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
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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.
Description
Thesis (Ph.D.)--Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics and Statistics