Page 80 - Introduction to Computational Fluid Dynamics

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3.4 UPWIND DIFFERENCE SCHEME
where ψ is sensitized to the sign and the magnitude of P c . Note that, in Equa- 10:54 59
tion 3.11, we took ψ = 0.5, an absolute constant.
3.4 Upwind Difference Scheme
The upwind difference scheme (UDS) was originally proposed in [8] but later in-
dependently developed by Runchal and Wolfshtein [60] among others. The scheme
simply senses the sign of P c but not its magnitude. Thus, instead of Equation 3.11,
we write
1 1
P e = [P +|P|] P + [P −|P|] E , (3.21)
2 2
1 1
P w = [P +|P|] W + [P −|P|] P . (3.22)
2 2
These expressions show that when P > 0, e = P and w = W . Similarly,
when P < 0, e = E and w = P . That is, the cell-face values always pick
up the upstream values of irrespective of the magnitude of P, hence, giving
rise to the name of this interpolation scheme as the upwind difference scheme. 1
Substituting these equations in Equation 3.10, we can show that Equation 3.14 again
holds with
1
AE = 1 + (|P c |− P c ), (3.23)
2
1
AW = 1 + (|P c |+ P c ), (3.24)
2
and AP = AE + AW. Equations 3.23 and 3.24 show that, irrespective of the mag-
nitude or sign of P (or P c ), AE and AW can never become negative. Also, AP
remains dominant. Therefore, obstacles to convergence are removed for all values
of P c . This was not the case with CDS. 2

1
Physically, the UDS can be understood as follows: Imagine standing at the middle of a long cor-
ridor at one end of which there is an icebox (at T ice ) and at the other end a firebox (at T fire ).
Then, neglecting radiation, the temperature experienced by you will be T m = 0.5(T ice + T fire )
when the air in the corridor is stagnant and heat transfer is only by conduction. Now, imag-
ine that there is air-flow over the firebox flowing through the corridor in the direction of the
icebox. You will now experience T m that weighs more in favour of T fire than T ice . The reverse
would be the case if the airflow was from the icebox end and towards the firebox end. The UDS
takes an extreme view of both situations and sets T m = T fire in the first case and T m = T ice in the
second case.
2
Incidentally, with respect to Equation 3.20, we may generalise AE and AW coefficients for both
CDS and UDS in terms of ψ as
AE = 1 − (1 − ψ) P c , AW = 1 + ψ P c , (3.25)

1 |P c |
ψ = 0.5 (CDS), ψ = 1 + (UDS). (3.26)
2 P c