In this dissertation, we develop Cartesian embedded boundary methods for flow around bodies with complicated geometries. These methods, also referred to as cut cell methods, cut the body out of a regular Cartesian grid. As a result, most of the grid is regular but special methods must be developed for the cut cells that intersect the solid boundary. The dissertation is divided in three parts. The first part describes a mixed explicit implicit scheme for solving the linear advection equation. It couples an explicit finite volume scheme to an implicit scheme on only the cut cells. This approach avoids the small cell problem -- that standard explicit finite volume methods are not stable on arbitrarily small cut cells. We examine two different ways of handling the coupling. We refer to them as cell bounding and flux bounding, depending on the kind of information that is used as input for the implicit scheme. For using flux bounding on our model problem, we prove a TVD result. We also examine different candidates for the implicit scheme. Numerical results in two dimensions for the resulting scheme indicate second-order accuracy in the L1 norm and between first- and second-order accuracy in the L∞ norm. The second part describes the extension of an existing projection algorithm for the incompressible Euler equations from Cartesian to Cartesian embedded boundary grids. The mixed explicit implicit scheme is the main ingredient for extending the advection step to cut cells. We also adjust all operators used in the projection step. We present initial results in two dimensions for a full projection algorithm. In the third part of this work we present a new slope limiter designed for cut cells. Standard slope limiters do not easily apply to cut cells. Most practitioners use a scalar limiter, where all components of a gradient are limited by the same scalar. We develop a new limiter based on solving a tiny linear program for each cut cell, which separately limits the x and y slope in a linearity preserving way. We present computational results in two dimensions showing significantly improved accuracy.
|Advisor:||Berger, Marsha J.|
|School:||New York University|
|School Location:||United States -- New York|
|Source:||DAI-B 75/03(E), Dissertation Abstracts International|
|Keywords:||Complex geometry, Cut cell method, Finite volume scheme, Projection method, Slope limiter, Small cell stability|
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