Local Rectangular Refinement in Three Dimensions (LRR3D): Development of a Solution-Adaptive Gridding Technique with Application to Convection-Diffusion Problems

Abstract

The local rectangular refinement (LRR) solution-adaptive gridding method, developed a decade ago to solve coupled nonlinear elliptic partial differential equations in two dimensions, has been extended to three dimensions. Like LRR2D, LRR3D automatically generates orthogonal unstructured adaptive grids, discretizes governing equations using multiple-scale finite differences, and solves the discretized system at all points simultaneously using Newton's method. The computational/programming challenges overcome in developing LRR3D are described. Next, a three-dimensional convection-diffusion-reaction problem with a known solution is used to demonstrate the accuracy and efficiency of LRR3D versus a Newton solver on a structured grid. Finally, natural convection within a tilted differentially heated cubic cavity with perfectly conducting nonisothermal walls is examined for a range of Rayleigh numbers (10³ to 4 × 10⁴) and for two inclinations (45° and 90°) of the isothermal walls. Excellent agreement with published computational and experimental results is observed.

The author gratefully acknowledges the financial support of the National Science Foundation under Grant 0137098.

Notes

^a LRR3D with linear (or bilinear or trilinear) interpolants at internal boundary points, in the spirit of several adaptive gridding techniques in the literature

^b Standard LRR3D: governing equations were discretized at internal boundary points using multiple-scale discretizations

^c For TPG3D, t _adapt represents the time required to double the number of points in each coordinate direction and to interpolate the previous grid's solution to the new grid

^a LRR3D with linear (or bilinear or trilinear) interpolants at internal boundary points, in the spirit of several adaptive gridding techniques in the literature. Results calculated on the final adaptive grid are presented

^b Standard LRR3D: governing equations were discretized at internal boundary points using multiple-scale discretizations. Results calculated on the final adaptive grid are presented

^c TPG3D on an 81³ equispaced grid. Spacing equals the finest spacing in the LRR3D grids

^d Experimental results

^a LRR3D with linear (or bilinear or trilinear) interpolants at internal boundary points, in the spirit of several adaptive gridding techniques in the literature. Results calculated on the final adaptive grid are presented

^b Standard LRR3D: governing equations were discretized at internal boundary points using multiple-scale discretizations. Results calculated on the final adaptive grid are presented

^c TPG3D on an 81³ equispaced grid. Spacing equals the finest spacing in the LRR3D grids

^d Experimental results

^e TASCflow3D (constant fluid properties) on nonuniform grids of sizes up to 60 × 28 × 60

^f TASCflow3D (variable fluid properties) on nonuniform grids of sizes up to 60 × 28 × 60