Page 369 - Computational Fluid Dynamics for Engineers

360 12. Compressible Navier-Stokes Equations

As in Section 5.4, where central differences were used for the spatial deriva-
tives in Eq. (5.4.12), central differences are again used in Eqs. (12.4.17) and
(12.4.18), and the resulting system is written in block tridiagonal form and
solved with the block elimination method described in Eqs. (4.4.32) through
(4.4.34).
As discussed in Section 5.7, in some implicit methods for hyperbolic equa-
tions, damping terms are necessary to suppress high frequency oscillations in the
solution. This can be accomplished by adding constant coefficient implicit and
explicit dissipation terms to the solution algorithm and, in the Beam-Warming
method, this is accomplished by adding an explicit fourth-order dissipation term
to the right-hand side of Eq. (12.4.17). If the steady-state is of interest, implicit
second order dissipation terms are added to the left-hand sides of Eqs. (12.4.17)
and (12.4.18) and the resulting equations become

Step 1:

d_
m + I 9 At dx [A] [P] ~ [A* Redx 2 [R] n - SiS x[I] AQ n - l / 2
2
Re
+
Z
At Vi + V 2^ n
dx Re dy Re
6At 1 l
l AV ^) l (AWr )
+ l + £Re { 2 + y
n
(AQ -')-e e(6Z + 6*)Q n
1 + C
(12.4.19)
Step 2:

9 At d_ 2
[B] 2 M eid y[I)
dy Re Redy

Here 8 2 and 6 4 denote central difference operators defined by
6%u = Ui+ij - 2uij + Ui-ij

and
6 xu = Ui+2, Au. «+ij + 6u 1,3 4u i - i j + u •i-2,3 (12.4.22)

On mesh points adjacent to the boundaries, Eq. (12.4.22) is approximated by
second-order four point finite differences rather than five point finite differences
to avoid requesting information from outside the computational domain. For
example, near the bottom boundary, Eq. (12.4.22) becomes

6yU « ^,4 - 4^,3 + 5ui t2 - 2i^ 5i (12.4.23)