A computational algorithm for cavitating bearings Requirement of boundary conditions which satisfy the principle of mass conservation W.B. Rowe and F.S. Chong* Many boundary conditions for solving lubrication problems can be found in the literature, among them the Swift-Stieber condition (film-breakdown), which is most frequently used by lubrication engineers, and the Jakobsson- Floberg condition (film-reformation), which is often not used because of the computing complexity involved. Most of these conditions, including the former, do not take proper account of mass conservation within the cavi- tated region and give rise to erroneous flow-rate predictions. Therefore, there is a need to develop a computational algorithm that does not violate the principle of mass conservation (like the Jakobsson--F Ioberg boundary condition) and yet is capable of being easily and economically accommo- dated into an existing bearing-program. This paper presents such a technique, and one which is generally applicable to a spectrum of liquid-film bearings, such as hydrodynamic, hydrostatic or hybrid bearings. The analysis demon- strates the importance of preserving mass conservation throughout the bearing fluid-film, especially for bearings operating at high speeds and low feed pressure-ratios. Furthermore, the necessity of a global check of the /7 n continuity requirement (ie ~ Aein = ~ AOout) is demonstrated. i=1 i=1 Keywords: bearings + cavitation, algorithms, mathematical analysis Since Reynolds I derived the differential equation governing the pressure distribution in a thin fluid-filrn, much work has been devoted to solving the equation, particularly the application of the correct boundary conditions. In most journal bearings, the fluid-film may be identified by two distinctive regions, the pressure region (full-f'dm) and the cavitation region. Therefore, once an equation governing the pressure distribution has been derived, the lubrication problem is basically a problem of: • determining the boundaries between the two regions (the film-breakdown and fdm-reformation boundaries), and • determining the pressure distribution within the pressure region. Numerous boundary conditions have been postulated for use with the Reynolds equation, such as the Sommerfeld, 2 Half-Sommerfeld or Gtimbel-Everling, a Swift-Stieber or Reynolds, 1,4's Jakobsson-Floberg, 6 Birkhoff-Hays 7 and Coyne-Ekod. a'9 The Swift-Stieber boundary condition (p = dp/dO = 0 at the breakdown boundary) is the most widely used and gives reasonably accurate static results, although the predicted flows may be in error by a large margin. The application of the Swift-Stieber boundary condition to the breakdown boundary without considera- tion of the reformation boundary leads to a violation of the principle of mass conservation in the cavitating region, *Department o f Mechanical, Marine and Production Engineering, Liverpool Polytechnic, Byrom Street, Liverpool, L3 3AF, UK and has consequences for the predicted flow and, in certain conditions, for the accuracy of the load and attitude angle. A common procedure is to put all negative pressures to zero, which does not correctly take account of the mass flow from the breakdown boundary towards the reforma- tion boundary. Although, under steady-running conditions, the position of the upstream boundary (film-breakdown) is widely accepted as the position where the pressure and pressure derivative vanishes, the downstream boundary (reforma- tion boundary) is somewhat more comphcated and can only be established with reasonable accuracy by prescrib- ing the condition of mass conservation throughout the cavitating region. The position of the reformation boundary is strongly influenced by the location of the oil groove (source) as well as by the supply pressure. Therefore, the reformation boundary is neither located at the maximum f'flm thickness nor does it start with a zero pressure derivative. Jakobsson and Floberg 6 were the first to treat the refor- mation boundary correctly. From a consideration of the continuity of flow from the breakdown boundary to the reformation boundary, they postulated a new boundary condition governing the reformation of oil film pressure, and also suggested a procedure for implementing the reformation boundary condition in association with the Reynolds equation. Due to the computing complexity involved, the Jakobsson-Floberg boundary condition is seldom used. Among all the boundary conditions that are TRIBOLOGY international 0301-679X/84/050243-08 $03.00 © 1984 Butterworth & Co (Publishers) Ltd 243