K. A. Cliffe

Three-dimensional thin film flow over and around an obstacle on an inclined plane

Steady Stokes flow driven by gravity down an inclined plane over and around an attached obstacle is considered. The effects of the obstacle are examined for various flow configurations and results produced for flow over hemispherical obstacles. Comparison is made with previously published papers that assume that the obstacle is small and/or the free surface deflection and disturbance velocity are small. Values for the unit normal and curvature of the free surface are found using both finite difference approximations and Hermitian radial basis function interpolations, with the resulting solutions compared. Free surface profiles for thin film flows over hemispherical obstacles that approach the film surface are produced and the effects of near point singularities considered. All free surface profiles indicate an upstream peak, followed by a trough downstream of the obstacle with the peak decaying in a “horseshoe” shaped surface deformation. Flow profiles are governed by the plane inclination, the Bond numbe...

A compressible flow model for the air-rotor–stator dynamics of a high-speed, squeeze-film thrust bearing

A compressible air-flow model is introduced for the thin film dynamics of a highly rotating squeeze-film thrust bearing. The lubrication approximation to the Navier-Stokes equations for compressible flow leads to a modified Reynolds equation incorporating additional rotation effects. To investigate the dynamics of the system, the axial position of the bearing stator is prescribed by a finite-amplitude periodic forcing. The dynamics of the squeeze-film are modelled in the uncoupled configuration where the axial position of the rotor is fixed. The coupled squeeze-film bearing dynamics are investigated when the axial position of the rotor is modelled as a spring-mass-damper system that responds to the film dynamics. Initially the uncoupled squeeze-film dynamics are considered at low operating speeds with the classical Reynolds equation for compressible flow. The limited value of the linearized small-amplitude results is identified. Analytical results indicate that finite-amplitude forcing needs to be considered to gain a complete understanding of the dynamics. Using a Fourier spectral collocation numerical scheme, the periodic bearing force is investigated as a nonlinear function of the frequency and amplitude of the stator forcing. High-speed bearing operation is modelled using the modified Reynolds equation. A steady-state analysis is used to identify the effect of rotation and the rotor support properties in the coupled air-flow-structure model. The unsteady coupled dynamics are computed numerically to determine how the rotor support structures and the periodic stator forcing influence the system dynamics. The potential for resonant rotor behaviour is identified through asymptotic and Fourier analysis of the rotor motion for small-amplitude, low-frequency oscillations in the stator position for key values of the rotor stiffness. Through the use of arclength continuation, the existence of resonant behaviour is identified numerically for a range of operating speeds and forcing frequencies. Changes in the minimum rotor-stator clearance are presented as a function of the rotor stiffness to demonstrate the appearance of resonance.