"Numerical Solution of Incompressible Viscous Flow Problems Using High-order Schemes"

Abstract

Numerical Solution of Incompressible Viscous Flow Problems Using High-order Schemes Numerical solution of the Navier-Stokes equations that describe incompressible viscous fluids has been a very active research field due to the rapid development of computational techniques and availability of high speed computers. It has motivated a very large number of researchers whose work provide an invaluable source of solution methods and test problems. Numerous computational methods have been developed and are in used today for steady and time-accurate computation of these equations. The motivation of this thesis is also a desire to develop an efficient, accurate and simple method for the numerical solution of incompressible viscous flow problems in primitive variables. For this purpose, a numerical method based on high-order compact finite difference schemes is developed in conjunction with the well-known artificial compressibility approach for solving incompressible Navier-Stokes equations. By adding pseudo time derivative terms to each equation, the coupled system becomes hyperbolic in time and the artificial compressibility method becomes applicable. We have also focused on the extension of the method for simulating two-phase flow by coupling phase-field model to the incompressible Navier-Stokes solver. This research work is divided into three steps in which each step focuses on an aspect of the development of a numerical method. In the first step, a third-order upwind compact finite difference scheme based on the fluxdifference splitting is developed and implemented with the implicit Beam-Warming approximate factorization scheme for solving the incompressible Navier-Stokes equations. The upwind compact scheme for the convective terms is preferred because of its high resolving efficiency with less numerical dissipation and truncation errors. The numerical scheme is applied to compute the flow inside the two sided lid driven cavity flow and compared with the finite difference alternating direction implicit scheme. In the second step, we implemented higher-order central compact finite difference scheme along with filtering procedure for steady and unsteady incompressible Navier- Stokes equations. The central compact scheme is also implemented under the framework of the artificial compressibility method in which convective terms of the governing x equations are approximated by using the high-order central compact schemes with filtering procedure and the viscous terms are discretized with a sixth-order central compact finite difference scheme. Dual-time stepping technique is employed for unsteady solutions at each physical time step. Computational efficiency and accuracy of the method is compared with upwind compact schemes by computing several benchmark flow problems. In the third step, the central compact scheme is applied successfully to incompressible two-phase flows both in two and three space dimensions. For this purpose, the modified Allen-Cahn type phase-field model is coupled with the incompressible Navier-Stokes equations. In the phase-field formulation, the classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of small but finite width, across which the composition of the one or two fluids changes continuously. The effectiveness of the method is demonstrated by computing several benchmark twophase incompressible flow problems. Finally, advantages and difficulties in solving incompressible viscous flow problems are discussed and future directions of the effort are proposed.