Masters Theses

Yanping Shi

Date of Award

5-1995

Degree Type

Thesis

Degree Name

Master of Science

Major

Aerospace Engineering

Major Professor

Robert L. Roach

Committee Members

Trevor Moulden, Gary Flandro

Abstract

A generalization of the Roscoe discretization for the unsteady, incompressible Navier- Stokes equations is studied in the present work. The distinction of this generalization lies in the systematic approach adopted for the discretization. The resulting discretized model generated from a linear differential system, which represents an approximation for the given nonlinear problem. A phase space study on one of the test problems, the nonlinear Burgers equation suggests that this approximate system of linear differential equations preserves the system stability of the original nonlinear equation. One of new features of the generalized method is that unique interpolation functions containing some of the characteristics and properties of the original system (in a approximate sense) are generated for use in the discretization. These unique interpolation functions are found to be a linear or quadratic function plus a exponential term. This is in contrast to most finite difference or finite volume methods where the interpolation functions are usually some low order polynomials. The interpolation functions are obtained by integrating locally linearized forms of the governing equations as a system of ordinary differential equations in one direction at a time. There is thus a unique function for each dependent variable in each coordinate direction. Since the equations are integrated as a system, interactions between the equations are retained in the resulting interpolation functions. Corresponding to the two constructions of the interpolation functions, two numerical schemes, a linear exponential spline and a quadratic exponential spline scheme, have been developed and applied to the steady state nonlinear Burgers equation and to the unsteady, incompressible Navier-Stokes equations for the Lamb vortex problem. The numerical results are compared with those from the original Roscoe scheme, second-order central differencing, a cubic spline method and a fourth-order compact method. A stability and accuracy analysis is also given with conclusions which fit well with the computational results. The two exponential spline schemes derived from the generalized Roscoe procedure are found to be unconditionally stable with respect to cell Reynolds number. Both of the numerical solutions and the analytical studies show that these exponential spline solving procedures, as developed, have second order accuracy with respect to the spatial grid size.

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