Skip to main content
SpringerLink
Log in

Refined Hydrodynamic Lubrication Model with a High-Accuracy Numerical Algorithm for Starved Lubrication with Free Surfaces

  • Original Paper
  • Published:
Tribology Letters Aims and scope Submit manuscript

Abstract

In this study, we propose a refined hydrodynamic lubrication model that incorporates a high-accuracy numerical algorithm for starved lubrication involving free surfaces. The performance of the proposed method is compared against that of the conventional model and numerical scheme. The results indicate that the Elrod-Adams model underestimates the hydrodynamic pressure due to unrealistic assumptions regarding the mass transport velocity of the lubricant beyond the contact region. Furthermore, the conventional numerical scheme fails to correctly predict the location of the rupture point and loses accuracy under conditions of severe starvation or high sliding speeds. In contrast, our proposed model ensures the conservation of mass flow between the outlet and inlet, while the new numerical algorithm restores the accuracy of the results.

Graphical Abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data Availability

Data generated during this study are available from the corresponding authors on reasonable request.

Abbreviations

A:

Location of reformation point

B:

Location of rupture point

C 1 i :

Coefficient

C 2 i :

Coefficient

F :

 = θH

G :

\(= - F\frac{\partial \alpha }{{\partial X}} + \frac{1}{{\Lambda }}\frac{\partial }{\partial X}\left( {H^3 \frac{\partial P}{{\partial X}}} \right)\)

H :

Film thickness (m)

h s :

Initial supplied film thickness (m)

h 0 :

Minimum film thickness (m)

h 1 :

\(= 4\delta x^2 /l^2 (m)\)

H :

Dimensionless film thickness, H = h/h0

L :

Length of upper surface (m)

l c :

Length of calculation zone (m)

M :

Mesh number

N T :

Time step number

P :

Pressure of fluid film (Pa)

p a :

Atmospheric pressure (Pa)

P :

Dimensionless fluid film pressure, P = p/pa

Q 1 i :

\({{H_{i + \frac{1}{2}}^3 }}/{(\Lambda \Delta X^2) }\)

Q 2 i :

\({{H_{i - \frac{1}{2}}^3 }}/{(\Lambda \Delta X^2) }\)

Q 3 i :

\(= (\alpha_{i + 1}^n - \alpha_{i - 1}^n )/(2{\Delta }X)\)

Q 4 i :

\(= Q_{1i} + Q_{2i}\)

T :

Time (s)

t 0 :

 = l/u, Unit time (s)

T :

 = t/t0, Dimensionless time

u :

Sliding speed (m/s)

v A :

Velocity of reformation point (m/s)

v B :

Velocity of rupture point (m/s)

V A = :

vA/u, Dimensionless velocity of reformation point

V B :

 = vB/u, Dimensionless velocity of rupture point

x :

coordinate in sliding direction

X :

Dimensionless coordinate, X = x/l

w :

Load-carrying capacity (N/m)

W :

 = w/(p0l), Dimensionless load-carrying capacity

α :

Mass transport velocity

δ :

Height of the upper surface (m)

Δ t :

 = Δx/(2u), Time step

ΔT :

 = Δt/t0, Dimensionless time step

Δ x :

 = l/m, Mesh size

ΔX :

 = 1/m, Dimensionless mesh size

ε :

Convergence tolerance

η :

Viscosity (Pa·s)

θ :

Saturation variable

ρ :

Cell-integrated quantity

ρ :

 = ∫f dx

Ξ :

 = -αT

Ωf :

Fully filled region

Ωp :

Partially filled region

Ωt :

Transition region

Λ :

 = 12ηlu/(pAh02)

-:

Left-hand limit

 + :

Right-hand limit

0:

Initial condition

n :

Time step

k :

Iteration step

*:

Temporary value

i :

Node index in x-direction

Iup:

\(= i - sgn\left( {\alpha_i } \right)\)

Icell:

 = i—½

A :

Reformation point

B :

Rupture point

References

  1. Elrod, H.G.: A cavitation algorithm. ASME J. Lubr. Technol. 103, 350–354 (1981). https://doi.org/10.1115/1.3251669

    Article  Google Scholar 

  2. Fowell, M.T., Medina, S., Olver, A.V., Spikes, H.A., Pegg, I.G.: Parametric study of texturing in convergent bearings. Tribol. Int. 52, 7–16 (2012). https://doi.org/10.1016/j.triboint.2012.02.013

