Page 83 - Introduction to Computational Fluid Dynamics

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1D CONDUCTION–CONVECTION
To compare CDS and UDS formulas, we modify Equation 3.32 to read as 3 10:54

P c | P c |
( E − W ) − 1 + [ E − 2 P + W ] = 0 (UDS). (3.33)
2 2
Comparison of Equation 3.33 with the CDS formula (3.31) raises several inter-
esting issues:
1. Recall that the first term in Equation 3.31 corresponds to the convective con-
tribution whereas the second term corresponds to the conductive contribution.
Further, since P is constant, we may view Equation 3.6 as
2
∂ ∂
P − = 0. (3.34)
∂ X ∂ X 2
If we discretise both the first and the second derivative through a Taylor series
expansion, it will be found that the CDS formula (3.31) represents both the
derivatives to second-order accuracy.
2. Equation 3.32, in contrast, suggests that UDS represents the convective contri-
bution to only first-order accuracy, whereas the conductive contribution is still
represented to second-order accuracy. Mathematically speaking, therefore, the
estimate of the convective contribution will have an error of O ( x).
3. In Equation 3.33, this error is reflected in the augmented conduction coefficient
because the convective term is now written to second-order accuracy as in the
CDS formula. Mathematically speaking, therefore, it may be argued that the
second-order-accurate UDS formula represents discretisation with augmented
or false conductivity k false = ρ C p |u | x/2. In fact, it can be shown that Equa-
tion 3.33 is nothing but a CDS representation of
∂ ρ C p |u | x ∂T
ρ C p uT − k + = 0. (3.35)
∂x 2 ∂x
Thus, if the last comment is given credence, then clearly the UDS represents
distortion of reality and is therefore a poor choice. Yet, the closeness of the UDS
result to the exact solution shown in Figure 3.2 suggests that the so-called false
conductivity is indeed needed. In fact, it is this false conductivity that reduces the
value of the effective Peclet number and thereby ensures convergence of the UDS
formula for all Peclet numbers.
Patankar [49] has therefore argued that to form a proper view of false diffusion,
it is necessary to compare the UDS with the exact solution rather than with the
second-order-accurate CDS formula. This is yet another example where the TSE
method is found wanting.
Of course, this is not to suggest that the UDS formula is the best representation
of reality. The properties embodied in the UDS formula suggest that one can derive
other variants that will sense not only the sign of P c but also its magnitude. Further

3 Equation 3.33 can also be derived for P c < 0.