#### Date of Award

1987

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mechanical Engineering

#### First Advisor

Sumanta Acharya

#### Abstract

This thesis deals with the formulation of a computationally efficient adaptive grid system for two-dimensional elliptic flow and heat transfer problems. The formulation is in a curvilinear coordinate system so that flow in irregular geometries can be easily handled. An equal order pressure-velocity scheme is formulated in this thesis to solve the flow equations. An adaptive grid solution procedure is developed in which the grid is automatically refined in regions of high errors and consecutive calculations are performed between the coarse grid and adapted grid regions in the same spirit as that of a Multi-Grid method. In orthogonal coordinate systems, checkerboard pressure and velocity fields are avoided by using staggered grids. In curvilinear coordinates however, the geometric complications associated with staggered grids are overwhelming and therefore a non-staggered grid arrangement is desirable. To this end, an equal order pressure-velocity interpolation scheme is developed in this thesis. This scheme is termed as the SIMPLEM algorithm and is shown to have good convergence characteristics, and to suppress checkerboard pressure and velocity fields. The adaptive grid technique developed flags the important regions in the calculation domain from an initial coarse grid calculation. Then, adaptation is performed by generating a nonuniform mesh in the flagged region using Poisson's equations in which the nonhomogeneous terms are chosen so that a denser clustering of grid points is obtained where needed most in the flagged region. Coarse grid calculations in the whole domain, and fine grid calculations in the flagged region are consecutively performed until convergence, with correction terms from the fine grid solution added to the coarse grid equations in the flagged region in every cycle of calculation. Thus, the solution in the non-refined regions improves due to the influence of the correction terms added to the coarse grid equations. The effectiveness of the method is demonstrated by solving a variety of test problems and comparing the results with those obtained on a uniform or fixed grid. The adaptive grid solutions are shown to be more accurate than the fixed uniform grid solutions for the same level of computational effort.

#### Recommended Citation

Moukalled, Fadl Hassan, "Adaptive Grid Solution Procedure for Elliptic Flows." (1987). *LSU Historical Dissertations and Theses*. 4464.

https://digitalcommons.lsu.edu/gradschool_disstheses/4464

#### Pages

375