Page 49 - Introduction to Computational Fluid Dynamics

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Table 2.2: Explicit procedure with ∆t = 20 s (unstable). 1D HEAT CONDUCTION
Time 0 mm 1 mm 3 mm 5 mm 7 mm 9 mm 10 mm
0 250 30 30 30 30 30 250
20 250 241.5 30 30 30 241.5 250
40 250 148.0 131.7 30 131.7 148.0 250
60 250 238.3 90.63 127.8 90.63 238.3 250
80 250 178.6 179.5 92.05 179.5 178.6 250
100 250 247.7 137.0 176.1 137.0 247.7 250

be pressed together can be determined by interpolation as (t − 90)/(100 − 90) =
(140 − 130.33)/(141.31 − 130.33) or at t = 98.8 s. This calculated time, of course,
need not be considered accurate. Its accuracy can be ensured by repeating calcu-
lations with increasingly smaller x (increasingly greater number of nodes) and
by taking ever smaller values of t. Further, note that the temperature distribu-
tions at any time t are symmetric about x = 5 mm. This is because of the sym-
metry of the boundary and the initial condition. Now, unlike in Table 2.1, the
results presented in Table 2.2 show zigzag or nonmonotonic evolution of temper-
ature. For example, at any x, the temperature first rises (as expected) and then
falls (against expectation). In fact, the reader is advised to carry the computations
well beyond 100 s or with larger values of t. Then, it will be found that the
evolved temperatures will show even more unexpected trends. That is, the interior
◦
◦
temperatures will exceed the bounds of 30 C and 250 C. Clearly, this is in vio-
lation of the second law of thermodynamics. Results of Table 2.2 are, therefore,
unacceptable.

2.5.2 Partially Implicit Procedure 0 <ψ < 1

o
In this case, if the condition of positiveness of the coefficient of T is invoked then
P
t must obey the following constraint:
ρ V i C
o
t < i . (2.36)
(1 − ψ)(AE i + AW i )
min
However, computations of the previous problem will show that stable (monotoni-
cally evolving) solutions can be obtained even with

o
ρ V i C
t < i for ψ< 0.5, (2.37)
(1 − 2ψ)(AE i + AW i )
min
and, for ψ ≥ 0.5, t can be chosen without any restriction. Clearly, therefore,
condition (2.36), though valid, is too restrictive on the time step. The reader
will appreciate this matter by solving Exercise 29. The more rigorous proof