Technical Report Wright Research & Development Center
IMPROVED ANALYTICAL CAPABILITIES FOR FOIL AIR BEARINGS
REVISED November 2024
The following is the report for SBIR
Aero Propulsion Laboratory
Wright Research and Development Center
Air Force system Command Wright-Patterson Air Force Base, Ohio 45433-6563
INTRODUCTION
Foil bearings are intended to operate at high speeds. For high speed bearings, in general, the effects of fluid inertia, turbulent flow and heat transfer become important issues in analyzing bearing performance characteristics. For the foil bearings, the thermal elasticity of the flexible foils adds complications to the analysis of the high speed bearings. A competent advanced analytical tool for the foil bearings should take the above-mentioned issues into consideration.
The conventional Reynolds equation approach, for several decades, has been a prevailing method for hydrodynamic lubrication analysis. The basic assumptions of the Reynolds equation are: (1) The Reynolds number is very small. (2) The flow across the fluid film is insignificant. And, (3) the fluid is isothermal. Apparently, these assumptions become inadequate as the Reynolds number is greater than the order of magnitude of one.
There have been technical papers and reports contributed to the understanding of the turbulent lubrication, the inertial and the thermal effects. For the turbulent fluid film, the
mixing-length [1] and the “wall of the law” [2] and [3] are the two most popular approaches. Launder and Leschziner [4] employed the low Reynolds number turbulence model to treat thrust bearings. Later Jones and Launder [5] used turbulent k-ε two equation models for the low Reynolds flows. The k-ε model [6,7] has been proved to be more in line with experimental observations than other turbulence models in various flow situations such as flat plate boundary layers, pipe flows and shear flows of jets. Although the k-ε model adds two more equations, the turbulence kinetic energy, k, and the dissipation rate of turbulent energy, ε, equations to the computation process, it relates the turbulent viscosity to the
local turbulent energy transport and provides a better tool to trace the history of the turbulent flow.
In dealing with the fluid inertia, represented by the convective terms in the momentum equations, most of previous studies integrated the momentum equations across the film gap, a process similar to the derivation of the Reynolds equation. Since the velocity profile is not
known beforehand in film gap, to carry out the integration, a velocity profile is assumed or borrowed from the non-inertial solution.
The runner surface and the bearing (or the foil) surface are not parallel, the inclination of the surfaces results in a transverse velocity component contributed by the runner’s tangential velocity, adding to the flow across the film gap. This transverse velocity component becomes more pronounced as the runner’s speed increases. Consequently, an assumed velocity profile may not be able to reveal the local flow behavior especially for the high speed bearings. The across-film integration, at best, can only catch the velocities at the two ends.
The present study adopts a three dimensional Navier-Stokes approach with the low Reynolds number k-ε turbulence model by Jones and Launder [5] for the turbulent flow case. The Navier-Stokes equations are written in cylindrical coordinates so that the curvature effects including the centrifugal and the coriolis accelerations can be taken into account. The momentum equations, the continuity equation, the energy equation and the k-ε equations
are solved by the SIMPLE (Semi-Implicit-Pressure-Linked-Equations) algorithm developed by Patankar and Spalding [8] and elaborated in the book by Patankar [9].
The thermohydrodynamic solution of the fluid film provides the pressure and the temperature distributions on the foil surfaces. The fluid pressure and the thermal effects cause the deformation of the foils. To analyze the thermoelasticity of the foil, a thin plate model is chosen to simulate the foil. The governing thermoelasticity equation, a fourth order partial differential equation, is solved by finite-difference approximations. The result of the numerical calculation gives the foil deflection which in turn changes the fluid film shape.
The interaction between the thermohydrodynamics of the fluid film and the thermoelasticity of the foil is dealt with combing the two parts of computations and performing iterations to reach a convergent solution.
For the Phase I study, two computer programs are developed, the journal bearing and the thrust bearing programs. The two kinds of bearings have quite different geometries. Therefore, the equations for the two bearings are not the same. these two programs can be
used to analyze the steady-state performances of the two kinds of bearings including the effects of inertia, the heat transfer and the turbulent flow. The programs can be utilized for the foil bearing as well as the solid-walled bearing analysis. The latter, is simply a special case as far as the program capability is concerned.
Computational examples are presented for both the thrust bearing and the journal bearings. The performance characteristics, such as the load carrying capacity, the frictional coefficient and the attitude angles (for the journal bearing) are calculated. Results are compared with
available solid-walled bearing data, [10] and [11]. Comparisons are also used to verify the accuracy and to reveal the inertial, the thermal and the turbulent flow effects.
The Navier-Stokes approach differs from the Reynolds equation approach not only in solving more complicated differential equations and realizing the above-mentioned three effects, but also in providing and advanced tool for visualizing how the fluid enter the bearing and how the flow behaves locally. The continuation of this study such as a Phase II study should focus on the transient case and the combination of the fluid dynamics and the
rotor dynamics. Hence, a better picture of the dynamic behaviors of the rotor-bearing can be obtained. The Phase I study has achieved the goal of establishing the foundation for pursuing further studies.
ANALYSIS
Basically, the fluid film in either a journal bearing or a thrust bearing is equivalent to the flow between two inclined surfaces. For the journal bearing, the flow is between two
eccentric cylinders, while for the thrust bearing, it is between two unparallel disks. To describe the three dimensional flow behaviors and to consider the curvature effects including the centrifugal and the coroilis forces, cylindrical coordinates are chosen to derive the governing equations. The main geometrical difference between the journal bearing and the thrust bearing lies in the direction of the film gap; for the former, the film gap is in the radial (r) direction, while for the latter, it is in the axial (z) direction. Nevertheless, both bearings have a common feature that is a main flow in the circumferential (θ) direction. Due to this dominating flow, the parabolization concept can be applied to the Navier-Stokes
equations and the so-called marching technique can be employed in the numerical computation.
Additionally, because of the eccentricity of the cylinders, in the journal bearing case and the unparallelness of the disks in the thrust bearing case, the main flow, Vθ , has components in the other two directions, i.e., the r and z directions. Furthermore, the shaft misalignment, if present in the journal bearing and the foil deflection will modify fluxes contributed by Vθ to the r and z directions. Consequently, film-shape fitted coordinates transformations are
very much desired to enable the realization of these additional fluxes in both of the r and z directions.
In the following, formulations coordinate transformations and numerical solution procedures will be elaborated: