Development of effective spectral methodologies for transient two-dimensional and three-dimensional nonlinear heat and fluid flow in complex domains

The study explores the applicability of the spectral method in heat transfer, and fluid flow problems in complex or periodic geometry usually found in the material processing industry. In the case of heat transfer, emphasis is on the heat conduction problems in periodic geometry with periodic boundary conditions. Both steady-state and transient heat conduction are considered. The studies are of relevance to coating, lubrication, and injection molding of thin parts in the case of fluid flow problems. The main advantage of the spectral method over conventional numerical method is its high, often exponential convergence order and relatively few grid points are needed. This makes the spectral method particularly attractive for multidimensional problems with high accuracy requirements arising in computational fluid dynamics and heat transfer.; Steady-state and transient linear heat conduction problems with periodic boundary conditions and periodic geometry, as well as nonlinear heat conduction are examined. The study consists of first mapping the complex geometry onto a rectangular domain. The temperature is then expanded by smooth orthogonal trigonometric functions and the Galerkin projection method is used to obtain a set of coupled ordinary differential equations (ODES). In the case of nonlinear problems, perturbation theory is used to linearize the problem, which is shown to reduce to a set of ODEs of the two-point-boundary-value type. The ODES are solved by a variable-step finite-difference scheme. It is found that a low number of modes are usually sufficient to capture an accurate solution. Both the spectral method and perturbation approach are validated upon comparison with conventional methods. Excellent agreement is obtained against the boundary element method and finite element method.; A low-order spectral scheme is then proposed to solve the two-dimensional and three-dimensional steady-state flows inside a thin cavity of arbitrary thickness. The influence of inertia and cavity topography is examined. The method is particularly effective for nonlinear flow, and its validity is here demonstrated for a flow with inertia. The problem is closely related to die casting and high-speed lubrication flow. The flow is determined by solving the thin-film equations, using a coupled spectral/finite difference method. The validity of the spectral representation in the case of 2D problem is assessed upon comparison with the fully two-dimensional finite volume solution, and depth-averaging method from shallow-water theory. (Abstract shortened by UMI.)...