Spectral methods solution of the Navier-Stokes equations for steady viscous flows

Abstract

A combination of Spectral Methods and Finite Differences will be used to solve the Navier-Stokes equations for a viscous flow past a circular cylinder and past symmetric Joukowski airfoils. Different discretizations of the physical problem will be explored, and the solution of the equations will be analyzed for different geometries and boundary conditions. This project is the continuation of our research started as a Master Thesis at Wichita State University under the advising of Professor Alan Elcrat; the project is a deep exploration of the solution of Navier-Stokes equations by implementing new methods of discretization including spectral differentiation. We will compare results previously obtained by Gauss-Seidel/Successive Over-Relaxation Methods (SOR) together with Finite Differences, with results using Newton’s Method, based on work by Bengt Fornberg, but implementing spectral differentiation. As we will see, due to the nature of the physical domain and the conformal map involved to transform it to a more tractable domain, the use of spectral methods in both directions of our two dimensional problem proved to be inefficient due to unnecessary concentration of points in areas of the domain of low gradients. However, to take advantage of spectral methods, we combined spectral methods in one direction with high order finite vii differences on the other direction, where different mesh densities were designed to have higher concentration of points where required. With this discretizations, spectral methods were approached as the limiting order of finite differences as presented in A Practical Guide to Pseudospectral Methods We will explore the solution for flows past more general geometries, symmetric Joukowski airfoils. Then we will study the implementation and effect of suction boundary conditions on the obstacle. In this text I have decided to include part of the introduction and theoretical background shown in my Master thesis to allow new readers to get familiarized with the subject, but the solution scheme, the different discretizations and results are all new explorations that we are proud to present.