Page 242 - Introduction to Computational Fluid Dynamics

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7.2 1D PROBLEMS FOR PURE SUBSTANCES
4. The interface location can be identified from the location of θ = 0, but, as 11:14 221
already explained, this will again predict a wavy interface history. Instead, one
may use variable to predict the interface history. This is because j is nothing
but the liquid fraction of the control volume surrounding phase-change node
j. Thus, at any time instant, one may simply add X for all nodes for which
j < 0 (i.e., solid nodes) and further add (1 − j ) X for the node for which
0 < j < 1 and ignore all nodes for which j > 1. The sum will readily predict
the instantaneous value of X i and this prediction will appear smooth but not
accurate on a coarse grid. This alternative procedure will again require book-
keeping.

These comments indicate that the simple procedure needs refinement in terms
of both economy and convenience.

7.2.3 Numerical Solution Using TDMA

To eliminate the bookkeeping requirements, the θ ∼ relations (7.15) to (7.17)
must be generalized [11] by writing

θ = + , (7.19)
where
1

= [|1 − | − | | − 1] . (7.20)
2
Equation 7.20 ensures that = 0 in solid ( < 0), =− during phase change

(0 < < 1), and =−1 in liquid ( > 1). Using Equation 7.19, we can reex-
press Equation 7.10 as
2
2
∂ ∂ ∂
= + (7.21)
∂τ ∂ X 2 ∂ X 2
and the discretised version will read as 1
τ τ

1 + 2 l+1 = l+1 + l+1
j−1
j+1
j
X 2 X 2
τ o

+ ( j+1 − 2 + j−1 ) + , (7.22)
j
j
X 2
where values lag behind values by one iteration. Thus, in step 4 of the simple

numerical procedure described in the previous subsection, (rather than θ j ) are
j
evaluated using Equation 7.20 and the bookkeeping requirement is eliminated. The
1 It is assumed that the reader will be able to make necessary changes to the discretised equation for
j = 2 and j = N − 1 nodes to account for any type of boundary condition.