Natural Convection in a Cubic Cavity: Implicit Numerical Solution of Two Benchmark Problems

Abstract

Natural convection of air within a cubic cavity, two opposite walls of which are differentially heated, is simulated numerically for Rayleigh numbers of 10³, 10⁴, 10⁵, and 10⁶. For each Rayleigh number, two benchmark problems are examined: all four remaining walls are either adiabatic or perfectly conducting. The conservation equations, written using a vorticity–velocity formulation, are discretized with second-order finite differences on uniform grids containing up to 81³ points. The equations are solved at all points simultaneously using Newton's method. Solutions on an 81³ nonuniform grid are also presented. Excellent agreement with published computational and experimental data is observed, and new benchmark data are reported for the second problem.

Acknowledgments

The authors gratefully acknowledge the financial support of the National Science Foundation under Grant 0137098.

Notes

^a Determined to be the benchmark solution. See text for more detail.

^b Solution obtained on 41³ grid by combining the 41³ and 81³ uniform grid solutions via Richardson extrapolation.

^c Nonuniform grid obtained by a geometric series; see Section 3.1 for details. All other grids are uniform.

^d Grid was too coarse to obtain a converged solution.

^a 32³ nonuniform grid.

^b 81³ uniform grid.

^c 81³ uniform grid for Ra = 10³; 61 × 45 × 45 nonuniform grid for Ra = 10⁴; and 91 × 45 × 45 nonuniform grid for Ra = 10⁵.

^d Richardson extrapolation using 80³ and 120³ uniform grids to get final data on 40³ uniform grid.

^e 81³ uniform grid for Ra = 10³ and Ra = 10⁴, and 81³ nonuniform grid for Ra = 10⁵ and Ra = 10⁶.

^a Determined to be the benchmark solution. See text for more detail.

^b Solution obtained on 41³ grid by combining the 41³ and 81³ uniform grid solutions via Richardson extrapolation.

^c Nonuniform grid obtained by a geometric series; see Section 3.1 for details. All other grids are uniform.

^d Grid was too coarse to obtain a converged solution.

^a Experimental data.

^b 42³ nonuniform grid.

^c 294,173 tetrahedral elements.

^d 40 × 40 × 20 nonuniform grid.

^e 40 × 40 × 20 uniform grid.

^f 30 × 30 × 30 and 80 × 80 × 40 nonuniform grids.

^g 61 × 61 × 31 nonuniform grid.

^h 2,048 nonuniform hexahedral elements.

ⁱ 81³ uniform grid for Ra = 10⁴, Ra = 4 × 10⁴, and Ra = 10⁵. 81³ nonuniform grid for Ra = 10⁶.