Page 79 - Introduction to Computational Fluid Dynamics

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1D CONDUCTION–CONVECTION
Clearly, the first term represents the net convection whereas the second term rep-
resents the net conduction. However, note that, unlike in the conduction term, P
does not appear in the convection term.
Equation 3.13 will now be rewritten in the familiar discretised form to
read as
AP P = AE E + AW W , (3.14)
where

P c
AE = 1 − , (3.15)
2

P c
AW = 1 + , (3.16)
2

AP = AE + AW = 2, (3.17)

and
uL x u x
P c = P X = = , (3.18)
α L α

where α = k/(ρ C p ) is the thermal diffusivity and P c is called the cell Peclet num-
ber. If we now invoke Scarborough’s criterion, it is clear that Equation 3.14 will be
convergent only when AE and AW are positive. This implies that the condition for
convergence is

|P c |≤ 2. (3.19)

Thus,whenconvectionisverylargecomparedtoconduction,tosatisfycondition
(3.19), one will need to employ very small values of X or a very fine mesh.
However, this can prove to be very expensive.
The more relevant question, however, is, Why do AE and/or AW turn neg-
ative when convection is dominant? The answer to this question can be found
in Equation 3.11, where, contrary to the advice provided by the exact solution,
the cell-face values are linearly interpolated between the values of at the adja-

cent nodes. Note that when P c > 2 and large, the exact solution gives e → P
and w → W . Similarly, when P c < −2, e → E and w → P . In Equa-
tion 3.11, we took no cognizance of either the direction of flow (sign of P c )orits
magnitude.
To obtain economic convergent solutions, therefore, one must write

e = ψ P + (1 − ψ) E , w = ψ W + (1 − ψ) P , (3.20)