Page 42 - Introduction to Computational Fluid Dynamics

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2.4 DISCRETISATION
must be evaluated at a fixed time. The choice of this fixed time, however, is not so 10:49 21
straightforward because over a time step t, one may evaluate the LHS at time t,
or t + t, or at an intermediate time between t and t + t. In general, therefore,
we may write Equation 2.5 as
n o
ψ (LH S) + (1 − ψ)(LH S) = RH S| P (2.6)
P P
where ψ is a weighting factor, superscript n refers to the new time t + t, and
superscript o refers to the old time t. If we choose ψ = 1 then the discretisation is
called implicit,if ψ = 0 then it is called explicit, and if 0 <ψ < 1, it is called semi-
implicit or semi-explicit. Each choice has a bearing on economy and convenience
with which a numerical solution is obtained. The choice of ψ is therefore made by
the numerical analyst depending on the problem at hand. The main issues involved
will become apparent following further developments.

2.4.1 TSE Method
To employ this method, Equation 2.5 is first written in a nonconservative form.
Thus,
2
∂ T ∂(kA) ∂T
LHS| P = kA + + q A, (2.7)

∂x 2 ∂x ∂x
∂(CT )
RHS| P = ρ A . (2.8)
∂t
Equation 2.7 contains first and second derivatives of T with respect to x.To
represent these derivatives we employ a Taylor series expansion:

2 2

e
∂T x ∂ T
T E = T P + x e + +· · · , (2.9)
∂x 2 ∂x 2
P P
2 2

w
∂T x ∂ T
T W = T P − x w + +· · · . (2.10)
∂x 2 ∂x 2
P P
From these two expressions, it is easy to show that
2 2 2 2

w
e
w
e
∂T x T E − x T W + ( x − x ) T P
= , (2.11)
∂x x e x w ( x e + x w )
P
2
∂ T x w T E + x e T W − ( x e + x w ) T P
= . (2.12)
∂x 2 P x e x w ( x e + x w )/2
Note that, in Equations 2.9 and 2.10, terms involving derivative orders greater
than 2 are ignored. Therefore, Equations 2.11 and 2.12 are called second-order-
accurate representations of first- and second-order derivatives with respect to x.