Page 343 - Computational Fluid Dynamics for Engineers

11.4 Artificial Compressibility Method: INS2D 333

The inverse matrices X x 1 and X 2 l are given by
2
—v —uv u 2 + /?' I" — u —v — (3 vu~
1
x; ci-u (3 0 XT =\ c 2-v 0 p
[_— c\ — u (3 0 [-C2-V 0 f3
(11.4.11)
As described in Section 5.5, A and B can be decomposed as the sum of two
parts, one part corresponding to positive eigenvalues and the other to negative
eigenvalues,
A = A + + A~ and B = B+ + B~ (11.4.12)
where
( 1 ) (1)
IA I
A A i + l i I
±
A
A^±|A?|
A± = Xi x;
AJ^IA^I
(11.4.13)
2 )
2 )
A< ±|A< |
£ d X 2 Ar^iA^i X. - l

A^IAfl

Application of the upwind scheme of Section 5.5 to the derivative of the
convective flux in the x- and ^-directions yields,

E
dE E i+l/2,j - i-l/2,j OF E ij+l/2 - E iJ-l/2
(11.4.14)
dx Ax dy Ay
where, referring to Fig. 11.1, E i+1/ 2j is a numerical flux at j and i + 1/2 is the
discrete spatial index for the x-direction. Similarly i^j+1/2 is a numerical flux
at i and j + 1/2. They are given by

E E D E D (11.4.15a)
i+l/2,j = «[ ( i+lj) + ( i,j) ~ <t>i+l/2j]

= -[F(D iJ+ 1) + F{D, (11.4.15b)
F iJ+ 1/2 1/2]
1,3) '\3 +
When
0z+i/2,i = 0 and fa j+1/2 = 0 (11.4.16)
the convective flux terms in Eq. (11.4.14) are represented by a second-order ac-
curate central difference scheme. A first-order accurate upwind difference scheme
is given by