    Article  Google Scholar 

  3. Yagi, K., Sugimura, J.: Performance of balancing wedge action in textured hydrodynamic pad bearings. ASME J. Tribol. (2017). https://doi.org/10.1115/1.4033128

    Article  Google Scholar 

  4. Jaramillo, A., Buscaglia, G.C.: An extended Elrod-Adams model to account for backpressure and blow-by inception. Tribol. Int. (2019). https://doi.org/10.1016/j.triboint.2019.105873

    Article  Google Scholar 

  5. Vladescu, S.C., Ciniero, A., Tufail, K., Gangopadhyay, A., Reddyhoff, T.: Looking into a laser textured piston ring-liner contact. Tribol. Int. 115, 140–153 (2017). https://doi.org/10.1016/j.triboint.2017.04.051

    Article  Google Scholar 

  6. Yagi, K., Matsunaka, W., Sugimura, J.: Impact of textured surfaces in starved hydrodynamic lubrication. Tribol. Int. (2020). https://doi.org/10.1016/j.triboint.2020.106756

    Article  Google Scholar 

  7. Venner, C.H.: Multilevel solution of the EHL line and point contact problems. PhD Thesis, University of Twente, Endschende (1991) https://doi.org/10.3990/1.9789090039749

  8. Giacopini, M., Fowell, M.T., Dini, D., Strozzi, A.: A mass-conserving complementarity formulation to study lubricant films in the presence of cavitation. ASME J. Tribol. (2010). https://doi.org/10.1115/1.4002215

    Article  Google Scholar 

  9. Woloszynski, T., Podsiadlo, P., Stachowiak, G.W.: Efficient solution to the cavitation problem in hydrodynamic lubrication. Tribol. Lett. 58, 18 (2015). https://doi.org/10.1007/s11249-015-0487-4

    Article  Google Scholar 

  10. Mezzadri, F., Galligani, E.: Splitting methods for a class of horizontal linear complementarity problems. J. Optim. Theory Appl. 180, 500–517 (2019). https://doi.org/10.1007/s10957-018-1395-1

    Article  Google Scholar 

  11. Holmes, M.J., Evans, H.P., Snidle, R.W., Zhu, D.: Discussion/Author’s reply. Tribol. Trans. 46, 282–288 (2003). https://doi.org/10.1080/10402000308982627

    Article  CAS  Google Scholar 

  12. Liu, Y., Wang, Q.J., Wang, W., Hu, Y., Zhu, D.: Effects of differential scheme and mesh density on EHL film thickness in point contacts. ASME J. Tribol. 128, 641–653 (2006). https://doi.org/10.1115/1.2194916

    Article  Google Scholar 

  13. Gümbel, L.: Das Problem der lagerreibung. Mbl. Berlin. Bez. Ver. Dtsch. Ing. 5(87–104), 109–120 (1914)

    Google Scholar 

  14. Swift, H.W.: The stability of lubricating films in journal bearings. Min. Proc. Inst. Civil. Eng. 233, 267–288 (1932). https://doi.org/10.1680/imotp.1932.13239

    Article  Google Scholar 

  15. Stieber, W.: Das Schwimmlager: Hydrodynamische theorie des gleitlagers. VDI-Verlag, Berlin (1933)

    Google Scholar 

  16. Ausas, R., Ragot, P., Leiva, J., Jai, M., Bayada, G., Buscaglia, G.C.: The impact of the cavitation model in the analysis of microtextured lubricated journal bearings. ASME J. Tribol. 129, 868 (2007)

    Article  Google Scholar 

  17. Qiu, Y., Khonsari, M.M.: On the prediction of cavitation in dimples using a mass-conservative algorithm. ASME J. Tribol. (2009). https://doi.org/10.1115/1.3176994

    Article  Google Scholar 

  18. Jakobsson, B., Floberg, L.: The finite journal bearing considering vaporization Trans. Chalmers Univ. Technol. 190, 1–117 (1957)

    Google Scholar 

  19. Olsson, K. O.: Cavitation in dynamically loaded bearings. Transcript 26, Transactions of Chalmers University of Technology. Sweden. (1965)

  20. Zhang, J., Meng, Y.: Direct observation of cavitation phenomenon and hydrodynamic lubrication analysis of textured surfaces. Tribol. Lett. 46, 147–158 (2012). https://doi.org/10.1007/s11249-012-9935-6

