Baxter, Steven J.
(2010)
Numerical simulation of three-dimensional free surface film flow over or around obstacles on an inclined plane.
PhD thesis, University of Nottingham.
Abstract
Within the bearing chamber of a gas turbine aero-engine, lubrication of the shaft and other bearings is achieved by an oil film which may become significantly disturbed by interacting with a range of chamber geometries which protrude from the chamber wall. Minimizing these disturbances and preventing possible dry areas is crucial in optimizing a bearing chambers design. In addition, multiple obstructions may be located close to one another, resulting in a more complex disturbed film profile than by individual obstacles. Prediction of the disturbance of the film is an important aspect of bearing chamber design.
For analysis of the film profile over or around a local obstacle, typical bearing chamber flows can be approximated as an incompressible thin film flow down an inclined wall driven by gravity. The Reynolds number of thin film flows is often small, and for the bulk of this thesis a Stokes flow assumption is implemented. In addition, thin films are often dominated by surface tension effects, which for accurate modelling require an accurate representation of the free surface profile. Numerical techniques such as the volume of fluid method fail to track the surface profile specifically, and inaccuracies will occur in applying surface tension in this approach. A numerical scheme based on the boundary element method tracks the free surface explicitly, alleviating this potential error source and is applied throughout this thesis. The evaluation of free surface quantities, such as unit normal and curvature is achieved by using a Hermitian radial basis function interpolation. This hermite interpolation can also be used to incorporate the far field boundary conditions and to enable contact line conditions to be satisfied for cases where the obstacle penetrates the free surface.
Initial results consider a film flowing over an arbitrary hemispherical obstacle, fully submerged by the fluid for a range of flow configurations. Comparison is made with previously published papers that assume the obstacle is small and / or the free surface deflection and disturbance velocity is small. Free surface profiles for thin film flows over hemispherical obstacles that approach the film surface are also 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 number and the obstacle geometry; effects of these key physical parameters on flow solutions are provided.
The disturbed film profiles over multiple obstacles will differ from the use of a single obstacle analysis as their proximity decreases. An understanding of the local interaction of individual obstacles is an important aspect of bearing chamber design. In this thesis the single obstacle analysis is extended to the case of flow over multiple hemispheres. For obstacles that are separated by a sufficiently large distance the flow profiles are identical to those for a single obstacle. However, for flow over multiple obstacles with small separation, variations from single obstacle solutions maybe significant. For flow over two obstacles placed in-line with the incident flow, variations with flow parameters are provided. To identify the flexibility of this approach, flows over three obstacles are modelled.
The calculation of flows around obstacles provides a greater challenge. Notably, a static contact line must be included such that the angle between the free surface and the obstacle is introduced as an extra flow parameter that will depend both on the fluid and the obstacle surface characteristics. The numerical models used for flow over hemispheres can be developed to consider film flow around circular cylinders. Numerical simulations are used to investigate flow parameters and boundary conditions. Solutions are obtained where steady flow profiles can be found both over and around a cylindrical obstacle raising the awareness of possible multiple solutions.
Flow around multiple obstacles is also analyzed, with profiles produced for flow around two cylinders placed in various locations relative to one another. As for flow over two hemispheres, for sufficiently large separations the flow profiles are identical to a single obstacle analysis. For flow around two obstacles spaced in the direction of the flow, effects of altering the four governing parameters; plane inclination angle, Bond number, obstacle size, and static contact angle are examined. The analysis of flow around three cylinders in two configurations is finally considered. In addition, for two obstacles spaced in-line with the incident flow, the numerical approaches for flow over and flow around are combined to predict situations where flow passes over an upstream cylinder, and then around an identical downstream cylinder.
The final section of this thesis removes the basic assumption of Stokes flow, through solving the full Navier-Stokes equations at low Reynolds number and so incorporating the need to solve nonlinear equations through the solution domain. An efficient numerical algorithm for including the inertia effects is developed and compared to more conventional methods, such as the dual reciprocity method and particular integral techniques for the case of a three-dimensional lid driven cavity. This approach is extended to enable calculation of low Reynolds number film profiles for both flow over and around a cylinder. Results are compared to the analysis from previous Stokes flow solutions for modest increases in the Reynolds number.
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