Page 288 - Computational Fluid Dynamics for Engineers

278 9. Grid Generation

coordinates only. For example, mapping can be done from the (r, 9) cylindrical
coordinates in the physical plane to the (77, £) cylindrical coordinates (i.e., 77 rep-
resenting the radial variables and £ the polar coordinate) in the computational
domain.
As discussed in Section 9.2, to map an irregularly shaped region from the
physical plane to the computational plane, the values of £ and 77 were specified at
every boundary segment of the physical region as 77 = constant, varying mono-
tonically and vice versa. As a result, the correspondence between the boundary
coordinates of the physical and computational regions is known. Then the prob-
lem becomes one of determining the correspondence between the coordinates
(x,y) and (£,77) at the interior points of the physical and computational re-
gions. The distribution of the points on the interior is determined by solving
the following two Poisson equations:

g 0 = FK,,) (..5.1.)
+

+
S §?-««••» <95ib)
where the non-homogeneous terms P(£, 77) and Q(£, 77) are called the control
functions. With proper selection of the P and Q terms, the coordinate lines £
and 77 can be concentrated towards a specified coordinate line or about a specific
grid point. In the absence of these functions, the coordinate lines will tend to
be equally spaced in the regions away from the boundaries regardless of the
concentration of the grid points along the boundaries. The boundary conditions
needed to solve Eqs (9.5.1) are determined from the requirement that the values
of £ and 77 are specified at every boundary segment of the physical domain.
While Eqs. (9.5.1) describe the basic coordinate transformation between the
(x,y) and (£,77) coordinate systems, all numerical calculations for the physical
problem are performed in the computational plane which has a simple regu-
lar geometry. Then, the problem becomes one of seeking the (x, y) values of
the physical plane corresponding to the (£,77) grid locations in the computa-
tional plane. For this reason Eqs. (9.5.1) are transformed to the computational
plane and the unknowns (x, y) are determined from the following two elliptic
equations:
2
2
2
d x _ <9 x d x 2 (~& x ~@ x
a 20 + + J P + Q = o (95 2a)
W- m, ^ { e ( a v ) '
+
+
<$-<^0 '(*! <$)=° <—>
The geometric coefficients a, (3 and 7 and the Jacobian J are given by

a = x'+y< (9.5.3a)