol., 104(2), pp. 157–163.
Elrod, H. , and Adams, M. , 1974, “ A Computer Program for Cavitation and Starvation Problems,” Cavitation and Related Phenomena in Lubrication, Mechanical Engineering Publications, New York, pp. 37–41.
Elrod, H. G. , 1981, “ A Cavitation Algorithm,” ASME J. Tribol., 103(3), pp. 350–354.
Miranda, A. , 1983, “ Oil Flow, Cavitation and Film Reformation in Journal Bearings, Including an Interactive Computer-Aided Design Study,” Ph.D. thesis, University of Leeds, Leeds, UK.
Hirani, H. , Athre, K. , and Biswas, S. , 2001, “ A Simplified Mass Conserving Algorithm for Journal Bearing Under Large Dynamic Loads,” Int. J. Rotating Mach., 7(1), pp. 41–51.
Vijayaraghavan, D. , and Keith, T. , 1990, “ An Efficient, Robust, and Time Accurate Numerical Scheme Applied to a Cavitation Algorithm,” ASME J. Tribol., 112(1), pp. 44–51.
Woods, C. M. , and Brewe, D. E. , 1989, “ The Solution of the Elrod Algorithm for a Dynamically Loaded Journal Bearing Using Multigrid Techniques,” ASME J. Tribol., 111(2), pp. 302–308.
Qiu, Y. , and Khonsari, M. , 2009, “ On the Prediction of Cavitation in Dimples Using a Mass-Conservative Algorithm,” ASME J. Tribol., 131(4), p. 041702.
Vijayaraghavan, D. , and Keith, T. , 1990, “ Grid Transformation and Adaption Techniques Applied in the Analysis of Cavitated Journal Bearings,” ASME J. Tribol., 112(1), pp. 52–59.
Almqvist, A. , Fabricius, J. , Larsson, R. , and Wall, P. , 2014, “ A New Approach for Studying Cavitation in Lubrication,” ASME J. Tribol., 136(1), p. 011706
Bayada, G. , and Chupin, L. , 2013, “ Compressible Fluid Model for Hydrodynamic Lubrication Cavitation,” ASME J. Tribol., 135(4), p. 041702.
Bertocchi, L. , Dini, D. , Giacopini, M. , Fowell, M. T. , and Baldini, A. , 2013, “ Fluid Film Lubrication in the Presence of Cavitation: A Mass-Conserving Two-Dimensional Formulation for Compressible, Piezoviscous and Non-Newtonian Fluids,” Tribol. Int., 67, pp. 61–71.
Sahlin, F. , Almqvist, A. , Larsson, R. , and Glavatskih, S. , 2007, “ A Cavitation Algorithm for Arbitrary Lubricant Compressibility,” Tribol. Int., 40(8), pp. 1294–1300.
Bayada, G. , 2014, “ From a Compressible Fluid Model to New Mass Conserving Cavitation Algorithms,” Tribol. Int., 71, pp. 38–49.
Braun, M. , and Hendricks, R. , 1984, “ An Experimental Investigation of the Vaporous/Gaseous Cavity Characteristics of an Eccentric Journal Bearing,” ASLE Trans., 27(1), pp. 1–14.
Giacopini, M. , Fowell, M. T. , Dini, D. , and Strozzi, A. , 2010, “ A Mass-Conserving Complementarity Formulation to Study Lubricant Films in the Presence of Cavitation,” ASME J. Tribol., 132(4), p. 041702.
Vijayaraghavan, D. , and Keith, T., Jr. , 1989, “ Development and Evaluation of a Cavitation Algorithm,” Tribol. Trans., 32(2), pp. 225–233.
Ståhl, J. , and Jacobson, B. O. , 2003, “ Compressibility of Lubricants at High Pressures,” Tribol. Trans., 46(4), pp. 592–599.
Rao, T. , and Sawicki, J. T. , 2002, “ Linear Stability Analysis for a Hydrodynamic Journal Bearing Considering Cavitation Effects,” Tribol. Trans., 45(4), pp. 450–456.
Yang, L.-H. , Wang, W.-M. , Zhao, S.-Q. , Sun, Y.-H. , and Yu, L. , 2014, “ A New Nonlinear Dynamic Analysis Method of Rotor System Supported by Oil-Film Journal Bearings,” Appl. Math. Modell., 38(21), pp. 5239–5255.
Ceze, M. , and Fidkowski, K. J. , 2015, “ Constrained Pseudo-Transient Continuation,” Int. J. Numer. Methods Eng., 102(11), pp. 1683–1703.
Fesanghary, M. , and Khonsari, M. , 2011, “ A Modification of the Switch Function in the Elrod Cavitation Algorithm,” ASME J. Tribol., 133(2), p. 024501.
Celik, I. B. , Ghia, U. , and Roache, P. J. , 2008, “ Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications,” ASME J. Fluids Eng., 130(7), p. 078001.
Jalali, A. , Sharbatdar, M. , and Ollivier-Gooch, C. , 2014, “ Accuracy Analysis of Unstructured Finite Volume Discretization Schemes for Diffusive Fluxes,” Comput. Fluids, 101, pp. 220–232.

## 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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