Linearized turbulent lubrication equation for finite self-acting porous bearings considering slip velocity over a hydrodynamically rough permeable boundary

Abstract

A first attempt to derive a new lubrication equation for the hydrodynamic pressure generation in a three-dimensional self-acting porous metal bearing operating in a turbulent regime (fully developed) with single-phase Newtonian incompressible lubricant is presented considering simultaneously the slip velocity and the hydrodynamically rough permeable surface (in terms of the median diameter of the graded constituent particles of the sintered porous matrix and their standard deviation) with a hydrodynamically rough rotor. The analysis is analogous to boundary layer theory wherein the lubricant motion is treated as a generalized turbulent channel flow in which the rough porous wall is stationary and the impervious wall (also considered rough) is in longitudinal motion so that the flow is a turbulent shear flow coupled with codirectional and transverse turbulent pressure flow. A linearization or perturbation technique is used to decouple the two orthogonal turbulent flows by assuming that the turbulent shear stress in a finite bearing is a small perturbation of the shear stress for turbulent couette flow. Using Boussinesq's eddy viscosity formulation and the experimentally established logarithmic law as the universal law of wall, the governing pressure distribution equation is obtained from consideration of the conservation of momentum and continuity. The whole treatment is approached from the point of view of fluid film design. No ad hoc slip model is used, and the lubrication equation is fully analytical and can be applied to a number of particular bearing problems by applying suitable simplifying restrictive conditions. The derived lubrication equation can be used to predict bearing performance characteristics even in situations where the bearing permeability is non-isotropic.