Upwind Schemes

The first-order upwind scheme has been described in Chapter 4, Section 4.2.3. Here, we concentrate on the formulation of the second-order upwind and third-order QUICK schemes as illustrated below. As an improvement to the first-order upwind scheme, the idea is to incorporate additional variables located at the neighboring grid nodal points indicated by the properties at points WW and E E as shown in Fig. B.1 in order to evaluate the interface values at the cell faces of w and e. Control volume

I

~Xww

0

....

I-

=1 0

WW

W

w

P

e

E

EE

FIGURE B.1 A schematic representation of a control volume around a node P in a one-dimensional domain with surrounding grid nodal points of WW, W, E, and EE

For the second-order upwind scheme, assuming uniform distribution of the grid nodal points, additional information of the fluid flow is introduced into the approximation by the consideration of an extra upstream variable point, that is, 3 '/'w

=

1 -

3 ~e = ~ P -

1 ~W

3

1

3

1

~e - "~qbE - "~qbEE

414

if Uw > 0 and Ue > 0

ifuw<0andue <0.

(B.1)

(B.2)

Appendix B / Upwind Schemes

~ 415

For the third-order QUICK scheme, a quadratic approximation is introduced across two variable points at the upstream and one at the downstream depending on the flow direction. The unequal weighting influence of this particular scheme still hinges on the knowledge biased toward the upstream flow information. The interface values ~bwand ~be based on a uniform grid nodal point distribution can be evaluated as: 1 1

6 6

~e = -gq~w + g~e 1

6

3 3

ifuw>0andue >0

(B.3)

if Uw < 0 and Ue < O.

(B.4)

+ g~E 3

g,w = -gg, e + g~P + g~bw 1 6 3 qbe = -- -~~EE + "~~E + -~qbp