Page 115 - Basic Structured Grid Generation

104 Basic Structured Grid Generation

calculate it. Stretching functions h AB (ξ) and h DC (ξ) can then be used to control the
location of grid-nodes along AB and DC. Here we present the stretching function

tanh[Q(1 − ξ)]
h = Pξ + (1 − P) 1 − , 0 ξ 1, (4.103)
tanh Q
originally due to Roberts (1971) and later modified by Eiseman (1979), where P and
Q are parameters to be chosen (for more details, see Section 4.6.1 below) on each
boundary. In principle we can then calculate the cartesian co-ordinates of grid-nodes
on the boundaries (x AB (h AB ), y AB (h AB )) and (x DC (h DC ), y DC (h DC )).
Stretching can also be used in the η-direction with a choice of similar stretching
functions h AD (η) and h BC (η). Linear interpolation could then lead to the following
expressions for cartesian co-ordinates:


˜
x(ξ, η) = 1 − h(ξ, η) x AB (h AB (ξ)) + h(ξ, η)x DC (h DC (ξ))
˜
(4.104)

y(ξ, η) = 1 − h(ξ, η) y AB (h AB (ξ)) + h(ξ, η)y DC (h DC (ξ)),
 ˜ ˜
where h(ξ, η) = h AD (η) + ξ(h BC (η) − h AD (η)). To achieve orthogonality at the
˜
two boundaries, however, it is possible to use the Hermite interpolation formula
(4.36). Since tangent vectors at the boundaries have direction dr/dh, with compo-

nents dx AB , dy AB and dx DC , dy DC , orthogonal directions will have components
dh AB dh AB dh DC dh DC

dy AB dx AB dy DC dx DC
, − and , − . Thus we can write the Hermite interpolation
dh AB dh AB dh DC dh DC
formula in component form as

dy AB dy DC
x(ξ, η) = 1 (η)x AB (h AB ) + 2 (η)x DC (h DC ) + T 1 3 (η) + T 2 4 (η)


dh AB dh DC
dx AB dx DC

y(ξ, η) = 1 (η)y AB (h AB ) + 2 (η)y DC (h DC ) − T 1 3 (η) − T 2 4 (η) ,

dh AB dh DC
(4.105)
where the two parameters T 1 , T 2 that have been introduced, while not affecting orthog-
onality at the boundaries, can be used to control the extent to which the grid in the
interior domain is orthogonal. In fact these parameters may have to be tuned to avoid
folding of the grid in the interior.
The accompanying floppy disk contains a numerical code for generating a grid
around an NACA-0012 airfoil using the two-boundary method. This may be found in
the file Two-boundary.f listed in Section 4.6.4 below.
4.5.2 Multisurface transformation
The multisurface method introduced by Eiseman (1979) is another unidirectional inter-
polation technique for generating a grid between two given curves (or surfaces), allow-
ing additional control over grid-node distribution by the use of intermediate curves (or
surfaces). In the two-dimensional case, suppose that a given inner boundary is the curve
r = r 1 (ξ), the given outer boundary is r n (ξ), and we construct (n-2) non-intersecting
intermediate curves r i (ξ), i = 2, 3,. ..,(n−1),where ξ is a parameter for each curve.
We assume that each surface is given by constant values of the independent co-ordinate