Page 294 - Computational Fluid Dynamics for Engineers

284 9. Grid Generation

in the x,y coordinate system. Noting that,

- = t a n 0 (9.6.7)
x
the polar angles in both coordinate systems are related by

(/) = 29 (9.6.8)
It follows from Eq. (9.6.3) and from the definition of r in Eq. (9.6.5) that the
radius vector in the computational plane is

r 2 2\Jx 2 + y 2 (9.6.9)

Since the radius vector R in the physical plane is

2 2
R = yjx + y (9.6.10)
the relation between r 2 and R is

r 2 = 2R (9.6.11)

The (£, rj) points in the computational plane given by Eq. (9.6.4) can now be
written as

£ = V2Rcos f | 1 , v = V2i?sin \ r \ ( 9 . 6 .12)
In conformal mapping, the singular point is defined to be the point where the
mapping function fails. In our case it is at the origin of the coordinate system,
x = 0, y = 0. It is usually avoided by choosing it to be at a distance half the
airfoil nose radius, r*o, from the leading edge in the physical plane. If the airfoil
contour is given analytically the nose radius can be obtained by calculating the
curvature at the nose. If only tabular values of airfoil coordinates are available,
the nose radius can be computed by fitting a circle to three points nearest to
the leading edge. An alternate way is to plot y/\/E vs. y/E and extrapolate the
resulting curve to y/fi = 0, in which case the nose radius TQ is computed from
the extrapolated y/y/fi value by:

Once the origin of the (x, y) coordinate system in the physical plane is fixed,
then the (£,77) points, corresponding to the airfoil surface coordinates (x p,y p)
can be computed as follows:
(a) Determine (j) from Eq. (9.6.7), that is

t a n 0 p = —
x P