A Semi-Analytical Method for the Solution of Entrance Flow Effects in Inherently Restricted Aerostatic Bearings

Abstract

The behaviour of aerostatic bearings is strongly influenced by the entrance flow effects nearby feeding sources. A basic understanding of these flow phenomena and accurate prediction of their relevant parameters are therefore essential in the design and optimisation process of any aerostatic bearing application. The subject matter of this paper has for long been the topic of extensive research. An overview of the different approaches found in literature shows mostly methods based on empirical formulas with a validity limited to the experimental conditions that produced them. The proposed solution method uses the technique of separation of variables to convert the boundary-layer equations describing the laminar channel flow into an initial value problem. This allows the exact calculation of the pressure profile from gap entrance up to the attainment of viscous flow. Knowledge of the pressure distribution near the gap entrance together with the mass flow rate leads to the determination of an expedient coefficient of discharge, allowing a more practical lumped parameter formulation. The results are compared with experimental data from various sources and the agreement is found to be remarkably good, indicating that a laminar flow model is adequate in characterising the entrance flow over a wide range of working parameters.

An Identification Method for Orifice-Type Restrictors Based on the Closed-Form Solution of Reynolds Equation

“…Figure 10 shows one of the experimental pressure profiles measured through the procedure described in Section 2.3. Here, as discussed in Section 1, it is possible to theoretically define the feed, entrance and viscous regions [3]. The proposed method is based on two main aspects:…”

“…On the basis of these two aspects, the authors' insight was to combine the measured experimental data with the analytical expression obtained for circular and centrally fed aerostatic bearings considering a laminar and isothermal fluid flow. The equations that are used in the proposed procedure are reported below (for further details on these equations readers can refer to Appendix A): Here, as discussed in Section 1, it is possible to theoretically define the feed, entrance and viscous regions [3]. The proposed method is based on two main aspects:…”

“…Among these, the prediction of the entrance flow effects near the bearing feeding sources is one of the most crucial [2]. It is well known that the flow field in the air gap film may be simplified by considering three subregions depending on their different characteristics [3,4]. Starting from the plenum chamber of the bearing, it is possible to distinguish a feed, entrance and a viscous flow region.…”

“…Along with CFD simulations, solving the laminar boundary layer equation through the method of "separation of variable" (MSV), proposed by Al-Bender and Van Brussel [14], is another numerical approach that was exploited to identify the discharge coefficients of aerostatic bearing restrictors. Through this method, Waumans et al [3] computed the discharge coefficients for different restrictors and air gap geometries. They used the pressure profile computed through the MSV method to identify the theoretical downstream pressures p th which are used to compute discharge coefficients.…”

Lumped parameters models of rectangular pneumatic pads: Static analysis

“…Recently, computational fluid dynamics (CFD) was used jointly with experimental activities to improve the description of the flow field near the supply holes [19][20][21][22][23]. In [24,25], by contrast, semi-analytical methods were employed to determine orifice discharge coefficients, which were then used in analytical formulations.…”