Page 56 - Introduction to Computational Fluid Dynamics

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2.7 DEALING WITH NONLINEARITIES
Comparing Equations 2.56 and 2.57, leads to May 25, 2005 10:49 35
−1
x i+1/2 − x i x i+1 − x i+1/2
k i+1/2 = x i+1/2 + . (2.58)
k i k i+1
If the cell face were midway between the nodes then this equation would read as
−1
1 1

k i+1/2 = 2 + . (2.59)
k i k i+1
These equations suggest that the conductivity at a cell face should be eval-
uated by a harmonic mean to accord with the physics of conductance. We shall
regard this as a general practise and extend it to the case when thermal conduc-
tivity varies with temperature. Thus, instead of using either Equation 2.52 or 2.53,
Equation 2.58 will be used with k i and k i+1 evaluated in terms of temperatures
T i and T i+1 , respectively. Further, note that if conductivity is constant, k i+1/2 =
k i = k i+1 .

2.7.3 Boundary Conditions
In practical problems, three types of boundary conditions are encountered:

1. Boundary temperatures T 1 and/or T N are speciﬁed.
2. Boundary heat ﬂuxes q 1 and/or q N are speciﬁed.
3. Boundary heat transfer coefﬁcients h 1 and/or h N are speciﬁed.

Our interest in this section lies in prescribing these boundary conditions by
employing Su and Sp for the near-boundary nodes.

Boundary Temperature Speciﬁed
For the purpose of illustration, consider the i = 2 node, where T 1 is speciﬁed. Then,
Equation 2.43 will read as

(AP 2 + Sp 2 ) T l+1 = AE 2 T l+1 + AW 2 T l+1 + Su 2 , (2.60)
2 3 1

where Su 2 and Sp 2 are already updated to account for any source term. Equation
2.60 can be left as it is but we alter it via a three-step procedure in which we set
Su 2 = Su 2 + AW 2 T 1 ,

Sp 2 = Sp 2 + AW 2 ,

AW 2 = 0.0. (2.61)

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