Page 43 - Introduction to Computational Fluid Dynamics

P1: IWV/ICD
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22
Now to evaluate the time derivative, we write 1D HEAT CONDUCTION

n
o
(CT ) = (CT ) + t ∂(CT ) +· · · , (2.13)
P
P
∂t P
or
n o
(CT ) − (CT )
∂(CT ) P P
= . (2.14)

∂t t
P
In Equation 2.13, derivatives of order higher than 1 are ignored; therefore,
Equation 2.14 is only a first-order-accurate representation of the time derivative. 2
Inserting Equations 2.11 and 2.12 in Equation 2.7 and Equation 2.14 in Equation
2.8 and employing Equation 2.6, we can show that
n
ρ VC n n n
+ ψ (AE + AW) T = ψ AE T + AW T + S, (2.15)
P E W
t
P
with
2 x w d (kA) x
AE = (kA) P + , (2.16)
x e 2 dx ( x e + x w )
P
2 x e d (kA) x
AW = (kA) P − , (2.17)
x w 2 dx ( x e + x w )
P
,n ,o o o
S = ψ q + (1 − ψ)q V + (1 − ψ) AE T + AW T
P P E W
o
ρ VC o
+ − (1 − ψ)(AE + AW) T , (2.18)
P
t P
where V = A x. Note that if the cell faces were midway between adjacent nodes,
2 x = x e + x w . Before leaving the discussion of the TSE method, we make
the following observations:
1. Calcuation of coefficients AE and AW requires evaluation of the derivative
d (kA)/dx | P . This derivative can be evaluated using expressions such as (2.11)
in which T is replaced by kA.
2. For certain variations of (kA) and choices of x e and x w , AE and/or AW can
become negative.
o
3. For certain choices of t, the multiplier of T in Equation 2.18 can become
P
negative.
o
4. In steady-state problems, t =∞ and T has no meaning. Therefore, in such
problems, ψ always equals 1.

2 Clearly, it is possible to represent the time derivative to a higher-order accuracy. However, the
0
00
n
resulting expression will involve reference to T , T , T , and so on.