Adaptive Numerical Modeling of Natural Convection, Conduction, and Solidification within Mold Cavities

Abstract

Time-dependent solidification of an aluminum alloy within a graphite mold cavity is adaptively modeled, including effects of latent heat convection, temperature-dependent variation in viscosity during phase change, and mold wall conduction. Three other simpler (but related) applications are also examined. The coupled, nonlinear conservation equations, written using the vorticity–velocity formulation, are solved simultaneously using Newton's method within the context of the local rectangular refinement (LRR) solution-adaptive gridding method. The LRR method automatically generates orthogonal unstructured adaptive grids and incorporates multiple-scale finite differences, producing considerable computational savings without loss of accuracy, compared to similar solution methods on structured grids. For the two (steady) applications for which published results exist, excellent agreement with published computational and experimental results is observed.

The author gratefully acknowledges financial support from the National Science Foundation (Grant 0137098), as well as Ms. Boma Brown-West's help in plotting early results.

Notes

^a Air properties are taken from [31].

^b Lexan properties are taken from [22]. For certain results reported here, the physical properties of the Lexan are altered slightly such that the thermal conductivity ratio is k∗ = k _m /k _c  = 10 and the thermal diffusivity ratio is α∗ = α _m /α _c  = k _m ρ _c c _c /k _c ρ _m c _m  = 0.005, as in some of the results reported in [22 23 29].

^c These quantities are not present in the problem formulation.

^d Thermal dilation coefficient β ≈ 1/T _ref, where T _ref = 294.15K.

^e Al alloy and graphite properties are taken from [29].

^f Dynamic viscosity μ _l of the molten aluminum alloy.

^g There is a typographical error in [29] that has been corrected here.

^a Results from [29] are finite-element data (first row) and finite-volume data (second row). Also, the |v|_max data from [29] are actually |v|_max, the maximum magnitude of nondimensionalized velocity throughout the cavity.

^b Each LRR adaptive grid contained 11,257, 11,079, 11,595, and 11,211 points for the cases of Ra = 10³, 10⁴, 10⁵, and 10⁶, respectively. Each LRR adaptive grid comprised an equispaced base grid of 41 × 41 = 1,681 points, plus two additional levels of adaptive refinement, resulting in a minimum grid spacing equal to that which would be found in an equispaced grid of 161 × 161 = 25,921 points.

^c TPG results were computed on an equispaced grid of 161 × 161 = 25,921 points.

^d Data not available.

^a Results from [29] are finite-element data (first row) and finite-volume data (second row). Also, is defined differently in [29] than in this article, so the values from [29] reported above have been multiplied by l _c /l _m  = 0.6.

^b Each LRR adaptive grid contained 15,861, 16,085, 16,073, and 15,179 points for the cases of Ra = 10³, 10⁴, 10⁵, and 10⁶, respectively. Each LRR adaptive grid comprised an equispaced base grid of 41 × 41 = 1,681 points, plus two additional levels of adaptive refinement, resulting in a minimum grid spacing equal to that which would be found in an equispaced grid of 161 × 161 = 25,921 points.

^c TPG results were computed on an equispaced grid of 161 × 161 = 25,921 points.

^d Data not available.

^a Results from [23] are experimental measurements taken with embedded thermocouples (first row) and finite-difference data (second row).

^b The LRR adaptive grid contained 15,667 points, and it comprised an equispaced base grid of 41 × 41 = 1,681 points, plus two additional levels of adaptive refinement, resulting in a minimum grid spacing equal to that which would be found in an equispaced grid of 161 × 161 = 25,921 points.

^c TPG results were computed on an equispaced grid of 161 × 161 = 25,921 points.