Page 53 - Introduction to Computational Fluid Dynamics

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4. The question of invoking explicit procedure arises only when unsteady-state
problems are considered. The implicit procedure, in contrast, can be invoked
for both unsteady-state as well as steady-state problems. In fact, in steady-state
problems ( t =∞) the implicit procedure is the only one possible. 6
5. The explicit procedure imposes restriction on the largest time step to obtain
stable solutions. The implicit procedure does not suffer from such a restriction.
In view of these comments, the best choice is to employ the IOCV method with
an implicit procedure. Throughout this book, therefore, this combination will be
preferred.

2.7 Dealing with Nonlinearities
Now that we have accepted a combination of IOCV with the implicit procedure, we
restate the main governing discretised equation (equations 2.38 and 2.39) but in a
slightly altered form:

(AP i + Sp i ) T l+1 = AE i T l+1 + AW i T l+1 + Su i , i = 2, 3,..., N − 1,
i i+1 i−1
(2.43)

AP i = AE i + AW i , (2.44)

kA
AE i = , (2.45)
x
i+1/2

kA
AW i = , (2.46)
x
i−1/2
ρ V i C i o o ρ V i C i n
Su i = T , Sp i = . (2.47)
i
t t
In these equations, the q term is deliberately ignored because it is a problem-

dependent term. The altered form shown in Equation 2.43 will be useful in dealing
with nonlinearities. Also, a generalised computer code can be constructed around
Equation2.43insuchawaythatpreservestheunderlyingphysics.Thenonlinearities
can emanate from three sources:

1. if q is a function of T

2. if conductivity k is a function of T or changes abruptly, as in a composite material
and/or
3. boundary conditions at x = 0 and x = L.

6 Some analysts employ an explicit procedure even for a steady-state problem. In this case, calcu-
lations proceed by introducing a false or imaginary time step. Hence, such procedures are called
false transient procedures.