Date of Award

Spring 2000

Document Type


Degree Name

Doctor of Philosophy (PhD)


Computational Analysis and Modeling

First Advisor

Ben Choi


The purpose of this study is to investigate one of the most interesting areas in computational fluid dynamics. The content of this paper is divided into three parts. The first part is the orthogonal grid generation. The second part is the non-reflecting boundary condition in curvilinear coordinates. The third part is parallel computation by Message Passing Interface.

The grid generation method is presented by solving elliptic partial difference equations. The elliptic grid generation method is based on the use of composite mapping, which consists of a non-linear algebraic transformation and an elliptic transformation. The elliptic transformation is based on the Laplace equations for the domains or on the Laplace-Beltrami equations for surfaces. The algebraic transformation maps the computational space one-to-one onto a parameter space, and the elliptic transformation maps the parameter space one-to-one onto the domain or the surfaces. The composition of these two mappings is a differentiable and one-to-one, which has a non-vanish Jacobian. Finally, some complicated test examples are given. Computation results show that the grids generated by these methods are smooth and orthogonal. The grid quality meets our requirement for high accuracy numerical simulation.

Numerical methods for time-dependent hyperbolic systems require time-dependent boundary conditions when the system is solved in a finite domain. The “correct” boundary conditions are crucial in solving such a system. The non-reflecting boundary conditions based on the Navier-Stokes equations have been derived for curvilinear coordinates. High order scheme is used to discretize the non-reflecting boundary conditions. Several examples of non-reflecting boundary conditions are tested. The results are compared with the reference method based on extrapolation or Rieman invariants. It is found that this non-reflecting boundary condition is much more accurate and effective than the tradition methods used to impose boundary conditions.

A parallel spatial direct numerical simulation code is developed to simulate the spatial evolving disturbances associated with the laminar-to-turbulent transition in a compressible boundary layer. MPI (Message Passing Interface) is employed to parallelize all processes for a distributed memory parallel computer. Explicit time-stepping is used in the DNS code on IBM/SP2 to simulate the flow transition. The machine-dependent phenomenon, which is always being considered as a problem for parallel computation, is successfully avoided. A fundamental breakdown on a flat plate boundary layer transition at Mach 0.5 is then studied using this code. The results demonstrate the optimistic future of MPI to direct numerical simulation.