Winslow Elliptic Smoothing Equations Extended to Apply to General Regions of an Unstructured Mesh

A Dissertation Presented for the Doctor of Philosophy in Computational Engineering, The University of Tennessee at Chattanooga

James S. Masters, December 2010

In any engineering endeavor, it is important to have the ability to efficiently and intelligently break up a region of interest in order to explore and investigate the interesting and dynamic aspects of the system enclosed in the region. In the field of fluid mechanics – and, in particular, computational fluid dynamics – this region breakup (known as discretization) has traditionally been done using structured meshes but, because of their flexibility with capturing real-world geometry, unstructured meshes are being increasingly utilized. Not surprisingly, techniques that have traditionally been reserved for structured meshes are migrating to the world of unstructured meshing as well. One such technique is the application of the Winslow elliptic smoothing equations to an unstructured mesh. The Winslow equations have proven to be powerful tools for smoothing unstructured meshes and some early difficulty using the Winslow equations with unstructured meshes was alleviated when it was determined that it was not necessary for the entire computational mesh to be constructed as an overarching system of nodes and elements, but rather, each node in computational space could be treated as an individual virtual control volume. Winslow equations have been shown to be ideal for smoothing non-boundary nodes in inviscid regions but of limited use in other situations. This presented two opportunities to improve and extend the usefulness of the Winslow equations in relation to unstructured meshes. Research was performed that allowed the Winslow equations to be applied to boundary nodes which, traditionally, have been held static while interior mesh points are smoothed. In addition, several methods were tested to make the Winslow equations applicable to highly anisotropic viscous regions of a mesh. The most successful methodology that was developed and explored in relation to a viscous region was to use an iteratively adapted computational space for each of the nodes in the viscous region. This technique allowed a mesh with an amenable connectivity structure to be smoothed in such a way that it could match a desired viscous profile based on an initial off-body spacing and the geometric progression of the viscous layers of the mesh.

James S. Masters Doctoral Dissertation