Integrated radial basis function methods for Newtonian and non-Newtonian fluid flows

Abstract

In this PhD thesis, one-dimensional integrated radial basis function networks (1D-IRBFNs) are further developed for the simulation of viscous and viscoelastic flows in two dimensions. The thesis consists of two main parts. In the first part, 1D-IRBFNs are incorporated into the Galerkin formulation to simulate viscous flows. The governing equations are taken in the streamfunction- vorticity formulation and in the streamfunction formulation. Boundary conditions are effectively imposed with the help of the integration constants. The proposed 1D-IRBFN-based Galerkin methods are validated through the numerical simulation of several benchmark test problems including free convection in a square slot and in a concentric annulus. In the second part, 1D-IRBFNs are incorporated into the Galerkin and collocation formulations to simulate viscoelastic flows. The momentum and continuity equations are taken in the streamfunction-vorticity formulation and two types of fluid, namely Oldroyd-B and CEF models, are considered. Flows in a rectangular duct and in straight and corrugated tubes are simulated to validate the proposed 1D-IRBFN-based Galerkin/Collocation methods. Main attractive features of the proposed methods include (i) easy implementation; (ii) avoidance of the reduction in convergence rate caused by differentiation; and (iii) effective treatment of derivative boundary conditions. Numerical results show that the proposed methods are stable, high-order accurate and converge well. This study further demonstrates the great potential of using RBFs in CFD.