Page 52 - Introduction to Computational Fluid Dynamics

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2.6 MAKING CHOICES
From the computed results, we make the following observations: 10:49 31
1. The temperature evolutions are monotonic irrespective of the time step since
there is no restriction on the time step in the implicit procedure.
2. With t = 10 s, the time for pressing is evaluated at 107.81 s and with t = 20 s
at 112.09 s. Again these times are not necessarily accurate. Accuracy can only
be established by repeating computations with ever smaller values of t and
x till the evaluated total time is independent of the choices made.
3. Comparison of results in Table 2.3 with those in Table 2.1 shows that temperature
evolutions calculated by the implicit procedure are more realistic. Note, for
example, that T 4 in the explicit procedure does not even recognise that heating
has started for the first 20 s. Of course, this lacuna can be nearly eliminated by
taking smaller time steps.
4. For the same time step, the explicit procedure reaches T 4 = 140 in 10 time
steps. The implicit procedure has, however, required 11 time steps. In addition,
at each time step, a few iterative calculations have been carried out. Thus,
in this example, the implicit procedure involves more arithmetic operations
than the explicit procedure. This, however, is not a general observation. When
x and t are reduced to obtain accurate solutions, or when coefficients AE
and AW are not constant but functions of temperature (through temperature-

dependent conductivity, for example), or when q = q (T ) is present, one
may find that an implicit procedure may yield more economic solutions than the
explicit procedure because the former enjoys freedom over the size of the time
step.

2.6 Making Choices

In the previous two sections, we have introduced TSE and IOCV methods as well as
explicit and implicit procedures. Here, we offer advice on the best choice of combi-
nation, keeping in mind the requirements of multidimensional problems (including
convection) to be discussed in later chapters. Further, we also keep in mind that
coefficients AE and AW are in general not constant. This makes the discretised
equations nonlinear.

1. Note that the TSE method casts the governing equations in non-conservative
form whereas the IOCV method uses the as-derived conservative form. As we
shall observe later, this matter is of considerable physical significance when
convective problems are considered.
2. In the TSE method, coefficients AE and AW carry little physical meaning. In
the IOCV method, they represent conductances.
3. In the TSE method, Scarborough’s criterion may be violated. In the IOCV
method, this can never happen.