## Electronic Theses and Dissertations

1994

Dissertation

Ph.D.

#### Department

Mathematics and Statistics

Barron, R.

Mathematics.

CC BY-NC-ND 4.0

#### Abstract

In this thesis, we address the problem of computing laminar incompressible viscous flows. For such flows, there are various possibilities for the formulation of the problem. These include primitive variable (velocity and pressure), velocity-vorticity and stream function-vorticity formulations. Each has its own strengths and weaknesses. In the stream function-vorticity formulation for two-dimensional flow, the difficulty is primarily associated with determination of vorticity at a boundary. In this thesis, we employ a variation of the stream function-vorticity formulation whereby incompressible, viscous flow in an internal complex geometry is formulated in terms of von Mises coordinates. That is, stream function is used as an independent rather than dependent variable. The formulation provides a rectangular computational domain with both Dirichlet and von Neumann boundary conditions for unknown functions, the vertical Cartesian coordinate and the vorticity, in terms of the horizontal Cartesian coordinate and the stream function. The governing second order nonlinear partial differential equations are solved by SLOR on uniform and, if required, clustered grids. A number of procedures for surmounting the problem of determining vorticity at a boundary are available. A novel approach to this problem is applied in this thesis. A difficult, but well-documented, test problem was chosen to study the applicability of the von Mises formulation in viscous flows. It has been shown that extreme care must be taken in applying von Mises coordinates to viscous flow situations. In particular, viscosity is known to generate vorticity in the flow field and to cause, under appropriate conditions, flow separation. Of these two phenomenon, rotational flow and viscous separation, it is shown that rotational effects can be handled with no more difficulty than experienced by conventional methods. However, separation cannot be handled directly by the von Mises formulation, and erroneous results may be obtained if not used carefully. This presents a challenging problem which has been overcome by developing an innovative way to predict the location of the streamline which divides the main flow from the recirculating region. In this way, the von Mises formulation can be used to study separated 2D viscous flows. This approach is used to predict the reattachment length for the flow over a backward facing step and the results, when compared to other numerical data, confirm the applicability and accuracy of the method.Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1994 .C376. Source: Dissertation Abstracts International, Volume: 56-11, Section: B, page: 6150. Adviser: Ron Barron. Thesis (Ph.D.)--University of Windsor (Canada), 1994.

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