Page 85 - Introduction to Computational Fluid Dynamics

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1 1D CONDUCTION–CONVECTION

+
+
P w = (P +|P|) f P + (1 − f ) WW
w
w
2
1
−
−
+ (P −|P|) f W + (1 − f ) E , (3.39)
w
w
2
where the fs are the appropriate weighting functions to be determined from

U − UU
f = f (ξ) = f (3.40)
D − UU
with suffix D referring to downstream, U to upstream, and UU to upstream of U.
The f , for example, is thus a function of ( P − W )/( E − W ) and f e − is a
+
e
function of ( E − EE )/( P − EE ). Here, EE refers to the node east of node E
and WW to the node west of node W.
It is interesting to note that if f equals its associated ξ then Equations 3.38 and
3.39 readily retrieve the UDS formula. Therefore, writing
f (ξ) = ξ + f c (ξ) (3.41)
we can show that
1
+
P e = P e | UDS + (P +|P|) f ( E − W )
ce
2
1
−
− (P −|P|) f ( EE − P ), (3.42)
ce
2
1
+
P w = P w | UDS + (P +|P|) f cw ( P − WW )
2
1
−
− (P −|P|) f cw ( E − W ). (3.43)
2
Substituting the last two equations in Equation 3.10, we can show that
AP P = AE E + AW W + S TVD , (3.44)
where AE, AW, and AP are the same as those for the UDS and the additional
source term S TVD contains the f c terms in Equations 3.42 and 3.43, which the
reader can easily derive. The f c (ξ) functions for some variants of TVD schemes
are tabulated in Table 3.2.
To appreciate the implications of the TVD scheme, consider the case in which
+
P c > 0. Then, from Equation 3.42, P e = P P + Pf ( E − W ) and ξ =
ce
( P − W )/( E − W ). Therefore, using the Lin–Lin scheme, for example, we get
⎧
P ,ξ (0, 1),
⎪
⎪
⎪
⎪
2 P − W ,ξ ∈ (0, 0.3),
⎪
⎪
⎨
e = 3 3 1 (3.45)
P + E − W ,ξ ∈ (0.3, 5/6),
⎪
⎪
⎪ 4 8 8
⎪
⎪
⎪
E ,ξ ∈ (5/6, 1.0).
⎩