Radial-basis-function calculations of heat and viscous flows in multiply-connected domains

Abstract

This PhD research project is concerned with the development of accurate and efficient numerical methods, which are based on one-dimensional integrated radial basis function networks (1D-IRBFNs), point collocation and Cartesian grids, for the numerical simulation of heat and viscous flows in multiply-connected domains, and their applications to the numerical prediction of the material properties of suspensions (i.e. particulate fluids). In the proposed techniques, the employment of 1D-IRBFNs, where the RBFN approximations on each grid line are constructed through integration, provides a powerful means of representing the field variables, while the use of Cartesian grids and point collocation provides an efficient way to discretise the governing equations defined on complicated domains. Firstly, 1D-IRBFN-based methods are developed for the simulation of heat transfer problems governed by Poisson equations in multiply-connected domains. Derivative boundary conditions are imposed in an exact manner with the help of the integration constants. Secondly, 1D-IRBFN based methods are further developed for the discretisation of the stream-function - vorticity formulation and the stream-function formulation governing the motion of a New- tonian fluid in multiply-connected domains. For the stream-function - vorticity formulation, a novel formula for obtaining a computational vorticity boundary condition on a curved boundary is proposed and successfully verified. For the stream-function formulation, double boundary conditions are implemented without the need to use external points or to reduce the number of interior nodes for collocating the governing equations. Processes of implementing cross derivatives and deriving the stream-function values on separate boundaries are presented in detail. Thirdly, for a more efficient discretisation, 1D-IRBFNs are incorporated into the domain embedding technique. The multiply-connected domain is transformed into a simply-connected domain, which is more suitable for problems with several unconnected interior moving boundaries. Finally, 1D-IRBFN-based methods are applied to predict the bulk properties of particulate suspensions under simple shear conditions. All simulated results using Cartesian grids of relatively coarse density agree well with other numerical results available in the literature, which indicates that the proposed discretisation schemes are useful numerical techniques for the analysis of heat and viscous flows in multiply-connected domains