Computer Vision Analysis of a Melting Interface Problem with Natural Convection - Visual Computing: Scientific Visualization and Imaging Systems - page 211

following equation, proposed by Gebhart and Mollendorf [ 16 ], provides very good
results for temperatures below 10 C:

j q q ð T Þ ¼q ref : 1 c : T T ref ð j Þ ð 12 : 7 Þ

The term c is the phenomenological coefficient given by c = 8 9 10 -6 C -2 ;
q = 2. In this case, q ref is equal to the maximum density of the fluid, also called
q M ; and T ref is equal to the temperature of the maximum density, which is also
called T M . For water T M = 3.98 C. A modified Grashof number based on the
cavity height and on DT max (maximum temperature interval considered) have been
defined:

Þ 2

Gr mod ¼ g : H 3 : c : DT max

ð

ð 12 : 8 Þ

t 2

The effect of variation of q is approximately symmetrical to the maximum
density. The relative density variation that causes the flow in each cell is directly
linked to the intervals between T M and the wall temperatures. Then:

DT 1 ¼ T H T M ð 12 : 9 Þ

DT 2 ¼ T M T 0 ð 12 : 10 Þ

The maximum temperature interval considered is:

DT max ¼ Max ð DT 1 ; DT 2 Þ ð 12 : 11 Þ

12.2.2 Numerical Procedure

The numerical method used in this work has been successfully compared to the
results of other researchers for the case of materials that do not present a maximum
density to a comparison exercise proposed by Gobin and Le Quéré [ 9 ]. The
numerical simulation technique is based on the hypothesis that the melting process
is a succession of quasi-stationary states. In order to map the irregular space
occupied by the liquid into a rectangular computational space, the dimensionless
coordinates were transformed. The curvilinear coordinate system adopted is given
by:

Z ¼ z; Y ¼ y = C ð Z Þ ð 12 : 12 Þ

where C(Z) = c*(z*)/L and L is the maximum width of the liquid cavity. Z and
Y are the computational dimensionless coordinates. Other details of the coordinate
transformation method are shown in the work of Vieira [ 17 ]. The transformed