    Article  CAS  Google Scholar 

  21. Yagi, K., Sato, H., Sugimura, J.: On the magnitude of load-carrying capacity of textured surfaces in hydrodynamic lubrication. Tribol. Online 10(3), 232–245 (2015). https://doi.org/10.2474/trol.10.232

    Article  Google Scholar 

  22. Elrod, H.G., Adams, M.L.: A computer program for cavitation and starvation problems. In: Proc First Leeds-Lyon Symposium on Cavitation and Related Phenomena in Lubrication, Leeds, UK 37–41 (1974)

  23. Ma, M.-T., Sherrington, I., Smith, E.H.: Implementation of an algorithm to model the starved lubrication of a piston ring in distorted bores: prediction of oil flow and onset of gas blow-by. Proc. Inst. Mech. Eng. Part J. J. Eng. Tribol. 210, 29–44 (1996)

    Google Scholar 

  24. Liu, L., Tian, T.: Implementation and improvements of a flow continuity algorithm in modeling ring/liner lubrication. SAE paper. 01, 1642 (2005)

    Google Scholar 

  25. Buscaglia, G.C., Ciuperca, I., Dalissier, E., Jai, M.: A new cavitation model in lubrication: the case of two-zone cavitation. J. Eng. Math. 83, 57–79 (2013). https://doi.org/10.1007/s10665-013-9629-6

    Article  CAS  Google Scholar 

  26. Dowson, D., Taylor, C.M.: Cavitation in bearings. Annu. Rev. Fluid Mech. 11, 35–65 (1979). https://doi.org/10.1146/annurev.fl.11.010179.000343

    Article  Google Scholar 

  27. Organisciak, M., Cavallaro, G., Lubrecht, A.A.: Starved hydrodynamic lubrication of the piston ring cylinder liner contact: preliminary study of the influence of surface texturing. In: Proc. 31st Leeds-Lyon Symp. Tribol., Leeds, UK, Tribology and Interface Engineering Series, 48, 573–583 Elsevier B.V., Amsterdam (2005). https://doi.org/10.1016/S0167-8922(05)80059-8

  28. Ausas, R.F., Jai, M., Ciuperca, I.S., Buscaglia, G.C.: Conservative one-dimensional finite volume discretization of a new cavitation model for piston-ring lubrication. Tribol. Int. 57, 54–66 (2013). https://doi.org/10.1016/j.triboint.2012.07.003

    Article  Google Scholar 

  29. Yabe, T., Aoki, T.: A universal solver for hyperbolic equations by cubic-polynomial interpolation I. One-dimensional solver. Comput. Phys. Commun. 66, 219–232 (1991). https://doi.org/10.1016/0010-4655(91)90071-r

    Article  Google Scholar 

  30. Yabe, T., Tanaka, R., Nakamura, T., Xiao, F.: An exactly conservative semi-lagrangian scheme (CIP–CSL) in one dimension. Mon. Weather Rev. 129(2), 332–344 (2001). https://doi.org/10.1175/1520-0493(2001)129%3C0332:AECSLS%3E2.0.CO;2

    Article  Google Scholar 

Download references

Funding

The funding was provided by the Research Committee RC282 of the Japan Society of Mechanical Engineers.

Author information

Authors and Affiliations

  1. Department of Hydrogen Energy Systems, Graduate School of Engineering, Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka, 819-0395, Japan

    Ke Zhang

  2. Department of Mechanical Engineering, Faculty of Engineering, International Institute for Carbon-Neutral Energy Research (WPI-I2CNER), Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka, 819-0395, Japan

    Kazuyuki Yagi

Authors
  1. Ke Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kazuyuki Yagi
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

K.Y. formulated the research concept and supervised the study. K.Z. developed the software for conducting the numerical simulation. K.Z. and K.Y. prepared the manuscript, including the texts, figures, and tables. K.Z. and K.Y. contributed to and approved the discussion of the results.

Corresponding authors

Correspondence to Ke Zhang or Kazuyuki Yagi.

Ethics declarations

Competing Interests

The authors have no competing financial interests or personal relationships that influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, K., Yagi, K. Refined Hydrodynamic Lubrication Model with a High-Accuracy Numerical Algorithm for Starved Lubrication with Free Surfaces. Tribol Lett 71, 122 (2023). https://doi.org/10.1007/s11249-023-01789-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11249-023-01789-2

Keywords

Search

Navigation