Page 291 - Computational Fluid Dynamics for Engineers

9.5 Differential Equation Methods 281

(9.5.12) are known on the boundaries, except f^ (On a £ = constant boundary,
the same equations for the control functions result, but with r^ as the unknown
quantity). The r m term must be solved for as part of the solution. An iterative
procedure is set up in order to evaluate f^ as the solution progresses. This
approach, first introduced by Sorenson [5] is now common, with modifications
of the basic concept introduced in various codes.
Once the form of the control functions defined, the iterative procedure for
the solution of the system (9.5.4) can proceed as follows:

1. A starting grid is generated by unidirectional linear interpolation from the
boundary points. However, since the final grid does not, in general, have
uniform spacing, the starting grid is stretched using appropriate algebraic
formulas according to input values of the cell spacing at the inner and outer
boundaries (ASi and AS2).
2. Once the starting grid is defined, the system (9.5.4) is solved numerically
using Successive Line Over-Relaxation. For the first iteration, zero values are
assumed for p, q, r, and s. All the fixed derivatives appearing in equations
(9.5.12) are computed.
3. Given the initial conditions or the results from the previous iteration, x^
and y vr], at j = 1 and j = j m a x are computed using special one-sided differ-
ence formulas suggested by Sorenson [5]. The functions p, q, r, and s are then
evaluated at the boundaries using equations (9.5.7) and (9.5.12). The con-
trol functions P and Q are then evaluated at all grid points using relations
(9.5.6). For numerical stability, these control functions are under-relaxed to
a degree specified by the user.
4. Another step of Successive Line Over Relaxation solution is then performed
on the system of equations (9.5.4).
5. Solutions steps 3 and 4 are repeated until convergence is attained.

A typical C-type algebraic grid is shown in Fig. 9.13a. The outer boundaries for
this type of grid consist of a semi-circle and a rectangle. The r\ — 0 boundary
moves forward from the rear boundary to the trailing edge, clockwise around
the airfoil, and then rearward again. The 77 = constant family of lines form open
curves resembling the letter C. The £ = constant lines join the inner (airfoil)
boundary to the outer boundary. This grid was used as the starting grid for
the elliptically generated grid shown in Fig. 9.13b. The grid consists of 70 x 30
nodes, with a spacing of 0.3 times the airfoil chord c at the outer boundary,
and 0.015c at the airfoil boundary. Fifty iterations were performed to obtain
this grid, with an over-relaxation factor of 1.3 applied on the main equations,
and an under-relaxation factor of 0.05 applied to the control functions. The
final grid obtained is relatively smooth and exhibits the proper characteristics.
Figure 9.13c provides as closer view of the grid in the vicinity of the airfoil,
where the orthogonality of the boundary cells can be observed.