Numerical analysis of steady-state performance of misaligned journal bearings with turbulent effect

  • Subrata DasEmail author
  • Sisir K. Guha
Technical Paper


A theoretical analysis has been carried out to investigate into the effect of turbulence and journal misalignment on the steady-state characteristics of hydrodynamic journal bearings lubricated with micropolar fluid. The governing non-dimensional Reynolds equation applicable to turbulent micropolar lubrication has been solved numerically to obtain the film pressure distribution which was then used to determine the load carrying capacity, attitude angle, misalignment moment, end flow rate and frictional parameter. The turbulent shear coefficients have been computed by using the turbulent model proposed by Ng and Pan. The results suggest that the effect of turbulence is to increase the load carrying capacity and misalignment moment of the misaligned journal bearings, and this effect is more pronounced for micropolar fluid as compared to Newtonian fluid.


Hydrodynamic lubrication Journal bearings Micropolar Misalignment Turbulence 



Radial clearance, m


Constant parameter of turbulent shear coefficient for axial flow


Journal diameter, m


Degree of misalignment, \( D_{m} = {{\xi_{e} } \mathord{\left/ {\vphantom {{\xi_{e} } {\xi_{m} }}} \right. \kern-0pt} {\xi_{m} }} \)


Steady-state eccentricity ratio at the mid plane of the bearing, \( \varepsilon_{0} = {{e_{0} } \mathord{\left/ {\vphantom {{e_{0} } C}} \right. \kern-0pt} C} \)

e′, ɛ

Magnitude of projection of the axis of misaligned journal onto the mid plane of the bearing, \( \varepsilon^{\prime } = {{e^{\prime } } \mathord{\left/ {\vphantom {{e^{\prime } } C}} \right. \kern-0pt} C} \)

\( e_{\hbox{max} }^{\prime } ,\varepsilon_{\hbox{max} }^{\prime } \)

Maximum possible value of e′ and ε′ respectively, \( \varepsilon_{\hbox{max} }^{\prime } = e_{\hbox{max} }^{\prime } /C \)


Frictional parameter, \( f(R/C) = {{\overline{F} } \mathord{\left/ {\vphantom {{\overline{F} } {\overline{W} }}} \right. \kern-0pt} {\overline{W} }} \)


Frictional force, N

\( \overline{F} \)

Non-dimensional frictional force, \( \overline{F} = {{FC^{2} } \mathord{\left/ {\vphantom {{FC^{2} } {\mu\Omega ^{2} R^{3} L}}} \right. \kern-0pt} {\mu\Omega ^{2} R^{3} L}} \)

h, \( \overline{h} \)

Film thickness, \( \overline{h} = {h \mathord{\left/ {\vphantom {h C}} \right. \kern-0pt} C} \)

hcav, \( \bar{h}_{\text{cav}} \)

Film thickness at the point of cavitation, \( \bar{h}_{\text{cav}} = {{h_{\text{cav}} } \mathord{\left/ {\vphantom {{h_{\text{cav}} } C}} \right. \kern-0pt} C} \)

\( k_{\theta } , { }k_{{\bar{z}}} \)

Turbulent shear coefficients in circumferential and axial directions, respectively


Non-dimensional characteristics length of micropolar fluid, lm = C


Bearing length, m

M, \( \overline{M} \)

Resultant misalignment moment, \( \overline{M} = {{MC^{3} } \mathord{\left/ {\vphantom {{MC^{3} } {\mu R^{3} L}}} \right. \kern-0pt} {\mu R^{3} L}} \)

Mi, \( \overline{M}_{i} \)

Misalignment moment, \( \overline{M}_{i} = {{M_{i} C^{3} } \mathord{\left/ {\vphantom {{M_{i} C^{3} } {\mu R^{3} L}}} \right. \kern-0pt} {\mu R^{3} L}} \), i = r- and \( \phi \)- for radial and transverse directions, respectively


Coupling number

p, \( \bar{p} \)

Steady-state film pressure in the film region, \( \bar{p} = {{pC^{2} } \mathord{\left/ {\vphantom {{pC^{2} } {\mu\Omega R^{2} }}} \right. \kern-0pt} {\mu\Omega R^{2} }} \)

Qi, \( \bar{Q}_{i} \)

Steady-state end flow rate, \( \bar{Q}_{i} = {{Q_{i} L} \mathord{\left/ {\vphantom {{Q_{i} L} {C\Omega R^{3} }}} \right. \kern-0pt} {C\Omega R^{3} }} \), i = Rear end, front end and z


Radius of the journal, m

\( Re \)

Mean or average Reynolds number defined by radial clearance, C,\( Re = {{\rho\Omega RC} \mathord{\left/ {\vphantom {{\rho\Omega RC} \mu }} \right. \kern-0pt} \mu } \)


Velocity of journal, U = ΩR, m/s

W, \( \bar{W} \)

Steady-state load in bearing, \( \bar{W} = {{WC^{2} } \mathord{\left/ {\vphantom {{WC^{2} } {\mu\Omega ^{2} R^{3} L}}} \right. \kern-0pt} {\mu\Omega ^{2} R^{3} L}} \)

Wi, \( \bar{W}_{i} \)

Steady-state load in bearing, \( \bar{W}_{i} = {{W_{i} C^{2} } \mathord{\left/ {\vphantom {{W_{i} C^{2} } {\mu\Omega ^{2} R^{3} L}}} \right. \kern-0pt} {\mu\Omega ^{2} R^{3} L}} \), i = r- and \( \phi \)- for radial and transverse directions respectively


Cartesian coordinate axis in the circumferential direction, x = , m

z, \( \bar{z} \)

Cartesian coordinate axis along the bearing axis, \( \bar{z} \, = \, {{ 2z} \mathord{\left/ {\vphantom {{ 2z} L}} \right. \kern-0pt} L} \)


Misalignment Angle

\( \phi_{0} \)

Steady-state attitude angle, rad

\( \Phi _{{\theta ,\bar{z}}} \)

Non-dimensional micropolar fluid functions along circumferential and axial directions


Angular velocity of journal, rad/s


Circumferential coordinate, rad


Misalignment ratio at either bearing ends, ξe = βmL/2C


Maximum possible value of ξe


Angle between the projection of the journal rear centre line onto the mid plane of the bearing and the eccentricity vector



The authors are grateful to Mechanical Engineering Department of Indian Institute of Science and Technology, Shibpur for the continuous encouragement and cooperation in executing this work.


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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringIndian Institute of Engineering Science and TechnologyHowrahIndia

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