Meshless radial basis function method for unsteady incompressible viscous flows

[Abstract]This thesis reports the development of new meshless schemes for solving timedependent partial differential equations (PDEs) and for the numerical simulation of some typical unsteady incompressible viscous flows. The new numerical schemes are based on the Idirect/Integrated Radial Basis Function Network (IRBFN) method which is fully meshless as no element-type mesh is required. The IRBFN method has been successfully applied to solve time-independent elliptic PDEs, some steady fluid flows and recently unsteady Navier-Stokes equations in streamfunction-vorticiy formulation using simple time integration methods (e.g. first-order backward Euler method). The main objective of the present research is to devise and implement meshless numerical schemes for unsteady problems in computational fluid dynamics where not only the accuracy but also the efficiency and stability of the numerical schemes are of primary concerns. In addition, the effects of different parameters of the IRBFN method on the accuracy, stability and efficiency of the proposed numerical schemes are extensively studied in this research. As the first step in extending the IRBFN method to various types of timedependent PDEs, two numerical schemes combining the IRBFN method with high-order time stepping algorithms are developed for solving parabolic, hyperbolic, and advection-diffusion equations. Sensitivity analysis of the method to point density, time-step size and shape parameter are extensively performed to study the influence of these parameters to the overall accuracy of the method. A further extension of the IRBFN method for incompressible fluid flows with moving interfaces, especially for passive transport problems is accomplished in this research with a novel meshless approach in which the level set method is coupled with the the IRBFN method for capturing moving interfaces in an ambient fluid flow without any explicit computation of the actual front location. Another contribution of this research is the development of two new meshless schemes based on the IRBFN method for the numerical simulation of unsteady incompressible viscous flows governed by the Navier-Stokes equations. In the new schemes, the splitting approach is used to deal with the momentum equation and the incompressibility constraint in a segregated manner. Numerical experiments on the new schemes in terms of accuracy and stability are performed for verification purposes. Finally, a novel meshless hybrid scheme is developed in this research to numerically simulate interfacial flows in which the motion and deformation of the interface between the two immiscible fluids are fully captured. Unlike the passive transport problems mentioned above where the influence of the moving interface on the surrounding fluid is ignored, the interfacial flows are studied here with the surface tension taken into account. As a result, a two-way interaction between the moving interface and the ambient flow is fully investigated. All numerical schemes developed in this research are verified through a wide range of transient problems including different kinds of time-dependent PDEs, typical passive transport problems and interfacial flows as well as unsteady incompressible viscous flows governed by Navier-Stokes equations.