Page 87 - Introduction to Computational Fluid Dynamics

P2: IWV
P1: KsF/ICD/GKJ
0 521 85326 5
10:54
May 25, 2005
CB908/Date
0521853265c03
66
1D CONDUCTION–CONVECTION
To understand the relevance of AR, let T P be the temperature at X P after the
first time step. Then, from Equation 3.48, it follows that
T P
= exp(− τ) = AR, (3.50)
T 0 sin(X P + )
where the wave propagation speed is given by
u t
exact =−P τ =− . (3.51)
λ
Finally, we note that the arbitrary length scale λ is nothing but the wave-
length and the propagation speed depends on λ. This dependence on λ is called
dispersion.
3.9.2 Explicit Finite-Difference Form

Since P > 0, using UDS, the explicit discretised form of Equation 3.47 will read as
o
o
o
T P = AE T + AW T + {1 − (AE + AW)} T , (3.52)
E W P
where
τ τ τ
AE = , AW = + P . (3.53)
X 2 X 2 X
o o
Now, consider the first time step. Then, T = T 0 sin(X P ), T = T 0 sin(X P +
P E
o
X), and T = T 0 sin(X P − X). Therefore, after some manipulation, it can be
W
shown that

T P tan ED
= [1 − (AE + AW)(1 − cos X)] × 1 + , (3.54)
T 0 sin(X P ) tan(X P )
where
(AE − AW)sin( X)
tan ED = . (3.55)
1 − (AE + AW)(1 − cos X)
In these equations, the suffix ED denotes explicit differencing. Now, consider the
identity

tan ED
sin(X P + ED ) = sin(X P )cos( ED ) 1 + . (3.56)
tan(X P )
Substituting Equation 3.56 in Equation 3.54, it follows that

T P 1 − (AE + AW)(1 − cos X)
AR ED = = . (3.57)
T 0 sin(X P + ED ) cos ED
Now, let us consider tendencies of AR ED and tan ED for fine ( X → 0) and
4
coarse ( X → π) grids. These are shown in Table 3.3.

2
4 Note that 1 − cos X = 2 sin ( X/2).