Page 117 - Basic Structured Grid Generation

106 Basic Structured Grid Generation

The multisurface formula becomes
1 2
η − η 1 2
η
r(ξ, η) = r 1 + 2 1 2 (r 2 − r 1 ) + 2 1 2 (r 3 − r 2 )
η 3 − η η
2 3 2 3
(η 3 − η)(2 − η 3 − η) 2η(η 3 − η) η 2
= r 1 + r 2 + r 3 . (4.111)
2
η 3 (2 − η 3 ) η (2 − η 3 ) η 2
3 3
Nowifwetakethe valueof η 3 to be very close to 1, this reduces (approximately)
to the interpolation formula:
2
2
r(ξ, η) = (1 − η) r 1 (ξ) + 2η(1 − η)r 2 (ξ) + η r 3 (ξ). (4.112)
When n = 4 and there are two intermediate surfaces, a similar method involves
taking parameter values η 1 = 0, η 2 = a, η 3 = 1, and η 4 unspecified, with 0 <a < 1
and η 4 > 1. Then we can take second degree Lagrange polynomials
1 η(1 − η) η(η − a)
ψ 1 = (a − η)(1 − η), ψ 2 = , ψ 3 = .
a a(1 − a) (1 − a)
Equation (4.110) now yields the formula

1 3 1 2
η − (1 + a)η + aη
3 2
(r 2 − r 1 )
r = r 1 +
1 3 1 2
η − (1 + a)η + aη 4
3 4 2 4

1 2 1 3 1 3 1 2
2 η − η 3 η − aη
3
2
(r 4 − r 3 ). (4.113)
+ (r 3 − r 2 ) +
1 2 1 3 1 3 1 2
η − aη
η − η
2 4 3 4 3 4 2 4
Again, if we take η 4 to be very close to 1, this expression reduces (approximately)
to the formula:
2
2
r(ξ, η) = (1 − η) (1 − a 1 η)r 1 (ξ) + (2 + a 1 )η(1 − η) r 2 (ξ)
2
2
+η (1 − η)(2 + a 2 )r 3 (ξ) + η (1 − a 2 + a 2 η)r 4 (ξ), (4.114)
where
2 2
a 1 = and a 2 = . (4.115)
3a − 1 2 − 3a
Clearly we must avoid taking the values 1 or 2 for a.
3 3
Note that eqn (4.110) can also be applied in the trivial case where n = 2, when
there are no intermediate curves. Taking ψ 1 to be a constant then leads to simple
linear interpolation between inner and outer curves:
r(ξ, η) = r 1 (ξ) + η{r 2 (ξ) − r 1 (ξ)}. (4.116)
4.5.3 Numerical implementation
Here we describe briefly the numerical implementation of the above methods for the
case of an airfoil NACA-0012. The relevant programs, Two-boundary.f and Multisur-
face.f, are listed at Section 4.6.4 and may be found in the directory Book/tfi.gds on