Page 365 - Computational Fluid Dynamics for Engineers

356 12. Compressible Navier-Stokes Equations

type boundary conditions are often obtained by simple application of the one-
dimensional characteristic theory described in subsection 10.11.2 parallel to the
flow direction.
When selecting the numerical conditions, it is important to select one which
does not introduce any added dissipation, since one objective of a Navier-Stokes
solution is to predict the wall shear-forces. A study on the choice of appropriate
boundary conditions can be found in [2].

12.3 MacCormack Method

In Section 5.3 the MacCormack method was discussed for a one-dimensional
problem and its extension to two-dimensional compressible viscous flows is dis-
cussed here. As in Section 11.4, the non-dimensional Navier-Stokes equations
are written in Cartesian coordinates,

where Q, E, F, E v and F v are given by Eqs. (2.2.32) and (2.2.33).
n+l
The predictor values at {t ,Xi,yi) are defined by Q™^ 1 (= Qij), and the
convective flux terms E and F are represented with forward differences followed
by a corrector step with backward differences for the same convective flux terms.
The viscous flux terms E v and F v are represented by central differences, as for
the incompressible case (subsection 11.4.3). The predictor step then becomes
/ rpn rpn rpn rpn
Wij - Wij at y A x + Ay

A~x + Ay ) ( 1 2 3 1 }
' -
and the corrector step

+ +
Rel Ax Ay j ^ ^

Updating gives
1
12 3 3
Q^^-^Ai) ( - - )
2
which remains unchanged from the D counterpart.
I