Page 89 - Introduction to Computational Fluid Dynamics

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1D CONDUCTION–CONVECTION
where is given by Equation 3.51. To derive an expression for tan ID (where the
subscript ID stands for implicit differencing), therefore, let X P = 0. Then, from
Equation 3.62, it can be shown that
(AE − AW)sin( X)
tan ID = . (3.63)
1 + (AE + AW)(1 − cos X)
Equation 3.63 again shows that, as X → 0, ID → exact . Also, notice that the
denominator of this equation with a plus sign before (AE + AW) is not the same
as the denominator in Equation 3.55. The plus sign indicates that the propagation
wave will be more severely damped than in the explicit procedure and this damping
will be greater for large X (small wavelength) than for small X. Now, to derive
an expression for AR ID , let X P = π/2. Then, using Equations 3.62 and 3.63, we
can show that

T P cos ID
AR ID = = . (3.64)
T 0 sin(x P + ID ) 1 + (AE + AW)(1 − cos X)
Again, this expression is different from Equation 3.57. Equation 3.64 shows
−1
2
that when X and ID are small, AR ID = 1 + (AE + AW) X /2 =
(1 + τ) −1 ∼ 1 + τ → exp(− τ)asrequired. When X = π (i.e.,for acoarse
grid), however, AR ID = cos ID /[1 + 2(AE + AW)].
These remarkable results show that AR ID can never be greater than 1 because
neither AE nor AW can be negative. Thus, the implicit discretisation is uncondi-
tionally stable and there is no restriction on the time step. Again, in pure conduction
(P = 0), we had demonstrated this result in Chapter 2 through a worked example.
The implicit discretisation is thus safe. The only disadvantage is that the discretised
equation must be solved iteratively rather than by a marching procedure, which is
possible in an explicit scheme.
The conclusions arrived at in this section apply equally to variables other than
T, to nonuniform grids, to -dependent coefficients, and to multiple dimensions.

EXERCISES

1. Derive Equation 3.7.
2. Show that the CDS formula (3.31) is second-order accurate for both the first
and the second derivatives.
3. Show that the UDS formula (3.32) represents convection to only first-order
accuracy.

4. Show that the UDS formula is a CDS representation of Equation 3.35.
5. Show correctness of the HDS (3.36) and power-law (3.37) expressions by
recalculating the P values shown in Table 3.1.